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finite temperature instantons calorons have a rich structure if one allows the polyakov loop xmath1 in the periodic gauge xmath2 to be non trivial at spatial infinity specifying the holonomy it implies the spontaneous breakdown of gauge symmetry for a charge one xmath3 caloron the location of the xmath4 constituent monopoles can be identified through i points where two eigenvalues of the polyakov loop coincide which is where the xmath5 symmetry is partially restored to xmath6 ii the centers of mass of the spherical lumps iii the dirac monopoles or rather dyons due to self duality as the sources of the abelian field lines extrapolated back to the cores if well separated and localised all these coincide xcite herewe study the case of two constituents coming close together for xmath7 with an example for xmath0 the eigenvalues of xmath8 can be ordered by a constant gauge transformation xmath9 3 mm ww 3mm1 nn111 with xmath10 the constituent monopoles have masses xmath11 where xmath12 using the classical scale invariance to put the extent of the euclidean time direction to one xmath13 in the same way we can bring xmath14 to this form by a local gauge function xmath15 we note that xmath16 unique up to a residual abelian gauge rotation and xmath17 will be smooth except where two or more eigenvalues coincide the ordering shows there are xmath4 different types of singularities called defects xcite for each of the neighbouring eigenvalues to coincide the first xmath18 are associated with the basic monopoles as part of the inequivalent xmath19 subgroups related to the generators of the cartan subgroup the xmath20 defect arises when the first and the last eigenvalue still neighbours on the circle coincide its magnetic charge ensures charge neutrality of the caloron the special status xcite of this defect also follows from the so called taubes winding xcite supporting the non zero topological charge xcite to analyse the lump structure when two constituents coincide we recall the simple formula for the xmath3 action density xcite 6mmf2x22 6mmmrmym ym1 0rm1 cmsm smcm with xmath21 the center of mass location of the xmath22 constituent monopole we defined xmath23 xmath24 xmath25 as well as xmath26 xmath27 we are interested in the case where the problem of two coinciding constituents in xmath3 is mapped to the xmath28 caloron for thiswe restrict to the case where xmath29 for some xmath30 which for xmath0 is always the case when two constituents coincide since now xmath31 one easily verifies that xmath32 describing a single constituent monopole with properly combined mass reducing eq 2 to the action density for the xmath28 caloron with xmath33 constituents the topological charge can be reduced to surface integrals near the singularities with the use of xmath34 where xmath35 if one assumes all defects are pointlike this can be used to show that for each of the xmath4 types the net number of defects has to equal the topological charge the type being selected by the branch of the logarithm associated with the xmath4 elements in the center xcite one might expect the defects to merge when the constituent monopoles do a triple degeneracy of eigenvalues for xmath0 implies the polyakov loop takes a value in the center yet this can be shown not to occur for the xmath0 caloron with unequal masses we therefore seem to have at least one more defect than the number of constituents when xmath36 we will study in detail a generic example in xmath0 with xmath37 we denote by xmath38 the position associated with the xmath22 constituent where two eigenvalues of the polyakov loop coincide in the gauge where xmath39 see eq 1 we established numerically xcite that p1pz1ei3 ei3e2i3 p2pz2e2i1 ei1ei1 p3pz3ei2 e2i2ei2this is for any choice of holonomy and constituent locations with the proviso they are well separated ie their cores do not overlap in which case to a good approximation xmath40 herewe take xmath41 xmath42 and xmath43 the limit of coinciding constituents is achieved by xmath44 with this geometryit is simplest to follow for changing xmath45 the location where two eigenvalues coincide in very good approximation as long as the first two constituents remain well separated from the third constituent carrying the taubes winding xmath46 will be constant in xmath45 and the xmath0 gauge field xcite of the first two constituents will be constant in time in the periodic gauge thus xmath47 for xmath48 greatly simplifying the calculations when the cores of the two approaching constituents start to overlap xmath49 and xmath50 are no longer diagonal but still block diagonal mixing the lower xmath51 components at xmath52 they are diagonal again but xmath50 will be no longer in the fundamental weyl chamber a weyl reflection maps it back while for xmath53 a more general gauge rotation back to the cartan subgroup is required to do so see fig 1 at xmath52 each xmath54 and xmath55 lies on the dashed line which is a direct consequence of the reduction to an xmath19 caloron to illustrate this more clearly we give the expressions for xmath54 which we believe to hold for any non degenerate choice of the xmath56 when xmath57 p1pz1e2i2 e2i2e4i2 p2pz2ei2 e2i2ei2 p3pz3ei2 e2i2ei2these can be factorised as xmath58 where xmath59 describes an overall xmath60 factor in terms of xmath61 xmath62 and xmath63the xmath19 embedding in xmath0 becomes obvious it leads for xmath64 to the trivial and for xmath65 to the non trivial element of the center of xmath19 appropriate for the latter carrying the taubes winding on