diff --git "a/9tFLT4oBgHgl3EQfui8A/content/tmp_files/load_file.txt" "b/9tFLT4oBgHgl3EQfui8A/content/tmp_files/load_file.txt"
new file mode 100644--- /dev/null
+++ "b/9tFLT4oBgHgl3EQfui8A/content/tmp_files/load_file.txt"
@@ -0,0 +1,685 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf,len=684
+page_content='epl draft  Towards a liquid-state theory for active matter  Yuting Irene Li1, Rosalba Garcia-Millan1,3, Michael E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Cates1 and ´Etienne Fodor2  1 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK  2 Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg  3 St John’s College, University of Cambridge, Cambridge CB2 1TP, UK  PACS 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='Ln – Nonequilibrium and irreversible thermodynamics  Abstract – In equilibrium, the collective behaviour of particles interacting via steep, short-ranged  potentials is well captured by the virial expansion of the free energy at low density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Here, we extend  this approach beyond equilibrium to the case of active matter with self-propelled particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Given  that active systems do not admit any free-energy description in general, our aim is to build the  dynamics of the coarse-grained density from first principles without any equilibrium assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Starting from microscopic equations of motion, we obtain the hierarchy of density correlations,  which we close with an ansatz for the two-point density valid in the dilute regime at small activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  This closure yields the nonlinear dynamics of the one-point density, with hydrodynamic coefficients  depending explicitly on microscopic interactions, by analogy with the equilibrium virial expan-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This dynamics admits a spinodal instability for purely repulsive interactions, a signature  of motility-induced phase separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Therefore, although our approach should be restricted to  dilute, weakly-active systems a priori, it actually captures the features of a broader class of active  matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Active matter is a wide category of systems out of equi-  librium, where a net flow of energy takes place at the local,  individual level [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Examples are swimming bacteria [4]  and active emulsions [5], amongst others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The realm of  motile active matter includes interacting, many-particle  systems, where particles move persistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Different the-  oretical approaches have been proposed to describe such  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' They include (i) particle-based models, such as  run-and-tumble particles (RTPs) [6], active Brownian par-  ticles (ABPs) [7], and active Ornstein-Uhlenbeck particles  (AOUPs) [8,9], (ii) microscopic field theories [10,11], and  (iii) coarse-grained field theories [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Self-propelled particles tend to slow down at high densi-  ties, due to either biochemical reasons or steric repulsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In contrast with passively diffusing particles, this slow-  down in turn increases the local density, creating a pos-  itive feedback loop that can result in a phase separation  between a dense and a dilute phase, known as motility-  induced phase separation (MIPS) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  To predict an-  alytically the emergence of MIPS, previous works have  relied on a local mean-field approximation to replace mi-  croscopic interaction with density-dependent motility [16],  and also on an adiabatic elimination of orientational de-  gree of freedom [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Other studies have developed some  aspects of a liquid-state theory for active matter with pair-  wise interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This is done, for instance, by expressing  thermodynamic observables, such as pressure [17–19] and  dissipation [20,21], in terms of correlations between hydro-  dynamic fields, typically density and polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Also,  exact results for density correlations have been obtained  at infinite dimensions [22,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  The success of equilibrium thermodynamics largely  stems from the ability to detect phase transitions by an-  alyzing the free-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In the dilute regime, the virial  expansion provides an approximate expression of the free-  energy that averages the effect of steep, short-ranged pair-  wise interactions [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  For passive Brownian particles  (PBPs) interacting with an isotropic pairwise potential  �  i,j<i Ψ(|xi − xj|) at temperature T, the free energy per  unit volume is given in terms of the uniform density ρ as  fvirial(ρ)/T = ρ log ρ + B(T)ρ2 + O(ρ3),  B = 1  2  �  dr  �  1 − e−Ψ(r)/T �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (1)  Here B is commonly referred to as the second virial coef-  ficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The major success of the virial expansion lies in  predicting the onset of liquid-gas phase separation in parti-  cle systems with a combination of strongly repulsive short-  ranged forces and weakly attractive long-range forces, such  as the Lennard-Jones potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' As temperature increases,  p-1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='12155v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='soft]  28 Jan 2023  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  the free energy, as a function of density ρ, goes from convex  (B > 0) to concave (B < 0): It yields a phase transition  from homogeneous to non-homogeneous density profiles,  with a separation between dilute and dense phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Im-  portantly, the virial expansion holds for a generic micro-  scopic potential (excluding those of such long range that  the excess free energy is not analytic in density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In classical thermodynamics, the virial expansion is of-  ten derived from the equilibrium partition function [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Interestingly, it is also possible to arrive at approx-  imate expressions of free energy by truncating the  hierarchy of density correlations,  known as the Bo-  goliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierar-  chy [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' For instance, using an ansatz for the two-particle  density, informed by the steady-state solution of the two-  body problem [26], the dynamics of the one-body density  follows a gradient flow with respect to fvirial, as shown  in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This type of derivation can potentially be ex-  tended to a large class of nonequilibrum systems, where  the dynamics does not derive from any free energy a pri-  ori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Such an extension requires proposing an ansatz for  the two-particle density, which should be appropriate to  the specific nonequilibrium system at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Interestingly, the two-body probability distribution can  be approached perturbatively for AOUPs, where the per-  sistence time is the small parameter in natural units [8,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  To first order, this solution already predicts that repul-  sive interaction yields effective attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This suggests  that the onset of MIPS can likewise already be captured  by approximating the dynamics of density at this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Therefore, inspired by the equilibrium virial expansion, it  is tempting to explore whether the density dynamics al-  ready contains any spinodal instability for dilute active  systems at weak persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Note that, although previ-  ous work refer to “active virial” as an expansion