diff --git "a/-dAyT4oBgHgl3EQfqfiK/content/tmp_files/load_file.txt" "b/-dAyT4oBgHgl3EQfqfiK/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/-dAyT4oBgHgl3EQfqfiK/content/tmp_files/load_file.txt" @@ -0,0 +1,881 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf,len=880 +page_content='Relativistic BGK hydrodynamics Pracheta Singhaa, Samapan Bhadurya,c, Arghya Mukherjeeb, Amaresh Jaiswala aSchool of Physical Sciences, National Institute of Science Education and Research, An OCC of Homi Bhabha National Institute, Jatni 752050, Odisha, India bDepartment of Physics and Astronomy, Brandon University, Brandon, Manitoba R7A 6A9, Canada cInstitute of Theoretical Physics, Jagiellonian University, ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Łojasiewicza 11, 30-348 Krakow, Poland Abstract Bhatnagar-Gross-Krook (BGK) collision kernel is employed in the Boltzmann equation to formulate relativistic dissipative hydro- dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In this formulation, we find that there remains freedom of choosing a matching condition that affects the scalar transport in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We also propose a new collision kernel which, unlike BGK collision kernel, is valid in the limit of zero chemical potential and derive relativistic first-order dissipative hydrodynamics using it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We study the effects of this new formulation on the coefficient of bulk viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Introduction Relativistic Boltzmann equation governs the space-time evo- lution of the single particle phase-space distribution function of a relativistic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Moreover, suitable moments of the Boltz- mann equation are capable of describing the collective dynam- ics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Therefore, it has been extensively used to de- rive equations of relativistic dissipative hydrodynamics and ob- tain expressions for the transport coefficients [1–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The col- lision term in the Boltzmann equation, which describes change in the phase-space distribution due to the collisions of particles, makes it a complicated integro-differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In order to circumvent this issue, several approximations have been sug- gested to simplify the collision term in the linearized regime [15–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Bhatnagar-Gross-Krook [15], and independently Welander [16], proposed a relaxation type model for the collision term, which is commonly known as the BGK model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' This model was further simplified by Marle [17] and Anderson-Witting [18] to calculate the transport coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In the non-relativistic limit, Marle’s formulation leads to the same transport coeffi- cient as the BGK model but fails in the relativistic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' On the other hand, the Anderson-Witting model, also known as the relaxation-time approximation (RTA), is better suited in the rel- ativistic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The RTA has been employed extensively in sev- eral areas of physics with considerable success and has been widely employed in the formulation of relativistic dissipative hydrodynamics [8–11, 20–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The RTA Boltzmann equation has provided remarkable in- sights into the causal theory of relativistic hydrodynamics as well as a simple yet meaningful picture of the collision mech- anism in a non-equilibrium system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' On the other hand, the Email addresses: pracheta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='singha@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='com (Pracheta Singha), samapan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='bhadury@niser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='in (Samapan Bhadury), arbp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='phy@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='com (Arghya Mukherjee), a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='jaiswal@niser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='in (Amaresh Jaiswal) BGK collision term ensures conservation of net particle four- current by construction, and is the precursor to RTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' While the RTA has been employed extensively, a consistent formulation of relativistic dissipative hydrodynamics with the BGK colli- sion term is relatively less explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' This may be attributed to the fact that the BGK collision kernel is ill defined for relativis- tic systems without a conserved net particle four-current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' This has limited the use of the BGK collision kernel to the studies related to flow of particle number and/or charge [33–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In this article, we take the first step towards formulating a consistent framework of relativistic dissipative hydrodynamics using the BGK collision kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Furthermore, we propose a modified BGK collisions kernel (MBGK), which is well de- fined even in the absence of conserved particle four-current and is better suited for the formulation of the relativistic dissipative hydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We find that there exists a free scalar parame- ter arising from the freedom of matching condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' This affects the scalar dissipation in the system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=', the coefficient of bulk viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We study the effect on bulk viscosity in several dif- ferent scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Relativistic dissipative hydrodynamics The conserved net particle four-current,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Nµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' and the energy- momentum tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' T µν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' of a system can be expressed in terms of the single particle phase-space distribution function and the hydrodynamic variables as [44],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Nµ = � dP pµ � f − ¯f � = n uµ + nµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (1) T µν = � dP pµpν � f + ¯f � = ϵ uµuν − (P0 + δP) ∆µν + πµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (2) where the Lorentz invariant momentum integral measure is de- fined as dP = g d3p/ � (2π)3E � with g being the degeneracy fac- tor and E = � |p|2 + m2 being the on-shell energy of the con- stituent particle of the medium with three-momentum p and Preprint submitted to Physics Letters B January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='00544v1 [nucl-th] 2 Jan 2023 mass m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Here f ≡ f(x, p) and ¯f ≡ ¯f(x, p) are the phase- space distribution functions for particles and anti-particles, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In the above equations, n is the net particle number density, ϵ is the energy density, P0 is the equilibrium pressure, nµ is the particle diffusion four-current, δP is the correction to the isotropic pressure, and πµν is the shear stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We note that the fluid four-velocity uµ has been defined in the Lan- dau frame, uµT µν = ϵuν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We also define ∆µν ≡ gµν − uµuν as the projection operator orthogonal to uµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In this article, we will be working in a flat space-time with metric tensor defined as, gµν = diag(1, −1, −1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Hydrodynamic equations are essentially the equations for conservation of net particle four current, ∂µNµ = 0, and energy- momentum tensor, ∂µT µν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Using the expressions of Nµ and T µν from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (1) and (2), the hydrodynamic equations can be obtained as, ˙n + nθ + ∂µnµ = 0 (3) ˙ϵ + (ϵ + P0 + δP) θ − πµνσµν = 0 (4) (ϵ + P0 + δP) ˙uα − ∇α (P0 + δP) + ∆α ν∂µπµν = 0 (5) where we use the standard notation, ˙A ≡ uµ∂µA for the co- moving derivatives, ∇α ≡ ∆αβ∂β for the space-like derivatives, θ = ∂µuµ for the expansion scalar, and σµν = 1 2 (∇µuν + ∇νuµ) − 1 3∆µνθ for the velocity stress-tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' To express the conserved net particle four-current and the energy-momentum tensor in terms of hydrodynamic variables in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (1) and (2), we chose Landau frame to define the fluid four-velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Additionally, the net-number density and energy density of a non-equilibrium system needs to be defined us- ing the so called matching conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We relate these non- equilibrium quantities with their equilibrium values as n = n0 + δn, ϵ = ϵ0 + δϵ, (6) where n0 and ϵ0 are the equilibrium net-number density and the energy density, respectively, and, δn, δϵ are the corre- sponding non-equilibrium corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' For a system which is out-of-equilibrium, the distribution function can be written as f = f0 + δ f, where f0 is the equilibrium distribution function and δf is the non-equilibrium correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In the present work, we consider the equilibrium distribution function to be of the classical Maxwell-Juttner form, f0 = exp(−β u · p + α), where β ≡ 1/T is the inverse temperature, α ≡ µ/T is the ratio of chemical potential to temperature and u · p ≡ uµpµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The equi- librium distribution for anti-particles is also taken to be of the Maxwell-Juttner form with α → −α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We can now express the equilibrium hydrodynamic quanti- ties in terms of the equilibrium distribution function as, n0 = � dP (u · p) � f0 − ¯f0 � (7) ϵ0 = � dP (u · p)2 � f0 + ¯f0 � (8) P0 = −1 3∆µν � dP pµpν � f0 + ¯f0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (9) Similarly, the non-equilibrium quantities can be expressed as δn = � dP (u · p) � δ f − δ ¯f � (10) δϵ = � dP (u · p)2 � δ f + δ ¯f � (11) δP = −1 3∆αβ � dP pαpβ � δ f + δ ¯f � , (12) nµ = ∆µ α � dP pα � δ f − δ ¯f � , (13) πµν = ∆µν αβ � dP pαpβ � δ f + δ ¯f � , (14) where ∆µν αβ ≡ 1 2(∆µ α∆ν β+∆µ β∆ν α)− 1 3∆µν∆αβ is a traceless symmetric projection operator orthogonal to uµ as well as ∆µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In order to calculate these non-equilibrium quantities, we require the out- of-equilibrium correction to the distribution function, δf and δ ¯f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' To this end, we consider the Boltzmann equation with BGK collision kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The Boltzmann equation and conservation laws The covariant Boltzmann equation, in absence of any force term or mean-field interaction term, is given by, pµ∂µ f = C[ f, ¯f], pµ∂µ ¯f = ¯C[ f, ¯f], (15) for a single species of particles and its antiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In the above equation, C[ f, ¯f] and ¯C[ f, ¯f] are the collision kernels that contain the microscopic information of the scattering pro- cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' For the formulation of relativistic hydrodynamics from the kinetic theory of unpolarized particles, the collision ker- nel of the Boltzmann equation must satisfy certain properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Firstly, the collision kernel must vanish for a system in equilib- rium, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=', C[ f0, ¯f0] = ¯C[ f0, ¯f0] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Further, in order to satisfy the fundamental conservation equations in the microscopic in- teractions, the zeroth and the first moments of the collision ker- nel must vanish, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=', � dPC = 0 and � dP pµ C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Vanishing of the zeroth moment and the first moment of the collision ker- nel follows from the net particle four-current conservation and the energy-momentum conservation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In the present work, we consider the BGK collision kernel which has the advantage that the particle four-current is con- served by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The relativistic Boltzmann equation with BGK collision kernel for particles can be written as [15], pµ∂µ f = −(u · p) τR � f − n n0 f0 � , (16) and similarly for anti-particles with f → ¯f and f0 → ¯f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Here, τR is a relaxation time like parameter1 which we assume to be the same for particles and anti-particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' It is easy to verify that the conservation of net particle four-current, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (1), follows from the zeroth moment of the above equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The 1A more conventional notation is the collision frequency which is defined as ν = 1/τR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 2 first moment of the above equations should lead to the con- servation of the energy-momentum tensor, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' However, we find that the first moment of the Boltzmann equa- tion, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (16), leads to, ∂µT µν = − 1 τR � ϵ − n n0 ϵ0 � , (17) which does not vanish automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In order to have energy-momentum conservation fulfilled by the Boltzmann equation with the BGK collision kernel, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (16), we require that ϵn0 = ϵ0n, (18) which we identify as one matching condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Note that two matching conditions are required to define the non-equilibrium net number density and the energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Along with the above equation, we are left with the freedom of one matching condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' It is important to observe that the RTA Boltzmann equation, pµ∂µ f = − (u·p) τR ( f − f0), is recovered from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (16) if the second matching condition is fixed as either ϵ = ϵ0 or equivalently n = n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' For the RTA collision term, both matching conditions ϵ = ϵ0 and n = n0, are necessary for net particle four- current and energy-momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' However, for the BGK collision kernel, both conservation equations are satisfied with only one matching condition, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (18), leaving the other condition free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We shall see later that this scalar freedom af- fects the coefficient of bulk viscosity, which is the transport co- efficient corresponding to scalar dissipation in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Note that the equilibrium net number density, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (7), vanishes in the limit of zero chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' This implies that the BGK collision term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (16) is ill defined in this limit, which is relevant for ultra-relativistic heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Therefore, it is desirable to modify the BGK col- lision kernel in order to extend its regime of applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' At this juncture, we are well equipped to propose a modification to BGK collision kernel that is well-defined for all values of chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' To this end, we rewrite the condition nec- essary for energy-momentum conservation from BGK collision kernel, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (18), in the form n n0 = ϵ ϵ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (19) Substituting the above equation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (16), we