the other hand xmath66 corresponds to xmath67 which for the xmath19 caloron is not related to coinciding eigenvalues for xmath44 fig 2 shows that xmath68 gets stuck at a finite distance 0131419 from xmath69 the xmath19 embedding determines the caloron solution for xmath70 with constituent locations xmath71 and xmath72 and masses xmath73 and xmath74 the best proof for the spurious nature of the defect is to calculate its location purely in terms of this xmath19 caloron by demanding the xmath19 polyakov loop to equal xmath75 for this we can use the analytic expression xcite of the xmath19 polyakov loop along the xmath76axis the location of the spurious defect xmath77 is found by solving xmath78 for our example xmath79 indeed verifies this equation with the xmath19 embedded result at hand we find that only for xmath80 the defects merge to form a triple degeneracy using xmath81 this is so for coinciding constituent monopoles of equal mass for unequal massesthe defect is always spurious but it tends to stay within reach of the non abelian core of the coinciding constituent monopoles except when the mass difference approaches its extremal values xmath82 see fig 2 bottom at these extremal valuesone of the xmath0 constituents becomes massless and delocalised which we excluded for xmath53 however the limit xmath44 is singular due to the global decomposition into xmath83 at xmath52 gauge rotations xmath84 in the global xmath19 subgroup do not affect xmath59 and therefore any xmath85 gives rise to the same accidental degeneracy in particular solving xmath86 corresponding to the weyl reflection xmath87 yields xmath88 for xmath89 isolated point in fig 2 top indeed xmath90 traces out a nearly spherical shell where two eigenvalues of xmath91 coincide note that for xmath80 this shell collapse to a single point xmath92 a perturbation tends to remove this accidental degeneracy abelian projected monopoles are not always what they seem to be even though required by topology topology can not be localised no matter how tempting this may seem for smooth fields i am grateful to andreas wipf for his provocative question that led to this work i thank jan smit and especially chris ford for discussions 9 t c kraan and p van baal nucl b 533 1998 627 hep th9805168 p van baal in lattice fermions and structure of the vacuum eds vmitrjushkin and g schierholz kluwer dordrecht 2000 p 269 hep th9912035 c ford t tok and a wipf nucl b 548 1999 585 hep th9809209 phys b 456 1999 155 hep th9811248 t c kraan and p van baal phys b 435 1998 389 hep th9806034 m n chernodub t c kraan and p van baal nucl 83 2000 556 hep lat9907001 c taubes in progress in gauge field theory eds gt hooft ea plenum press new york 1984 p563 mgarca prez agonzlezarroyo a montero and p van baal jhep 9906 1999 001 hep lat9903022
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we analyse what happens with two merging constituent monopoles for the xmath0 caloron identified through degenerate eigenvalues the singularities or defects of the abelian projection of the polyakov loop it follows that there are defects that are not directly related to the actual constituent monopoles 1 cm
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introduction
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example
resolution
lesson
acknowledgements
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the first two exact solution of einstein s field equations were obtained by schwarzschild 1 soon after einstein introduced general relativity gr the first solution describes the geometry of the space time exterior to a prefect fluid sphere in hydrostatic equilibrium while the other known as interior schwarzschild solution corresponds to the interior geometry of a fluid sphere of constant homogeneous energy density xmath0 the importance of these two solutions in gr is well known the exterior solution at a given point depends only upon the total mass of the gravitating body and the radial distance as measured from the centre of the spherical symmetry and not upon the type of the density distribution considered inside the mass however we will focus on this point of crucial importance later on in the present paper on the other hand the interior schwarzschild solution provides two very important features towards obtaining configurations in hydrostatic equilibrium compatible with gr namely i it gives an absolute upper limit on compaction parameter xmath1 mass to size ratio of the entire configuration in geometrized units xmath2 for any static and spherical solution provided the density decreases monotonically outwards from the centre in hydrostatic equilibrium 2 and ii for an assigned value of the compaction parameter xmath3 the minimum central pressure xmath4 corresponds to the homogeneous density solution see eg 3 regarding these conditions it should be noted that the condition i tells us that the values higher than the limiting maximum value of xmath5 can not be attained by any static solution but what kinds of density variations are possible for a mass to be in the state of hydrostatic equilibrium the answer to this important question could be provided by an appropriate analysis of the condition ii and the necessary conditions put forward by exterior schwarzschild solution despite the non linear differential equations various exact solutions for static and spherically symmetric metric are available in the literature 4 tolman 5 obtained five different types of exact solutions for static cases namely type iii which corresponds to the constant density solution obtained earlier by