of pres-  sure [27, 28], here we are instead interested in deriving  dynamical equations of motion for the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Indeed, at  variance with equilibrium, phase transitions in active sys-  tems are usually not thermodynamically controlled by any  equation of state, but they can still be directly detected  as instability in the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In this Letter, we start with the particle-based descrip-  tion of AOUPs, and consider their steady-state probability  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Our roadmap is essentially a “small density-  small persistence” bottom-up derivation of the density dy-  namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' To this end, we first introduce the corresponding  hierarchy of density correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Inspired by equilibrium  thermodynamics [25, 26], we then close the hierarchy to  arrive at a nonlinear dynamics for the one-point density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  It allows us to identify the MIPS spinodal instability for  any microscopic repulsive interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' An advantage of  our approach is that does not impose an equilibrium map-  ping a priori (see [8,9] for a discussion of closures to the  AOUP equations that do so), instead retaining the non-  equilibrium structure of the microscopic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' A disad-  vantage is that extensions to higher order in density would  involved increasingly complicated calculations of higher  virial coefficients, which we do not attempt here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Interacting AOUPs:  Joint distribution func-  tion – We consider a system of N AOUPs at positions  xi interacting with pairwise potential U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  The particles  are subject to stochastic self-propulsion velocities vi with  persistence time τ and diffusivity D [8,9]:  ˙xi = vi − ∂xiU,  U =  N  �  i=1  N  �  j<i  Ψ(rij),  τ ˙vi = −vi +  √  2DΛi,  (2)  where rij = |xi − xj|, and we have set the mobility to  unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Here Λi is a Gaussian white noise, with correlation  ⟨(Λi)α(t)(Λj)β(0)⟩ = δijδαβδ(t), where latin and Greek  letters respectively denote particle index and spatial co-  ordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' It follows that vi is a colored Gaussian noise,  with correlation ⟨(vi)α(t)(vj)β(0)⟩ = δijδαβ(D/τ)e−|t|/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In the limit of vanishing persistence (τ → 0), the corre-  lation of vi becomes white, so that the system reduces to  a set of overdamped PBPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Note that, at finite τ, if the  potential term −∂xiU was in the rhs of the vi-equation  (instead of the xi-equation), the system would represent  underdamped PBPs in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' For this reason, over-  damped AOUPs and underdamped PBPs coincide in the  noninteracting limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Following  standard  procedures  [29],  the  Fokker-  Planck equation of the joint probability distribution  pN(x1, v1, x1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' , xN, vN, t) associated with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (2) is  ∂tpN =  N  �  i=1  �  Lf,ipN + ∂xi · (pN ∂xiU)  �  ,  Lf,i = D  τ 2 ∂2  vi +  �1  τ ∂vi − ∂xi  �  vi,  (3)  where Lf,i is the non-interacting Fokker-Planck operator  governing the free-particle motion of the i-th particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Al-  though it cannot be solved exactly, its steady state can be  computed to lowest orders in the persistence time τ in a  similar manner to [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Such a perturbative calculation  can be performed in arbitrary dimension, though we now  restrict ourselves to 1D for simplicity, as a proof of prin-  ciple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' After scaling units as v → v  �  τ/D , x → x/  √  τD,  t → t/τ, and Ψ → Ψ/D, we find the many-body steady  state probability [see SM for details],  pN ∝ e−U−  v2  1+v2  2  2  �  1 + √τ  N  �  i=1  vi∂xiU  + τ  N  �  i=1  �v2  i  2  �  (∂xiU)2 − ∂2  xiU  �  − (∂xiU)2 + 3  2∂2  xiU  �  + o(τ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (4)  Note that the steady state probability here is given in  (x, v) space, whereas [8] gives the corresponding probabil-  ity in (x, ˙x) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  p-2  Towards a liquid-state theory for active matter  Nonequilibrium density correlations: Hierarchy  of equations – Although the joint distribution function  pN in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (4) contains all information about the steady  state density, it is generally difficult to predict the emer-  gence of phase transitions from the perspective of the full  phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Instead, our aim is to obtain a reduced de-  scription of the systel in terms of density correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The  n-particle density pn is found by marginalising the joint  distribution function pN as  pn(x1, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' , xn, vn, t)  = PN,n  �  pN(x1, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' , xN, vN, t)  N  �  j=n+1  dxjdvj,  (5)  where PN,n = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='/(N − n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' is the permutation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Thereafter, for simplicity, we use the shorthand notation  pn = pn(x1, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' , xn, vn, t) for arbitrary n, where the  dependence on positions, self-propulsion velocities, and  time will be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Integrating eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (3) over the positions  and velocities of all particles but one yields an equation for  the one-particle density that depends on the two-particle  density:  ∂tp1 = Lf,1p1 + F1[p2],  F1[p2] = ∂x1 ·  �  p2 ∂x1Ψ(r12) dx2dv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (6)  The first term Lf,1p1 corresponds to the free single-particle  dynamics, whereas the second term F1[p2] represents the  effects of pairwise interaction between one particle and all  the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Following the same marginalisation procedure for the  n-particle density pn, we obtain a hierarchy of equations,  each depending on the density pn+1, analoguous to the  BBGKY hierarchy of PBPs [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The resulting equation  for the two-particle density reads  ∂tp2 = (Lf,1 + Lf,2 + Lint)p2 + F2[p3],  Lint =  �  i=1,2  ∂xi ·  �  ∂xiΨ(r12)  �  ,  F2[p3] =  �  i=1,2  ∂xi ·  �  p3 ∂xiΨ (ri3) dx3dv3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (7)  Here Lint governs the pairwise interactions between any  two particles, and F2[p3] represents the interactions be-  tween either of the first two particles and all the other  particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' As we will show now, this is a higher-order term  in the average number density ρ = N/V compared to the  rest, and hence can be neglected in the dilute limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  To determine the scaling of pn with respect to ρ, one  can start with their normalisation properties [24]  �  pn  n  �  i=1  dxidvi = PN,n,  (8)  from which follows pn ∼ ρn for small n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Physically, this  is indeed expected as p1(x1, v1) is the probability density  of finding n particles at a given set of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' With  this scaling in mind, we estimate the scaling of each term  in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (7) as  (∂t − Lf,1 − Lf,2 − Lint  ����  ∼1/τint  ) p2  ����  ∼ρ2  =  �  i=1,2  ∂xi ·  �  p3  ����  ∼ρ3  ∂xiΨ(ri3)  �  ��  �  ∼1/τint  dx3  ����  ∼(r0)d  dv3,  (9)  where τint is the typical timescale of interaction, r0 is the  lengthscale of the potential, and d denotes the sptial di-  mension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' One can see that the rhs of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (9) is small if  rd  