obtain Boltz- mann equation for particles with a modified BGK (MBGK) col- lision kernel, pµ∂µ f = −(u · p) τR � f − ϵ ϵ0 f0 � , (20) and similarly for anti-particles with f → ¯f and f0 → ¯f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The advantage of the above modification is that the collision ker- nel conserves energy-momentum by construction and is appli- cable to systems even without any conserved four-current, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=', in the limit of vanishing chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In the case of finite chemical potential, the matching condition, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (18), ensures net particle four-current conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' It is important to note that BGK and MBGK are completely equivalent for the purpose of the derivation of hydrodynamic equations at finite chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In the following, we consider the MBGK Boltzmann equation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (20), to obtain non-equilibrium correction to the distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Non-equilibrium correction to the distribution function In order to obtain the non-equilibrium correction to the dis- tribution function, we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (6) to rewrite the MBGK Boltz- mann equation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (20), as pµ∂µ f = −(u · p) τR � δ f − δϵ ϵ0 f0 � , (21) and similarly for anti-particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The next step is to solve the above equation, order-by-order in gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In this work, we intend to obtain the non-equilibrium correction to the distribu- tion function up to first-order in derivative, which we repre- sent by δ f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' However, obtaining the expressions for δf1 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (21) is not straightforward because it contains δϵ which is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (11) as an integral over δ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Therefore, to solve for δ f1, we examine each term individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Up to first-order in gradients, the structure of the term on the left-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (21) has the form, pµ∂µ f0 = � AΠθ + Anpµ∇µα + Aπpµpνσµν � f0, (22) and similarly for anti-particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Here, AΠ = − � (u · p)2 � χb − β 3 � − (u · p) χa + βm2 3 � , (23) An = 1 − n0 (u · p) (ϵ0 + P0), Aπ = − β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (24) The coefficients χa and χb appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (23) are defined via the relations ˙α = χa θ, ˙β = χb θ, ∇µβ = n0 ϵ0 + p0 ∇µα − β˙uµ (25) χa = I− 20(ϵ0 + P0) − I+ 30n0 I+ 30I+ 10 − I− 20I− 20 , χb = I+ 10(ϵ0 + P0) − I− 20n0 I+ 30I+ 10 − I− 20I− 20 , (26) where, the thermodynamic integrals are given by, I± nq = (−1)q (2q + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' � dP (u · p)n−2q � ∆αβpαpβ�q � f0 ± ¯f0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (27) With the above definition, we identify n0 = I− 10, ϵ0 = I+ 20 and P0 = I+ 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We assume δ f1 to have the same form as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (22), δ f1 = τR � BΠθ + Bnpµ∇µα + Bπpµpνσµν � f0, (28) and similarly for anti-particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In the above expression, the co- efficients BΠ, Bn and Bπ needs to be determined using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (21), up to first order in derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' To that end, we substitute the expression for δ f1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (11) to obtain δϵ = τR � dP (u · p)2 � BΠ f0 + ¯BΠ ¯f0 � θ (29) 3 Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (22), (28) and (29) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (21) and comparing both sides, we get − AΠ (u · p) = BΠ − 1 ϵ0 � dP (u · p)2 � BΠ f0 + ¯BΠ ¯f0 � (30) Bn = − An (u · p) , Bπ = − Aπ (u · p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (31) Another set of equations in terms of ¯AΠ, ¯An and ¯Aπ can be obtained by considering the MBGK equation, analogous to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (21), for anti-particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Note that the coefficients Bn, ¯Bn, Bπ and ¯Bπ are easily determined but BΠ and ¯BΠ require further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' To obtain their expressions, we consider BΠ to be of the gen- eral form, BΠ = �+∞ k=−∞ bk (u · p)k and ¯BΠ = �+∞ k=−∞ ¯bk (u · p)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Substituting these in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (30) and its corresponding equation for anti-particles, we can conclude that the only non-zero bk and ¯bk are the ones with k = −1, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We obtain BΠ = 1 � k=−1 bk (u · p)k , ¯BΠ = 1 � k=−1 ¯bk (u · p)k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (32) Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (23) and (32) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (30), we find b1 = ¯b1 = χb − β 3 and, b−1 = ¯b−1 = m2β 3 , (33) where we have also used the relation analogous to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (30) for anti-particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' On the other hand, for b0 and ¯b0 we find two coupled equations, which are identical and can be simplified to the relation, ¯b0 = b0 + 2 χa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (34) Hence, we see that a unique solution for b0 and ¯b0 can not be obtained but they are constrained by the above relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We need to provide one more condition, which we recognize as the second matching condition, to fix b0 and ¯b0 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Nevertheless, at this stage, we can determine δf1 and δ ¯f1 up to a free parameter, b0, by using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (30)-(34) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (28), and similarly for anti-particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We obtain, δf1 = τR f0 � � m2β 3 (u · p) + b0 + (u · p) � χb − β 3 �� θ − � 1 (u · p) − n0 (ϵ0 + P0) � pµ � ∇µα � + βpµpµσµν (u · p) � , (35) δ ¯f1 = τR ¯f0 � � m2β 3 (u · p) + b0 + 2 χa + (u · p) � χb − β 3 �� θ + � 1 (u · p) + n0 (ϵ0 + P0) � pµ � ∇µα � + βpµpµσµν (u · p) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (36) Note that for vanishing chemical potential, we have α = χa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In this case, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (35) and (36) coincide to give δf1 ����µ=0 = τR βf0 �� m2 3 (u·p) + b0 β +(u·p) � c2 s− 1 3 �� θ+ pµpµσµν (u·p) � , (37) where we have used χb = βc2 s, with c2 s being the squared of the speed of sound, given by, c2 s = (ϵ0 + P0) 3ϵ0 + �3 + z2� P0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (38) Here z ≡ m/T is the ratio of particle mass to temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' First order dissipative hydrodynamics The first-order correction to the phase-space distribution functions of the particles and anti-particles at finite µ are given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (35) and (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Substituting them in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (10)-(14), we obtain the first-order expressions for non-equilibrium hydrody- namic quantities as δn = νθ, δϵ = eθ, δP = ρθ, nµ = κ∇µα, πµν = 2ησµν, (39) where, ν = τR (χa + b0) n0, e = τR (χa + b0) ϵ0, (40) ρ = τR � (χa + b0)P0 + χb (ϵ0 + P) β − 5 3βI+ 32 − χan0 β � , (41) κ = τR ������I+ 11 − n2 0 β(ϵ0 + P) ������ , η = τR β I+ 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (42) Note that the parameter b0 appears in the expressions of ν, e and ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Of these, ν and e vanishes for b0 = −χa which cor- responds to the Landau matching condition and RTA collision kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Conductivity κ and the