schwarzschild 1 type iv type v type vi and type vii the solution independently obtained by adler 6 adams and cohen 7 and kuchowicz 8 buchdahl s solution 9 for vanishing surface density the gaseous model the solution obtained by vaidya and tikekar 10 which is also obtained independently by durgapal and bannerji 11 the class of exact solutions discussed by durgapal 12 and also durgapal and fuloria 13 solution knutsen 14 examined various physical properties of the solutions mentioned in references 6 8 10 11 and 13 in great detail and found that these solutions correspond to nice physical properties and also remain stable against small radial pulsations upto certain values of xmath3 tolman s v and vi solutions are not considered physically viable as they correspond to singular solutions infinite values of central density that is the metric coefficient xmath6 at xmath7 and pressure for all permissible values of xmath3 except tolman s v and vi solutions all other solutions mentioned above are known as regular solutions finite positive density at the origin that is the metric coefficient xmath9 at xmath10 which decreases monotonically outwards which can be further divided into two categories i regular solutions corresponding to a vanishing density at the surface together with pressure like tolman s vii solution mehra 15 durgapal and rawat 16 and negi and durgapal 17 18 and buchdahl s gaseous solution 9 and ii regular solutions correspond to a non vanishing density at the surface like tolman s iii and iv solutions 5 and the solutions discussed in the ref6 8 and 10 13 respectively the stability analysis of tolman s vii solution with vanishing surface density has been undertaken in detail by negi and durgapal 17 18 and they have shown that this solution also corresponds to stable ultra compact objects ucos which are entities of physical interest this solution also shows nice physical properties such as pressure and energy density are positive and finite everywhere their respective gradients are negative the ratio of pressure to density and their respective gradients decrease outwards etc the other solution which falls in this category and shows nice physical properties is the buchdahl s solution 9 however knutsen 19 has shown that this solution turned out to be unstable under small radial pulsations all these solutions with finite as well as vanishing surface density discussed above in fact fulfill the criterion i that is the equilibrium configurations pertaining to these solutions always correspond to a value of compaction parameter xmath3 which is always less than the schwarzschild limit i e xmath11 but this condition alone does not provide a necessary condition for hydrostatic equilibrium nobody has discussed until now whether these solutions also fulfill the condition ii which is necessary to satisfy by any static and spherical configuration in the state of hydrostatic equilibrium recently by using the condition ii we have connected the compaction parameter xmath3 of any static and spherical configuration with the corresponding ratio of central pressure to central energy density xmath12 and worked out an important criterion which concludes that for a given value of xmath13 the maximum value of compaction parameter xmath14 should always correspond to the homogeneous density sphere 20 an examination of this criterion on some well known exact solutions and equations of state eoss indicated that this criterion in fact is fulfilled only by those configurations which correspond to a vanishing density at the surface together with pressure 20 or by the singular solutions with non vanishing surface density section 5 of the present study this result has motivated us to investigate in detail the various exact solutions available in the literature and disclose the reason s behind non fulfillment of the said criterion by various regular analytic solutions and eoss corresponding to a non vanishing finite density at the surface of the configuration in this connection in the present paper we have examined various exact solutions available in the literature in detail it is seen that tolman s vii solution with vanishing surface density 15 17 18 buchdahl s gaseous solution 9 and tolman s v and vi singular solutions pertain to a value of xmath3 which always turns out to be less than the value xmath15 of the homogeneous density sphere for all assigned values of xmath13 on the other hand the solutions having a finite non zero surface density that is the pressure vanishes at the finite surface density do not show consistency with the structure of the general relativity as they correspond to a value of xmath3 which turns out to be greater than xmath15 for all assigned values of xmath13 and thus violate the criterion obtained in 20 one may ask for example what could be in fact the reasons behind non fulfillment of the criterion obtained in 20 by various exact solutions corresponding to a finite non zero density at the surface we have been able to pin point which is discussed under section 3 of the present study the main reason namely the actual total mass xmath16 which appears in the exterior schwarzschild solution in fact can not be attained by the configurations corresponding to a regular density variation with non vanishing surface density the metric inside a static and spherically symmetric mass distribution corresponds to xmath17 where xmath18 and xmath19 are functions of xmath20 alone the resulting field equations for the metric governed by eq1 yield in the following form xmath21 1r2 8pi t11 8pi p elambda nur 1r2 1r2 8pi