0ρ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Since rd  0 is effectively the volume of a particle,  this condition is satisfied if the volume fraction of parti-  cles in the system is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Therefore, as expected from the  analogy with the BBKY hierarchy [25], the interaction of  two particles with respect to others in the bath can be  neglected at small volume fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Closure of hierarchy:  Insight from quasistatic  approximation – We now propose a closure of the hierar-  chy of equation for density correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' At small volume  fraction, neglecting the rhs in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (9) yields  ∂tp2 = (Lf,1 + Lf,2 + Lint) p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (10)  The dynamics in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (10) is equivalent to the Fokker-  Planck equation of two AOUPs interacting through the  pairwise potential Ψ(r12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Recall from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (4) the N-  particle stationary probability, calculated order by order  in the persistence time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The two-particle probability is  hence,  p2,ss(r12, v1, v2) ∝ e−Ψ(r12)−  v2  1+v2  2  2  g(r12, v1 − v2),  (11)  where  g(r, w) = 1 + √τwΨ′ + (τw2/2)((Ψ′)2 − Ψ′′)  + τ(3Ψ′′ − 2(Ψ′)2) + O(τ 3/2),  (12)  and Ψ′ = dΨ/dr [see SM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' In the quasistatic limit, where  the interaction timescale is much shorter than the persis-  tence (τint ≪ τ), we assume that every pair of particles  reaches steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Inspired by previous works [26, 30],  this assumption motivates the following ansatz to close the  hierarchy of density correlations:  p2(x1, v1, x2, v2, t) = p1(x1, v1, t) p1(x2, v2, t)  × e−Ψ(r12)g(r12, v1 − v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (13)  There is evidence that the above ansatz works well for  strong short-ranged interactions in the equilibrium case  [see SM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We assume it would also work in the non-  equilibrium case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Indeed, when the interparticle distance  is larger than the interaction range (r12 > r0), the two-  point function p2 reduces to the product of one-point func-  tions p1, as expected from a mean-field approach for non-  interacting systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The second line of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (13) accounts  p-3  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  for corrections due to interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' For repulsive poten-  tial, the factor e−Ψ captures the reduction of the two-  point density caused by the interaction, as expected from  equilibrium, whereas g(r12, v1 − v2) encapsulates nonequi-  librium effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Substituting the ansatz from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (13) into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (6), we  arrive at the following dynamics for the one-particle den-  sity:  (∂t−Lf) p1(x, v, t)  = τ∂x  �  p1(x, v, t)  �  Lvp1(x, v − w, t)dw  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (14)  Here, Lf is the free Fokker-Planck operator in scaled units,  and the operator Lv effectively takes into account interac-  tions:  Lf = ∂2  v +  �  ∂v − √τ∂x  �  v,  Lv =  �  Ψ′(r)e−Ψ(r)g(r, w)e−r∂xdr,  (15)  where we have introduced the translation operator e−r∂x,  which shifts the position of the function acted upon as  e−r∂xp1(x, v) = p1(x − r, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Hence, Lv corresponds to an  infinite series of the gradient ∂x, with series coefficients set  by the microscopic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Equation (14) is the central  result of this Letter: It provides the dynamics of the one-  particle density in a closed form for small average density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Importantly, this closed form depends explicitly on the  details of the pairwise potential Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  To obtain a more explicit expression of Lv, we next  substitute our perturbative result for g [eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (12)] into the  definition of Lv [eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (15)], yielding  Lv = 2wLa + 2Lb∂x + w2Lc∂x,  (16)  where, after integration by parts, we get  La = √τ  � ∞  0  dr(Ψ′)2 e−Ψe−r∂x,  Lb = −  � ∞  0  drf0(r)e−r∂x,  Lc = −  � ∞  0  dr f1(r)e−r∂x,  (17)  and  f0(s) =  � ∞  s  dr Ψ′e−Ψ �  1 − τ  �  2(Ψ′)2 − 3Ψ′′��  ,  f1(s) = τ  � ∞  s  dr Ψ′ �  (Ψ′)2 − Ψ′′�  e−Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (18)  The integrands featuring in the definition of the operators  {La, Lb, Lc} [eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (17)] are even with respect to r, provided  that Ψ(r) = Ψ(−r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' It follows that these operators can be  expressed as a series of ∂2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  The functions f0 and f1 are analoguous to the Mayer  function which appears in the equilibrium virial expan-  sion [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Indeed, in the equilibrium limit (τ → 0), the  only surviving term in Lv stems from the leading order in  f0, in agreement with the derivation for equilibrium over-  damped PBPs [see SM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This term is responsible for the  B coefficient in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (1) when that result is derived dynam-  ically for equilibrium systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' In that respect, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (16)  can be regarded as a direct generalization of the equilib-  rium virial expansion to the AOUP setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Importantly,  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (16) shows that activity produces additional velocity-  dependent terms that are absent in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' In what  follows, we analyze in detail the corresponding dynam-  ics, in search for the onset of instability as a signature of  MIPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Linear stability analysis: Eigenvalue problem –  The stationary solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (14) is given by p(0)  1 (v) =  (ρ/  √  2π)e−v2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This solution corresponds to a uniform  density for the position and a Gaussian distribution for  the self-propulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In addition, as we will see shortly,  this is also the ground state of the free operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' In the  following, we expand perturbatively around p(0)  1  to find its  instability regions in parameter space: If the uniform state  is not stable, then we argue that the system undergoes  spinodal decomposition via a MIPS mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  We consider p1(x, v, t) = p(0)  1 (v) + ε(x, v, t), with ε a  small perturbation about the ground state p(0)  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Expand-  ing eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (14) to linear order in ε, we get  (∂t− ¯Lf) ε(x, v, t) = τp(0)  1 (v)  �  Lv∂xε(x, v−w, t)dw, (19)  where Lv is defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (16), and  ¯Lf = ∂2  v + (∂v − α∂x)v,  α = √τ − 2ρτ 3/2  � ∞  0  (Ψ′)2e−Ψdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (20)  To analyze the time evolution of the perturbation given in  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (20), the difficulty lies in treating the effect of the op-  erator Lv, which contains information about microscopic  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Then, it is convenient to expand ε in the  eigenfunctions of the bare theory, namely in the absence  of Lv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Interestingly, as stated previously, the bare theory  maps into underdamped passive Brownian motion, and  one can readily find the solution in the literature [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  The corresponding eigenfunctions, which we here call the  Fourier-Hermite basis, are  ψnk(x, v) = e−ik(x+αv)Un(v − 2iαk),  (21)  where Un is the Hermite function  Un(z) =  1  √  2π Hn(z) e−z2/2 = (−1)n  √  2π ∂n  z e−z2/2,  (22)  with the property that (∂2  z + ∂zz)Un(z) = −nUn(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' In-  deed, acting on the Fourier-Hermite basis with the modi-  fied free operator ¯Lf yields  ¯Lfψnk = −λknψnk,  λkn = (αk)2 + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (23)  p-4  Towards a liquid-state theory for active matter  As the operator ¯Lf is not Hermitian, the conjugate basis is  not simply the complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Instead, we introduce  the following conjugate basis  ¯ψnk(x, v) = eik(x+αv) ¯Un(v − 2iαk),  ¯Un(z) = 1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='Hn(z),  (24)  yielding  �  dxdv ¯ψmk′(x, v)ψnk(x, v) = 2πδ(k − k′)δmn,  (25)  so that the orthogonality relation between the basis ψnk  and its conjugate ¯ψnk indeed holds as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Having obtained the eigenvectors and eigenvalues of the  bare theory, we decompose the perturbation ε in eigen-  functions of ¯Lf as  ε(x, v, t) =  �  n  �  εkn(t)ψnk(x, v)dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (26)  Multiplying eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (19) with the conjugate basis ¯ψkn and inte-  grating over {x, v} yields the dynamics of the perturbation  ε, in the Fourier-Hermite basis, as  ˙εkm = −λkmεkm+  �  n  �  M (1)  kmn+M (2)  kmn+M (3)  kmn  �  εkn, (27)  where  M (1)  kmn = 2τρLa,k  (−iαk)n+m  αm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  e(αk)2(m − n),  M (2)  kmn = −2τρLb,kk2 (−iαk)n+m  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  e(αk)2,  M (3)  kmn = τρLc,k  (−iαk)n+m  α2m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  e(αk)2  ×  �  m(m − 1) + n(n − 1) − 2mn − 2(αk)2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (28)  Here, La,k =  �  eikxLadk, with similar definitions for Lb,k  and Lc,k, where the operators {La, Lb, Lc} are defined in  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  If the matrix Mknm = −λkmδmn + M (1)  kmn +  M (2)  kmn +M (3)  kmn, which controls the growth rate of the per-  turbation ε, has no positive eigenvalue, the uniform so-  lution is stable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' otherwise the system undergoes spinodal  decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Hence we only need the largest eigenvalue  of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' As shown in the SM, the matrix M can be diagnon-  alized exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This calculation actually only amounts to  finding the maximum eigenvalue of a 3 × 3 matrix, which  can be done straightforwardly, while also perturbatively  keeping track of orders of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Spinodal instability – We now illustrate how our ap-  proach, valid for an arbitrary pairwise potential, can be  deployed to detect the onset of instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We consider the  following short-ranged, weakly repusive interaction poten-  tial:  Ψ(r) = ν exp  �  −  1  1 − (r/r0)2  �  ,  (29)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='00  k  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='10  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='10  λmax  τ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='00  k  τ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='50  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' 1: The value of the largest eigenvalue λmax as a function of  the wavevector k for ν = 10, r0 = 2, varying τ and φ (blue solid  line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The orange dotted line represents the short length-scale  cut-off as we are only concerned with length-scales beyond the  particle size r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  characterised by its strength ν and range r0, whose deriva-  tives are continuous at any order for r within [0, r0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The  maximum eigenvalue of the corresponding growth rate ma-  trix M, expanded analytically to first order in the persis-  tence time τ, is plotted against wavevector k in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' 1  for various values of τ and volume fraction φ = 2r0N/V ,  where 2r0 is taken as the size of the particle in 1D and  V = L is the system volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' As expected [15], the sys-  tem exhibits a spinodal instability at high τ and packing  fraction φ, for which the linear perturbation ε is unstable,  namely limk→0 λmax(k) = 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' 2 shows the stabil-  ity diagram based on the sign of the longest wavelength  perturbation, λmax(2π/L), as a function of τ and φ, show-  ing the transition between a uniform stable and a phase-  separated state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (Note that we could equally have chosen  the fastest growing mode to locate the spinodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=')  It is notable that our theory of interacting AOUPs  shows a spinodal instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' As in equilibrium systems  with attractions, it does this even at lowest order in the  virial-type expansion in powers of density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Just as holds  there, finding the instability requires using a low-density  theory at finite densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' However the purpose of this ap-  proach is not to gain quantitative predictions about the ex-  act location of the spinodal curve, but rather to establish  that the macroscopic conditions for phase separation can  be satisfied, by considering microscopic interaction laws  and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Notably, unlike in equilibrium, the spin-  odal instability happens here for purely repulsive interac-  tions – a key feature of MIPS [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Our microscopic calcu-  lation complements previous viewpoints based on effective  quasi-equilibrium attractions [31], and/or collisional slow-  ing down [16,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Nonetheless, there are several caveats and limitations to  our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Firstly, the chosen potential is not strictly  hard-core, but depends on the strength parameter ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' In  fact it is not possible to implement a true hard-core po-  tential due to the nature of the small τ expansion for the  two-particle density, as there are terms directly propor-  tional to Ψ or its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Unsurprisingly then, the  p-5  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='0  φ  10−1  100  τ  Phase separation  Uniform  0  λmax  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' 2: Stability diagram in τ − φ space showing λmax(2π/L)  for φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='6, L = 200, ν = 10, r0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The white region in the  top-right corner corresponds to when α < 0, where the method  breaks down, as it is no longer in the small τ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The solid  black line is the spinodal instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  results presented here do depend on ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' While overly large  ν will break the small-τ expansion, we have checked that a  range of moderate values produce stability diagrams qual-  itatively similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Secondly, for some parameters,  we found that the largest eigenvalue stays negative at low  wavenumber but turns positive (unstable) at large ones,  resembling an upside-down version of the right frame of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' At first sight this might be taken as evidence of  new physics in the form of microphase separation at finite  wavenumber, an outcome generically predicted by some  field-theoretic models [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' However, when this happens  the wavenumbers in question (at or around the orange