coefficient of shear viscosity η does not contain the parameter b0, and the expressions for these two transport coefficients, given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (42), matches with those derived using RTA collision kernel [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Next, we ana- lyze entropy production in the MBGK setup in order to identify dissipative transport coefficients in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (40)-(42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' To study entropy production, we start from the kinetic theory definition of entropy four-current, given by the Boltzmann’s H- theorem, for a classical system S µ = − � dPpµ� f (ln f − 1) + ¯f � ln ¯f − 1 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (43) The entropy production is determined by taking four- divergence of the above equation, ∂µS µ = − � dPpµ� � ∂µ f � ln f + � ∂µ ¯f � ln ¯f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (44) Using the MBGK Boltzmann equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (21), and keep- ing terms till quadratic order in deviation-from-equilibrium, we obtain ∂µS µ = 1 τR � dP (u·p) �� δ f − δϵ ϵ0 f0 � φ + � δ ¯f − δϵ ϵ0 ¯f0 � ¯φ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (45) where we have defined φ ≡ δ f/ f0 and ¯φ ≡ δ ¯f/ ¯f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (35) and (36) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (45), we obtain, ∂µS µ = −β Π θ − nµ∇µα + βπµνσµν, (46) 4 where, Π = δP − χb β δϵ + χa β δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (47) It is important to note that the right-hand-side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (46) repre- sents entropy production due to dissipation in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Here the shear stress tensor πµν is the tensor dissipation, the parti- cle diffusion four-current nµ is the vector dissipation and Π is the scalar dissipation, referred to as the bulk viscous pressure2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (47), we observe that δP, δϵ, and δn, all contribute to the bulk viscous pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Comparing with the Navier-Stokes relation of bulk viscous pressure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=', Π = −ζ θ, we obtain the coefficient of bulk viscosity as, ζ = −τR �χb β (ϵ0 + P0) − 5βI+ 32 3 − χan0 β + (χa + b0) β ( βP0 − χbϵ0 + χan0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (48) Demanding that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (46) does not violate the second law of thermodynamics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=', ∂µS µ ≥ 0, leads to the following con- straints [45], ζ ≥ 0, κ ≥ 0, η ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (49) These three transport coefficients represent the three dissipative transport phenomena of the system related to the transport of momentum and charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We see that out of the three transport coefficients, only ζ depends on the parameter b0 and the second matching condition is necessary to uniquely determine ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' This is to be expected because the matching conditions are scalar conditions and should only affect the scalar dissipation in the system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=', bulk viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In the following, we specify the second matching condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' With the parameter, b0 still not specified, the hydrodynamic equations obtained using the MBGK Boltzmann equation forms a class of hydrodynamic theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' A specific hydrodynamic the- ory is determined by a specific b0 parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We can access different hydrodynamic theories by varying the b0 parameter, which is solely controlled by the second matching condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Thus, picking a specific second matching condition will fix b0 and hence the hydrodynamic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' To this end, we define a function A± r as [19, 46], A± r = � dP (u · p)r � δf ± δ ¯f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (50) The second matching condition then amounts to assigning a value for a given A± r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' For instance, the RTA matching condi- tions can be recovered by setting A− 1 = A+ 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' It is apparent that the choice of a second matching condition is vast, and de- termination of the full list of the allowed ones is a non-trivial task that goes beyond the scope of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Presently, for the second matching condition, we shall restrict our analysis to a special set A+ r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' These matching conditions ensures that 2We can further identify that, � ∂P ∂ϵ � n = χb β and, � ∂P ∂n � ϵ = − χα β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='10 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='3 Figure 1: Dependence of the parameter b0 on z for different matching condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The red region corresponds to negative values of ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The plot is for zero chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' the homogeneous part of δ f vanishes3 [47] and are also valid in the zero chemical potential limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (35) and (36) in our proposed matching condition A+ r = 0, we obtain b0 = − � 1/I+ r,0 � � χbI+ r+1,0 − βI+ r+1,1 + χa � I+ r,0 − I− r,0 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (51) In the next section, we explore the effect of different b0 on the coefficient of bulk viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Results and discussions In this section, we study the effect of MBGK collision kernel on transport coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In the previous Section, we found that the effect of MBGK collision kernel manifests in the parameter b0 which affects only the scalar dissipation, namely bulk vis- cous pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' On the other hand, the vector (net particle dif- fusion) and tensor (shear stress tensor) dissipation remain unaf- fected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Therefore, we study only the properties of bulk viscous coefficient in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Before we proceed to quantify the effect of varying the sec- ond matching condition on the coefficient of bulk viscosity, we must establish the allowed values for the parameter b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' To this end, we note that the second law of thermodynamics demands that the coefficient of bulk viscosity must be positive, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 1, we plot b0 vs z for different values of r required to 3It must be noted that this is not the only class of matching conditions that guarantee the zero value of the homogeneous part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='100 1 10 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='04 Figure 2: Dependence of ζ/ (s0τRT) on the T/m for various α = µ/T values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The curves labelled RTA corresponds to r = 2 and those labelled MBGK cor- responds to r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' define the second matching condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (51), at zero chem- ical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The red region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 1 corresponds to the part of b0-z plane where the coefficient of bulk viscosity becomes negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Therefore all values of r for which the curves for b0 lies in the red zone are not physical and must be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The boundary of the red region corresponds to the ζ = 0 line and is given by b0 = −χa + ������ χb (ϵ0 + P0) − χan0 − (5/3) β2I+ 32 χbϵ0 − χan0 − βP0 ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (52) We find the b0 parameter with