t22 8pi t33 8pi p elambda nu2 nu24nulambda4nu lambda2r endaligned where the primes represent differentiation with respect to xmath20 the speed of light xmath22 and the universal gravitation constant xmath23 are chosen as unity that is we are using the geometrized units xmath24 and xmath0 represent respectively the pressure and energy density inside the perfect fluid sphere related with the non vanishing components of the energy momentum tensor xmath25 0 1 2 and 3 respectively eqs2 4 represent second order coupled differential equations which can be reduced to the first order coupled differential equations by eliminating xmath26 from eq 3 with the help of eqs 2 and 4 in the well known form namely tov equations tolman 5 oppenheimer volkoff 21 governing hydrostatic equilibrium in general relativity xmath27rr 2mr xmath28 and xmath29 where prime denotes differentiation with respect to xmath20 and xmath30 is defined as the mass energy contained within the radius xmath31 that is xmath32 the equation connecting metric parameter xmath19 with xmath30 is given by xmath33 1 8pi rint0r er2 dr the three field equations or tov equations mentioned above involve four variables namely xmath34 and xmath19 thus in order to obtain a solution of these equations one more equation is needed which may be assumed as a relation between xmath24 and xmath0 eos or can be regarded as an algebraic relation connecting one of the four variables with the radial coordinate xmath20 or an algebraic relation between the parameters for obtaining an exact solution the later approach is employed notice that eq9 yields the metric coefficient xmath35 for the assumed energy density xmath0 as a function of radial distance xmath31 once the metric coefficient xmath35 or mass xmath30 is defined for assumed energy density by using eqs9 or 8 the pressure xmath24 and the metric coefficient xmath36 can be obtained by solving eqs5 and 6 respectively which yield two constants of integration these constants should be obtained from the following boundary conditions in order to have a proper solution of the field equations b1 in order to maintain hydrostatic equilibrium throughout the configuration the pressure must vanish at the surface of the configuration that is xmath37 where xmath38 is the radius of the configuration b2 the consequence of eq10 ensures the continuity of the metric parameter xmath39 belonging to the interior solution with the corresponding expression for well known exterior schwarzschild solution at the surface of the fluid configuration that is xmath40 where xmath41 is the total mass of the configuration however the exterior schwarzschild solution guarantees that xmath42 which means that the matching of the metric parameter xmath43 is also ensured at the surface of the configuration together with xmath39 that is xmath44 irrespective of the condition that the surface density xmath45 is vanishing with pressure or not that is xmath46 together with eq 10 or xmath47 where xmath48 is the compaction parameter of the configuration defined earlier and xmath16 is defined as eq 8 xmath49 thus the analytic solution for the fluid sphere can be explored in terms of the only free parameter xmath48 by normalizing the metric coefficient xmath43 yielding from eq11 at the surface of the configuration that is xmath50 at xmath51 after obtaining the integration constants by using eqs10 that is xmath52 at xmath53 and 11 xmath54 at xmath51 respectively however at this place we recall the well known property of the exterior schwarzschild solution which follows directly from the definition of the mass xmath55 appears in this solution namely at a given point outside the spherical distribution of mass xmath55 it depends only upon xmath55 and not upon the type of the density variation considered inside the radius xmath56 of this sphere it follows therefore that the dependence of mass xmath55 upon the type of the density distribution plays an important role in order to fulfill the requirement set up by exterior schwarzschild solution the relation xmath57 immediately tells us that for an assigned value of the compaction parameter xmath3 the mass xmath55 depends only upon the radius xmath56 of the configuration which may either depend upon the surface density or upon the central density or upon both of them depending upon the type of the density variation considered inside the mass generating sphere we argue that this dependence should occur in such a manner that the definition of mass xmath55 is not violated we infer this definition as the type independence property of the mass xmath55 which may be defined in this manner the mass xmath55 which appears in the exterior schwarzschild solution should either depend upon the surface density or upon the central density and in any case not upon both of them so that from an exterior observer s point of view the type of the density variation assigned for the mass should remain unidentified we may explain the type independence property of mass xmath55 mentioned above in the following manner the mass xmath55 is called the coordinate mass that is the mass as measured by some external observer and from this observer s point of view if we are measuring a sphere of mass xmath55 we can not know by any means the way in which the matter is distributed from the centre to the surface of this sphere that is if we are measuring xmath55 with the help of non vanishing surface density obviously by calculating the coordinate radius xmath56 from the expression connecting the surface density and the compaction parameter and by using the