line  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' 1) are comparable to the inverse particle separation  1/r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' As the theory proposed here is a macroscopic one,  it may not operate reliably in this large k region, so we  do not take this as evidence of microphase separation in  repulsive AOUPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Conclusion – Inspired by liquid-state theory of equi-  librium systems, we have started with the hierarchy of  density correlations for AOUPs, and closed it at second  order using a quasistatic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This approach is  closely related to the virial expansion in the equilibrium  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' It yields a mean field theory for the one-point den-  sity from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We have analysed the stability  of the uniform solution by looking at the largest eigen-  value of the growth matrix, and determined the onset of  a spinodal instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Our method deals directly with particle velocities as well  as positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Perhaps more importantly, it is able to cap-  ture (to lowest order in density) the physics of steep, short-  ranged interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In that respect, it is distinct from  standard coarse-graining methods for non-equilibrium sys-  tems, such as those based on Dean’s equation [33], which  starts as an exact representation but upon assuming a  smooth (coarse-grained) density becomes an expansion in  weak or slowly varying interaction forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  The Dean’s  approach neglects strong, short-range interactions that  change the statistics of close encounters even when the  density, and hence the average interaction force, is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In equilibrium statistical mechanics, the virial expansion  is designed to handle exactly this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We have pre-  sented the leading order counterpart of this, for an active  system comprising interacting AOUPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  There are some limitations to our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Firstly, the  two-particle stationary solution used in the quasistatic  ansatz is not exact, but rather a small-persistence approx-  imation, unlike the equilibrium case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Secondly, in truncat-  ing the density-correlation hierarchy at the second order,  we assumed that the number density is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Hence, even  assuming small persistence, our approach is quantitatively  accurate only in the dilute limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Nonetheless, our method  gives bottom-up confirmation of how phase separation can  occur in purely repulsive active systems, and it can be used  to study how the spinodal density varies with interaction  parameters and with the persistence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  ∗ ∗ ∗  The authors acknowledge insightful discussions with  Alexander Grosberg, Jean-Francois Joanny and Marius  Bothe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  YIL acknowledges support from Royal Society  grant (RP\\R1\\180165).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  ´EF acknowledges support from  the Luxembourg National Research Fund (FNR), grant  reference 14389168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Work funded in part by the Euro-  pean Research Council under the EU’s Horizon 2020 Pro-  gramme (Grant number 740269).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  RG-M acknowledges  support from a St John’s College Research Fellowship,  University of Cambridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  REFERENCES  [1] Marchetti M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Joanny J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Ramaswamy S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Liv-  erpool T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Prost J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Rao M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Simha R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 85 (2013) 1143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [2] O’Byrne J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Kafri Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Tailleur J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and van Wijland  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 4 (2022) 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [3] Fodor E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Jack R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Cates M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Matter Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 13 (2022) 215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [4] Elgeti J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Winkler R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Gompper G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 78 (2015) 056601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [5] Weber C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Zwicker D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', J¨ulicher F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Lee C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=',  Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 82 (2019) 064601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [6] Tailleur J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Cates M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 100 (2008)  218103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [7] Fily Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Marchetti M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 108  (2012) 235702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [8] Fodor ´E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Nardini C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Cates M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Tailleur J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=',  Visco P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and van Wijland F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 117  (2016) 038103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [9] Martin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', O’Byrne J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Cates M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Fodor ´E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=',  Nardini C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Tailleur J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and van W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  E, 103 (2021) 032607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  p-6  Towards a liquid-state theory for active matter  [10] Garcia-Millan R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Pruessner G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' :  Theory Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 2021 (2021) 063203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [11] Pruessner G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Garcia-Millan R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Field theories  of active particle systems and their entropy production  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [12] Nardini C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Fodor E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Tjhung E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', van Wijland F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=',  Tailleur J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Cates M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' X, 7 (2017)  021007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [13] J¨ulicher F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Grill S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Salbreux G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 81 (2018) 076601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [14] Tjhung E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Nardini