non-negative values of r respects the requirement of the second law of thermodynamics Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The black line with r = 2 represents the b0 for which the MBGK reduces to the RTA, where b0 vanishes for all z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' From numerical analysis, we find that large negative values of r leads to b0 which corresponds to negative ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 1, we see that the curve for b0, which corresponds to r = −4, passes through the physically forbidden region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Having determined the allowed range of r and equivalently, the allowed values of b0, we will restrict ourselves to b0 cor- responding to r ≥ 0 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 2 we plot the dimen- sionless quantity ζ/ (s0τRT) for MBGK with r = 0, and RTA (r = 2) against T/m for different values of chemical potential, where s0 ≡ (ϵ0 + P0 − µ n0)/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We observe that ζ/ (s0τRT) is a non-monotonous function of temperature, having a maxi- mum for each r for MBGK case, similar to the behavior known from RTA [19, 46, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We also note that the dependence of ζ/ (s0τRT) on α is also non-monotonous, which can be realized by observing that not only the position of the peak for α = 1 is at higher T/m values than for α = 0 and α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='5, but the peak value is also higher for α = 1 compared to α = 0 and α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' To better understand the effect of changing matching condi- tions on the behavior of the bulk viscosity for the MBGK col- lision kernel, we focus on the zero chemical potential limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In this limit, we study the scaling behavior of the ratio of the coef- ficient of bulk viscosity to shear viscosity, ζ/η, with conformal- ity measure 1/3−c2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 3, we plot the ratio (ζ/η)/(1/3−c2 s)2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='100 1 10 100 0 20 40 60 80 Figure 3: Variation of the dimensionless quantity (ζ/η)/(1/3−c2 s)2 with respect to z for various matching conditions determined by r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' as a function of z for different r values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We observe that this ratio saturates in both small-z and large-z limits indicating a squared dependence of ζ/η on the conformality measure, char- acteristic to weakly coupled systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We also observe that in the small-z limit, this ratio saturates to different values whereas in the large-z limit, they all converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In order to better un- derstand the behavior of ζ/η in these regimes, we separately analyze the small-z and large-z limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Small-z behaviour : The small-z limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=', m/T ≪ 1, is the ultra-relativistic limit where the mass of the particles can be ignored compared to the temperature of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' At zero chemical potential, the small-z limiting behavior of the confor- mality measure is given by � 1 3 − c2 s � = z2 36 + O � z3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' On the other hand, the small-z behavior of the ratio ζ/η is found to be ζ η = Γ(r) �1 3 − c2 s �2 + O � z5� , (53) for all r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We find the r-dependence of the coefficient to be, Γ(r) ≡ lim z→0 ζ/η � 1 3 − c2s �2 = 15(r2 + 23r + 10) 4(r + 1) , (54) for r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Thus, while the ratio ζ/η shows a z4 dependence in the same small-z limit, the coefficient Γ depends on the match- ing condition through b0, and equivalently r, as is evident from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 4, we show the variation of the coefficient Γ as a function of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We observe that for r = 2, we recover the RTA value, Γ = 75, marked with a red dot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Large-z behaviour : On the opposite end, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=', at the large-z limit where m/T ≫ 1, we have the non-relativistic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In this limit, the conformality measure is expanded in powers of 1/z and is given by, 1 3 − c2 s = 1 3 − 1 z + O � 1 z2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The behaviour of the ratio ζ/η in the same limit is given by, ζ η = 2 3 − 3 z + O � 1 z2 � , (55) 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='0 40 50 60 70 80 Figure 4: Variation of the scaling coefficient Γ, defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' (53) and (54), with respect to parameter r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The red dot represents the RTA value of Γ = 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' for all r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Considering the leading terms in this expansion, we find, ζ η = 6 �1 3 − c2 s �2 , (56) which is independent of r and hence the second matching con- dition, as is evident from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In this limit, the MBGK and RTA results coincide implying that in the non-relativistic limit, the properties of the fluid are independent of the nature of col- lision with BGK collision kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Summary and outlook In this work, we have provided the first formulation of rela- tivistic dissipative hydrodynamics from BGK collision kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We first propose a modified BGK collision kernel which we ad- vocate to be better suited for derivation of hydrodynamic equa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We show that at finite chemical potential, where BGK collision kernel is defined, the formulation of relativistic hy- drodynamics with MBGK collision kernel becomes identical to that obtained using BGK collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The advantage of MBGK is that it is well defined even in the limit of vanishing chemi- cal potential, and represents a generalization of RTA collision kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' In the formulation of relativistic BGK hydrodynamics, we found the theory is controlled by a free parameter related to freedom of a matching condition, which affects the coeffi- cient of bulk viscous pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' It is important to note that the BGK or MBGK collision kernels are affected by the matching conditions, which in turn affects the dissipative processes in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We have identified a class of matching conditions for which the homogeneous part of the solution to the relativistic Boltzmann equation vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' We examined the effect of choice of matching condition on dissipative coefficients and also stud- ied some scaling properties of the ratio of coefficients of bulk viscosity to shear viscosity on the conformality measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The present formulation of hydrodynamics with a modified BGK collision kernel opens up several possibilities for future investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' This MBGK collision kernel may also find po- tential applications in non-relativistic physics domain where BGK collision is widely used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The formulation of causal hydro- dynamics with MBGK collision kernel is an immediate possible extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Formulation of higher-order hydrodynamic theories may be affected more significantly as the evolution equations of scalar, vector, and tensor dissipative quantities contain cross- terms giving rise to the possibility of them being controlled by the matching conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Higher-order theories also exhibit in- teresting features like fixed points and attractors [49, 50], which could also be studied