C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Cates M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Physical Re-  view X, 8 (2018) 031080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [15] Cates M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Tailleur J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Matter Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 6 (2015) 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [16] Stenhammar J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Tiribocchi A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Allen R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Maren-  duzzo D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Cates M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 111 (2013)  145702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [17] Takatori S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Yan W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Brady J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 113 (2014) 028103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [18] Yang X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Manning M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Marchetti M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Soft  Matter, 10 (2014) 6477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [19] Solon A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Fily Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Baskaran A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Cates M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=',  Kafri Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Kardar M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Tailleur J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Nature Physics,  11 (2015) 673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [20] Tociu L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Fodor ´E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Nemoto T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Vaikuntanathan  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Physical Review X, 9 (2019) 041026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [21] Fodor E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Nemoto T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Vaikuntanathan S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', New  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 22 (2020) 013052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [22] Arnoulx de Pirey T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Lozano G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and van Wijland  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 123 (2019) 260602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [23] Arnoulx de Pirey T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Manacorda A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', van Wijland  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Zamponi F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 155 (2021) 174106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [24] Kardar M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Statistical physics of particles (Cambridge  University Press) 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [25] Hansen J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and McDonald I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Theory of simple  liquids: with applications to soft matter (Academic press)  2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [26] Grosberg A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Joanny J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' E, 92  (2015) 032118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [27] Winkler R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Wysocki A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Gompper G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Soft  matter, 11 (2015) 6680.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [28] Falasco G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Baldovin F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Kroy K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Baiesi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=',  New Journal of Physics, 18 (2016) 093043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [29] Risken H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Fokker-planck equation (Springer) 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [30] Ilker E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Joanny J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Research, 2  (2020) 023200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [31] Farage T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Krinninger P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Brader J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=',  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' E, 91 (2015) 042310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [32] Takatori S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Brady J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' E, 91  (2015) 032117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [33] Dean D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' A, 29 (1996) L613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  p-7  epl draft  Supplementary material:  “Towards a liquid-state theory for active matter”  Yuting Irene Li1, Rosalba Garcia-Millan1,3, Michael E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Cates1 and ´Etienne  Fodor2  1 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road,  Cambridge CB3 0WA, UK  2 Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxem-  bourg, Luxembourg  3 St John’s College, University of Cambridge, Cambridge CB2 1TP, UK  PACS 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='Ln – Nonequilibrium and irreversible thermodynamics  Abstract – Supplementary material to “Towards a liquid-state theory for active matter”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Overdamped passive Brownian particles: Quasistatic ansatz for density cor-  relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' –  In this section, we demonstrate that, for overdamped passive Brownian par-  ticles (PBPs), one recovers the virial expansion of the free energy when using the quasistatic  ansatz p2(x1, x2, t) = p1(x1, t)p1(x2, t)pss(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We consider the following equations of  motion for N PBPs with pair-interaction potential Ψ:  ˙xi = −∂xiU +  √  2TΛi,  U =  N  �  i=1  N  �  j<i  Ψ(|xi − xj|),  (S1)  where we have set the mobility to unity, so that T is the diffusion constant, and Λi is a  Gaussian white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Similar to the main text, we write down the corresponding hierarchy  for one- and two-particle densities, assuming that the density is sufficiently low to ignore  contributions from three-particle density:  ∂tp1(x1, t) = T∇2  x1p1(x1, t) + ∇x1 ·  �  p2(x1, x2, t)∇x1Ψ(r)dx2,  ∂tp2(x1, x2, t) = T  �  ∇2  x1 + ∇2  x2  �  p2(x1, x2, t) +  �  i=1,2  ∇xi · (p2(x1, x2, t)∇xiΨ(r)),  (S2)  where r = |x1 −x2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' In the quasistatic limit, where the interaction time is much faster than  the free particle travel time, it is reasonable to suppose that every pair of particles reach  steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This inspires the following ansatz to close the density hierarchy [1,2]:  p2(x1, x2, t) = p1(x1, t) p1(x2, t) exp(−Ψ(r)/T),  (S3)  where we have used that the Boltzmann distribution is the the steady-state distribution of  two particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Substituting this ansatz into the p1 equation, we have  ∂tp1(x1, t) = T∇2  x1p1(x1, t) + ∇x1 ·  �  p1(x1, t)  �  p1(x2, t)e−Ψ(r)/T ∇x1Ψ(r)dx2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S4)  p-1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='12155v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='soft]  28 Jan 2023  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Notice that e−Ψ(r)/T ∇x1Ψ(r) = −T∇x1f(r), where f(r) = exp(−Ψ(r)/T) − 1 is usually  called the Mayer-f function [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Changing variables from x2 to r = x1 −x2 in the integrand  of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S4), and integrating by parts, we get  1  T ∂tp1(x1, t) = ∇2  x1p1(x1, t) − ∇x1 ·  �  p1(x1, t)  �  f(r)∇x1p1(x1 − r, t)dr  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S5)  Thus far, we have closed the density hierarchy and obtained a mean-field equation for the  one-particle density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' In the following, we seek to connect eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S5) with existing knowledge  of equilibrium physics of interacting gases, in particular, the virial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Virial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  We assume that the density distribution ρ does not vary dramatically  over the length-scale of the pair-potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Therefore, expanding p1(x1−r) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S5) around  x1 yields  ∂tp1 = ∇ · (p1∇µ),  µ = T(log p1 + 2Bp1),  B = −1  2  �  f(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S6)  Here B(T) is the second virial coefficient in equilibrium statistical physics [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The stationary  solution minimises the following free energy:  F = T  � �  p1 log p1 + Bp2  1  �  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S7)  The lack of gradient terms in the free energy is a result of neglecting higher-order terms in  the expansion of p1(x1 − r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Assuming that the stationary state is uniform with density  ρ = N/V , we obtain  F(N, V, T) = TN log(N/V )  �  ��  �  ideal gas  +  TBN 2/V  �  ��  �  virial correction  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S8)  This is exactly the equilibrium virial expansion to second order [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  All-gradient expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  To explore the effects of higher-gradient terms, we expand  p1(x − r) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S5) to all orders in gradient:  p1(x − r) = e−r·∇xp1(x),  (S9)  where e−r·∇x = �  n  (−r·∇x)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Substituting into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S5), we obtain  1  T ∂tp1 = ∇2p1 + 2∇ ·  �  p1∇  � ¯Bp1  ��  ,  ¯B = −1  2  �  f(r)e−r·∇xdr,  (S10)  where the second virial coefficient has the same form as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S6) but ¯B is now an operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Proceeding as in the previous section, the chemical potential µ has the same form as eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  In Fourier space, ¯Bk =  �  eik·x ¯Bdx is the Fourier transform of the Meyer-f function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Note  that ¯Bk=0 = B, since we only captured the zeroth mode when truncating the gradient  expansion at the lowest order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' The corresponding free energy is  ¯F = T  �  p1 log p1dx  �  ��  �  ideal gas  + T  �  p1,k ¯Bk p1,−k  dk  (2π)d  �  ��  �  interactions  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S11)  Overall, the gradient expansion is a natural extension of the equilibrium virial expansion  for non-uniform densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We note that one cannot trust the high-k behaviour of B(k), as  it is sensitive to the details of the potential at close range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' However, since we are always  interested in length-scales much larger than the range of the potential (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  size of the  individual particles), we do not need to be concerned with such a high-k behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  p-2  Supplementary material: “Towards a liquid-state theory for active matter”  Steady state of AOUPs: Small-τ expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' –  In this section, we derive the sta-  tionary probability as an expansion at small τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' As opposed to [4], where the expansion was  performed in the (x, ˙x) space, we consider here expansion in (x, v) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This is necessary  as we need the steady-state two-particle distribution in (x, v) space in the quasistatic ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  We start from the Fokker-Planck equation for two particles in one spatial dimension:  ∂tP =  �  i=1,2  LiP,  Li = D  τ 2 ∂2  vi +  �1  τ ∂vi − ∂xi  �  vi + ∂xi(∂xiΨ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S12)  Scaling units as v → v  �  τ/D, x → x/  √  τD, t → t/τ, and Ψ → Ψ/D, we obtain  Li = Li,0 + √τLi,1 + τLi,2,  (S13)  where  Li,0 = ∂2  vi + ∂vivi,  Li,1 = −∂xivi,  Li,2 = ∂xi(∂xiΨ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S14)  We assume the following τ-expansion for the stationary distribution:  p2,ss = e− 1  2(v2  1+v2  2)−Ψ  �  1 +  �  n  (√τ)nAi({xi, vi})  �  ,  (S15)  where Ai is the i-th order term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Solving order by order, we get  p2,ss ∝ exp  �  −Ψ(r) − 1  2  �  v2  1 + v2  2  ��  ×  �  1 + √τ(v1 − v2)Ψ′ + τ  �(v1 − v2)2  2  ((Ψ′)2 − Ψ′′) − 2(Ψ′)2 + 3Ψ′′  �  + o(τ)  �  ,  (S16)  where Ψ′ = dΨ/dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' This result is used to derive the quasistatic approximation in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (12)  of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  The interaction matrix in Fourier-Hermite basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' –  In this section, we show the  details of the change of basis to the Fourier-Hermite basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Recall that the perturbation  density field ε can be decomposed in the Fourier-Hermite basis as  ε(x, v) =  �  n  �  εkn(t)ψnk(x, v)dk,  ψnk = e−ik(x+αv)Un(v − 2iαk),  (S17)  where the Hermite function Un is defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (22) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Substituting the  decomposition into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (19) in the main text, we get  �  n  �  ( ˙εkn + εknλnk)ψnk(x, v)dk = −τρ  �  n  �  ikεknU0(v)dk  �  ψnk(x, v − w)dw  ×  �  2La,kw + 2Lb,k(−ik) + Lc,k(−ik)w2�  ,  (S18)  where  La,k = √τ  � ∞  0  dr (Ψ′)2 e−Ψeirk,  Lb,k = −  � ∞  0  drf0(r)eirk,  f0(s) =  � ∞  s  dr Ψ′e−Ψ �  1 − τ2(Ψ′)2 + τ3Ψ′′�  ,  Lc,k = −  � ∞  0  dr f1(r)eirk,  f1(s) = τ  � ∞  s  drΨ′((Ψ′)2 − Ψ′′)e−Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S19)  p-3  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  We then multiply both sides of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S18) by the dual basis ¯ψmk′, as defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (24) of  the main text, and integrate over the (x, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' By orthogonality of the basis, as outlined in  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (25) of the main text, we have  ˙εkm = −λkmεkm +  �  n  �  M (1)  kmn + M (2)  kmn + M (3)  kmn  �  εkn,  (S20)  where  M (1)  kmn = 2τρLa,k(−ik)  �  dv U0(v) ¯Um(v − 2iαk)  �  dw w eiαkwUn(v − w − 2iαk),  M (2)  kmn = 2τρLb,k(−ik)2  �  dv U0(v) ¯Um(v − 2iαk)  �  dw eiαkwUn(v − w − 2iαk),  M (3)  kmn = τρLc,k(−ik)2  �  dv U0(v) ¯Um(v − 2iαk)  �  dw w2eiαkwUn(v − w − 2iαk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S21)  We first perform all the w-integrals:  In  0 (v, k) ≡  �  dw eikαwUn(v − w − 2iαk) = e  3  2 α2k2+iαkv(−iαk)n  In  1 (v, k) ≡  �  dw  √  2π w eiαkwUn(v − w − 2iαk) = −e  3  2 α2k2+iαkv(−iαk)n−1 �  iαkv + n + α2k2�  In  2 (v, k) ≡  �  dw  √  2π w2eiαkwUn(v − w − 2iαk)  = e  3  2 α2k2+iαkv(−iαk)n−2 �  n(n − 1) − α2k2(1 + (v − iαk)2) + 2inαk(v − iαk)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S22)  Performing next the v-integrals, we find:  M (1)  kmn = 2τρLa,k(−ik)  �  dv  √  2π e− 1  2 v2Hm(v − 2iαk)In  1 (v, k)  = 2τρLa,k  (−iαk)n+m  αm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  eα2k2(m − n),  M (2)  kmn = −2τρLb,kk2  �  dv  √  2π e− 1  2 v2Hm(v − 2iαk)In  0 (v, k)  = −2τρLb,kk2 (−iαk)n+m  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  eα2k2,  M (3)  kmn = τρLc,k(−ik)2  �  dv  √  2π e− 1  2 v2Hm(v − 2iαk)In  2 (v, k)  = τρLc,k  (−iαk)n+m  α2m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  eα2k2 �  m(m − 1) + n(n − 1) − 2mn − 2α2k2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S23)  If