within the MBGK hydrodynamics frame- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' The present article forms the basis for all these studies which we leave for future explorations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Acknowledgements The authors acknowledge Sunil Jaiswal for sev- eral useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' was supported in part by the DST-INSPIRE faculty award under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' DST/INSPIRE/04/2017/000038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' References [1] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' M¨uller, Zum Paradoxon der Warmeleitungstheorie, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 198 (1967) 329–344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1007/BF01326412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Chapman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Cowling, The Mathematical Theory of Non- Uniform Gases, 3rd ed, Cambridge University Press, Cambridge (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [3] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Israel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Stewart, Transient relativistic thermodynamics and ki- netic theory, Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 118 (1979) 341–372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1016/ 0003-4916(79)90130-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Muronga, Relativistic Dynamics of Non-ideal Fluids: Viscous and heat-conducting fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Transport properties and microscopic descrip- tion of relativistic nuclear matter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' C 76 (2007) 014910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv: nucl-th/0611091, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='014910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' York, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Moore, Second order hydrodynamic coefficients from kinetic theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' D 79 (2009) 054011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:0811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='0729, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='054011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [6] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Betz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Henkel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rischke, From kinetic theory to dissipative fluid dynamics, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 62 (2009) 556–561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:0812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 1440, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='ppnp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [7] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Romatschke, New Developments in Relativistic Viscous Hydrodynam- ics, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' E 19 (2010) 1–53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:0902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='3663, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1142/S0218301310014613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [8] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Denicol, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Koide, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rischke, Dissipative relativistic fluid dy- namics: a new way to derive the equations of motion from kinetic the- ory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 105 (2010) 162501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='5013, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='162501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Denicol, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Niemi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Molnar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rischke, Derivation of transient relativistic fluid dynamics from the Boltzmann equation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' D 85 (2012) 114047, [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='D 91, 039902 (2015)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv: 1202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='4551, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='114047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Jaiswal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Ryblewski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Strickland, Transport coefficients for bulk viscous evolution in the relaxation time approximation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' C 90 (4) (2014) 044908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='7231, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='044908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Jaiswal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Friman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Redlich, Relativistic second-order dissipative hydrodynamics at finite chemical potential, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' B 751 (2015) 548–552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='02849, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Gabbana, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Mendoza, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Succi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Tripiccione, Kinetic approach to relativistic dissipation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' E 96 (2) (2017) 023305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv: 1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='02523, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='023305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Blaizot, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Yan, Onset of hydrodynamics for a quark-gluon plasma from the evolution of moments of distribution functions, JHEP 11 (2017) 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='10694, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1007/JHEP11(2017)161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 7 [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Jaiswal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Blaizot, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Bhalerao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Jaiswal, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Yan, From moments of the distribution function to hydrodynamics: The nonconfor- mal case, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' C 106 (4) (2022) 044912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='02750, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='044912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [15] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Bhatnagar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Gross, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Krook, A Model for Collision Processes in Gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Small Amplitude Processes in Charged and Neutral One- Component Systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 94 (1954) 511–525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/ PhysRev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [16] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Welander, On the temperature jump in a rarefied gas, Arkiv Fysik 7 (1954) 507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='osti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='gov/biblio/4395580 [17] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Marle, Sur l’´etablissement des ´equations de l’hydrodynamique des fluides relativistes dissipatifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' - L’´equation de Boltzmann relativiste, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 10 (1969) 67–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' URL http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='numdam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='org/item/AIHPA_1969__10_1_67_0/ [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Anderson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Witting, A relativistic relaxation-time model for the boltzmann