the overall growth rate matrix Mkmn = −λkmδmn + M (1)  kmn + M (2)  kmn + M (3)  kmn has no  positive eigenvalue, the uniform solution is stable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' otherwise the system undergoes spinodal  decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Hence, all we need is the sign of the largest eigenvalue of the M matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Despite the appearance of M as an infinite-dimensional matrix, it is in fact possible to  find another set of basis where M is block diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Indeed, {M (1), M (2), M (3)} are all  linear combinations of the following matrices:  (−iαk)m+n  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  , (−iαk)m+n  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  m, (−iαk)m+n  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  n, (−iαk)m+n  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  m(m − 1), (−iαk)m+n  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  n(n − 1),  (S24)  p-4  Supplementary material: “Towards a liquid-state theory for active matter”  so that they can be written as  M (1) = 2τρLa,k  α  eα2k2 (u1v⊺  0 − u0v⊺  1) ,  M (2) = −2τρLb,kk2eα2k2u0v⊺  0,  M (3) = τρLc,k  α2  eα2k2 �  u2v⊺  0 + u0v⊺  2 − 2u1v⊺  1 − 2α2k2u0v⊺  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S25)  where ⊺ denotes matrix transpose, and  (u0)m = (−iαk)m  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  ,  (u1)m = (−iαk)m  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  m,  (u2)m = (−iαk)m  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  m(m − 1),  (v0)n = (−iαk)n,  (v1)n = (−iαk)nn,  (v2)n = (−iαk)nn(n − 1),  (S26)  Observe that when acting on an arbitrary vector y, uv⊺y = (v⊺y)u, meaning that a matrix of  this form projects all vectors onto the direction of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We introduce ∆ = M (1) +M (2) +M (3)  in this section, and, grouping the terms, we have  e−(αk)2∆ = h0u0v⊺  0 + h10(u1v⊺  0 − u0v⊺  1) + h11u1v⊺  1 + h2(u2v⊺  0 + u0v⊺  2),  (S27)  where  h0 = −2k2τρ (Lb,k + Lc,k) ,  h10 = 2τρLa,k/α,  h11 = 2τρLc,k/α2,  h2 = τρLc,k/α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S28)  We now consider the following spanning set: (v0, v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='), where (vj)n = (−iαk)nPn,j  (recall Pn,j = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='/(n − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' is the permutation coefficient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' One can easily check that it is  a spanning set by spotting that the first j elements of vj are zeros, and hence no vector  in this set can be written as a linear combination of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Furthermore, let us introduce  (q0, q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='.) as the dual basis of {vj}, such that v⊺  i qj = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We proceed to write ∆ in the  new basis: ˜∆ij ≡ v⊺  i ∆ qj, where  ˜∆ = eα2k2  �  �  �  �  �  �  �  �  �  �  �  h0v⊺  0u0 + h10v⊺  0u1 + h2v⊺  0u2  −h10v⊺  0u0 + h11v⊺  0u1  h2v⊺  0u0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  h0v⊺  1u0 + h10v⊺  1u1 + h2v⊺  1u2  −h10v⊺  1u0 + h11v⊺  1u1  h2v⊺  1u0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  h0v⊺  2u0 + h10v⊺  2u1 + h2v⊺  2u2  −h10v⊺  2u0 + h11v⊺  2u1  h2v⊺  2u0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  0  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  �  �  �  �  �  �  �  �  �  �  �  ,  (S29)  which is nonzero only in the top 3 × 3 due to the orthogonality relation between v’s and q’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  p-5  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Computing the inner products and setting ˜k = αk for convenience, we have  v⊺  0u0 =  �  n  (−˜k2)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  = e−˜k2,  v⊺  1u0 = v⊺  0u1 =  �  n  (−˜k2)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  n = −˜k2 �  n  (−˜k2)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  = −˜k2e−˜k2,  v⊺  2u0 = v⊺  0u2 =  �  n  (−˜k2)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  n(n − 1) = ˜k4e−˜k2,  v⊺  1u1 =  �  n  (−˜k2)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [n(n − 1) + n] = (˜k4 − ˜k2)e−˜k2,  v⊺  1u2 = v⊺  2u1 =  �  n  (−˜k2)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [n(n − 1)(n − 2) + 2n(n − 1)] = (−˜k6 + 2˜k4)e−˜k2,  v⊺  2u2 =  �  n  (−˜k2)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [n(n − 1)(n − 2)(n − 3) + 4n(n − 1)(n − 2) + 2n(n − 1)]  = (˜k8 − 4˜k6 + 2˜k4)e−˜k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S30)  The resulting 3 × 3 matrix appearing in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' (S29) can then be written as  �  �  �  �  �  h0 − h10˜k2 + h2˜k4  −h10 − h11˜k2  h2  −h0˜k2 + h10  �  ˜k4 − ˜k2�  + h2  �  −˜k6 + 2˜k4�  h10˜k2 + h11  �  ˜k4 − ˜k2�  −h2˜k2  h0˜k4 + h10  �  −˜k6 + 2˜k4�  + h2  �  ˜k8 − 4˜k6 + 2˜k4�  −h10˜k4 + h11  �  −˜k6 + 2˜k4�  h2˜k4  �  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S31)  To compute the eigenvalues of the full matrix M = D + ∆, where Dkmn = (−n − ˜k2)δmn,  we also need to write D in the new basis:  ˜Djl = v⊺  j D ql = −δj+1,l − (j − ˜k2)δjl,  (S32)  where we have used n(vj)n = (vj+1 + jvj)n, or explicitly  ˜D =  �  �  �  �  �  �  �  �  −˜k2  −1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  0  −1 − ˜k2  −1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  0  0  −2 − ˜k2  −1  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  �  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S33)  Notice that ˜D is upper triangular and therefore the eigenvalues are simply the diagonal  elements −n − ˜k2, which are the same as for D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Having written both the diagonal and non-diagonal parts in the new basis, we proceed  to solve for the eigenvalues λ of �  M = ˜D + ˜∆, satisfying det  �  �  M − λI  �  = 0, where I is the  identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Observe that �  M is of the following form,  �  M =  �  �X  Y  0  Z  �  � ,  (S34)  where X is a 3 × 3 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' One can prove that det  �  �  M − λI  �  = det(X − λI) det(Z − λI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  So we only need to solve for the eigenvalues of X and Z separately, but we already know  p-6  Supplementary material: “Towards a liquid-state theory for active matter”  the eigenvalues of Z as it is upper-triangular with diagonal elements (−n − ˜k2) for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Hence what remains are the eigenvalues of X, which can be solved numerically easily as it  is only 3 × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Overall, we have reduced the problem of finding the maximum eigenvalue of  an infinite-dimensional matrix to diagonalising the 3 × 3 matrix ˜∆ + ˜D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' We then compute  the largest eigenvalue as  λmax = −k2  �  1 + τ 1 + 2(B2(k) − 2A1(k)2 + C2(k)) − 4k2B0(k)(B2(k) + C2(k))  1 − 2B0(k)k2  �  + o(τ),  (S35)  where  B0(k) = Lb,k  ��  τ=0,  B2(k) = (d/dτ)Lb,k,  A1(k) = (1/√τ)La,k,  C2(k) = (1/τ)Lc,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  (S36)  REFERENCES  [1] Grosberg A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Joanny J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' E, 92 (2015) 032118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [2] Ilker E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and Joanny J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Research, 2 (2020) 023200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [3] Kardar M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Statistical physics of particles (Cambridge University Press) 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  [4] Fodor ´E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Nardini C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Cates M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Tailleur J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Visco P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' and van Wijland F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content=', 117 (2016) 038103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}
+page_content='  p-7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFLT4oBgHgl3EQfui8A/content/2301.12155v1.pdf'}