equation, Physica 74 (3) (1974) 466–488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [19] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rocha, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Denicol, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Noronha, Novel Relaxation Time Approx- imation to the Relativistic Boltzmann Equation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 127 (4) (2021) 042301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='07489, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='042301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [20] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Florkowski, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Maj, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Ryblewski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Strickland, Hydrodynamics of anisotropic quark and gluon fluids, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' C 87 (3) (2013) 034914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='3671, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='034914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [21] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Florkowski, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Ryblewski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Strickland, Anisotropic Hydrodynam- ics for Rapidly Expanding Systems, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' A 916 (2013) 249–259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='0665, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='nuclphysa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [22] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Florkowski, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Ryblewski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Strickland, Testing viscous and anisotropic hydrodynamics in an exactly solvable case, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' C 88 (2013) 024903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='7234, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 024903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [23] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Denicol, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Florkowski, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Ryblewski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Strickland, Shear-bulk coupling in nonconformal hydrodynamics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' C 90 (4) (2014) 044905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='4767, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='044905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [24] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Florkowski, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Maksymiuk, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Ryblewski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Strickland, Exact so- lution of the (0+1)-dimensional Boltzmann equation for a massive gas, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' C 89 (5) (2014) 054908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='7348, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/ PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='054908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [25] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Florkowski, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Ryblewski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Strickland, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Tinti, Leading-order anisotropic hydrodynamics for systems with massive particles, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' C 89 (5) (2014) 054909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1223, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/ PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='054909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [26] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Tinti, Anisotropic matching principle for the hydrodynamic expan- sion, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' C 94 (4) (2016) 044902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='07164, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='044902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [27] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Czajka, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Hauksson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Shen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Jeon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Gale, Bulk viscosity of strongly interacting matter in the relaxation time approximation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' C 97 (4) (2018) 044914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='05905, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/ PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='044914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Kurian, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Chandra, Bulk viscosity of a hot QCD medium in a strong magnetic field within the relaxation-time approximation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' D 97 (11) (2018) 116008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='07904, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/ PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='116008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [29] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Chattopadhyay, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Heinz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Pal, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Vujanovic, Higher order and anisotropic hydrodynamics for Bjorken and Gubser flows, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' C 97 (6) (2018) 064909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='07755, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='064909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [30] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Chattopadhyay, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Jaiswal, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Du, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Heinz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Pal, Non-conformal attractor in boost-invariant plasmas, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' B 824 (2022) 136820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='05500, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='136820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [31] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Jaiswal, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Chattopadhyay, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Du, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Heinz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Pal, Nonconformal kinetic theory and hydrodynamics for Bjorken flow, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' C 105 (2) (2022) 024911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='10248, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 024911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [32] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Liyanage, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Everett, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Chattopadhyay, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Heinz, Prehydrody- namic evolution and its impact on quark-gluon plasma signatures, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' C 105 (6) (2022) 064908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='00964, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/ PhysRevC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='064908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [33] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Carrington, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Fugleberg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Pickering, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Thoma, Dielectric functions and dispersion relations of ultrarelativistic plasmas with col- lisions, Can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 82 (2004) 671–678.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:hep-ph/0312103, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1139/p04-035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [34] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Schenke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Strickland, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Greiner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Thoma, A Model of the effect of collisions on QCD plasma instabilities, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' D 73 (2006) 125004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:hep-ph/0603029, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='125004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Mandal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Roy, Wake potential in collisional anisotropic quark-gluon plasma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' D 88 (7) (2013) 074013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='4660, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='074013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [36] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Jiang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Hou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='-r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Li, Refractive index of quark-gluon plasma: Kinetic theory with a Bhatnagar-Gross-Krook collisional kernel, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' D 94 (7) (2016) 074026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='074026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [37] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Han, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Hou, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='-r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Li, Jet energy loss in quark- gluon plasma: Kinetic theory with a Bhatnagar-Gross-Krook collisional kernel, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' A 53 (10) (2017) 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1140/epja/ i2017-12400-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [38] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Kumar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Jamal, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Chandra, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Bhatt, Collective excitations of a hot anisotropic QCD medium with the Bhatnagar-Gross-Krook col- lisional kernel within an effective description, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' D 97 (3) (2018) 034007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='01032, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='034007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [39] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Khan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Patra, Cumulative effects of collision integral, strong magnetic field, and quasiparticle description on charge and heat transport in a thermal QCD medium, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' D 104 (5) (2021) 054024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv: 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='02682, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='054024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [40] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Formanek, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Grayson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rafelski, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' M¨uller, Current-conserving rel- ativistic linear response for collisional plasmas, Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 434 (2021) 168605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='07897, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='aop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='168605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [41] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Khan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Patra, Transport coefficients in thermal QCD: A probe to the collision integral, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' D 106 (9) (2022) 094033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv: 2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='00317, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='094033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [42] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Khan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Patra, Seebeck and Nernst coefficients of a magnetized hot QCD medium with number conserving kernel (11 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv: 2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='10779.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Shaikh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rath, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Dash, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Panda, Flow of charge and heat in thermal QCD within the weak magnetic field limit: A BGK model approach (10 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='15388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [44] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' De Groot, Relativistic Kinetic Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Principles and Applications, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [45] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Kovtun, First-order relativistic hydrodynamics is stable, JHEP 10 (2019) 034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='08191, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1007/JHEP10(2019)034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [46] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Biswas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Mitra, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Roy, Is first-order relativistic hydrodynamics in a general frame stable and causal for arbitrary interactions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' D 106 (1) (2022) L011501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='08685, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/ PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='L011501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [47] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Hoult, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Kovtun, Causal first-order hydrodynamics from kinetic theory and holography, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' D 106 (6) (2022) 066023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv: 2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='14042, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='066023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [48] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Dash, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Bhadury, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Jaiswal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Jaiswal, Extended relaxation time approximation and relativistic dissipative hydrodynamics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' B 831 (2022) 137202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='14581, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='137202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [49] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Heller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Spalinski, Hydrodynamics Beyond the Gradient Ex- pansion: Resurgence and Resummation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 115 (7) (2015) 072501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='07514, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='072501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' [50] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Blaizot, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Yan, Fluid dynamics of out of equilibrium boost invariant plasmas, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' B 780 (2018) 283–286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='03856, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content='058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'} +page_content=' 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfqfiK/content/2301.00544v1.pdf'}