relation xmath58 we can not measure it by any means from the knowledge of the central density because if we can not know by any means the way in which the matter is distributed from the centre to the surface of the configuration then how can we know about the central density and this is possible only when there exist no relation connecting the mass xmath55 and the central density that is the mass xmath55 should be independent of the central density meaning thereby that the surface density should be independent of the central density for configurations corresponding to a non vanishing surface density however if we are measuring the mass xmath55 by using the expression for central density in the similar manner as in the previous case by calculating the radius xmath56 from the expression of central density and using the relation xmath59 we can not calculate it by any means from the knowledge of the surface density in view of the type independence property of the mass xmath55 and this is possible only when there exist no relation connecting the mass xmath55 and the surface density meaning thereby that the central density should be independent of the surface density from the above explanation of type independence property of mass xmath16 it is evident that the actual total mass xmath16 which appears in the exterior schwarzschild solution should either depend upon the surface density or depend upon the central density of the configuration and in any case not upon both of them however the dependence of mass xmath16 upon both of the densities surface as well as central is a common feature observed among all regular solutions having a non vanishing density at the surface of the configuration see for example eqs21 25 29 and 33 respectively belonging to the solutions of this category which are discussed under sub sections a d of section 5 of the present study thus it is evident that the surface density of such solutions is dependent upon the central density and vice versa that is the total mass xmath55 depends upon both of the densities meaning thereby that the type of the density distribution considered inside the sphere of mass xmath55 is known to an external observer which is the violation of the definition of mass xmath55 defined as the type independence property of mass xmath16 above such structures therefore do not correspond to the actual total mass xmath16 required by the exterior schwarzschild solution to ensure the condition of hydrostatic equilibrium this also explains the reason behind non fulfillment of the compatibility criterion by them which is discussed under section 5 of the present study however it is interesting to note here that there could exist only one solution in this regard for which the mass xmath16 depends upon both but the same value of surface and centre density and for regular density distribution the structure would be governed by the homogeneous constant density throughout the configuration that is the homogeneous density solution note that the requirement type independence of the mass would be obviously fulfilled by the regular structures corresponding to a vanishing density at the surface together with pressure because the mass xmath16 will depend only upon the central density surface density is always zero for these structures see for example eqs37 and 41 discussed under sub section e and f for buchdahl s gaseous model and tolman s vii solution having a vanishing density at the surface respectively furthermore the demand of type independence of mass xmath55 is also satisfied by the singular solutions having a non vanishing density at the surface because such structures correspond to an infinite value of central density and consequently the mass xmath16 will depend only upon the surface density see for example eqs46 and 50 discussed under sub section g and h for tolman s v and vi solutions respectively both types of these structures are also found to be consistent with the compatibility criterion as discussed under section 5 of the present study the discussion regarding various types of density distributions considered above is true for any single analytic solution or equation of state comprises the whole configuration at this place we are not intended to claim that the construction of a regular structure with non vanishing surface density is impossible it is quite possible provided we consider a two density structure in such a manner that the mass xmath16 of the configuration turns out to be independent of the central density so that the property type independence of the mass xmath16 is satisfied examples of such two density models are also available in the literature see eg ref 22 but in the different context however it should be noted here that the fulfillment of type independence condition by the mass xmath16 for any two density model will represent only a necessary condition for hydrostatic equilibrium unless the compatibility criterion 20 is satisfied by them which also assure a sufficient and necessary condition for any structure in hydrostatic equilibrium this issue is addressed in the next section of the present study the above discussion can be summarized in other words as although the exterior schwarzschild solution itself does not depend upon the type of the density distribution or eos considered inside a fluid sphere in the state of hydrostatic equilibrium however it puts the important condition that only two types of the density variations are possible inside the configuration in order to fulfill the condition of hydrostatic equilibrium 1 the surface density of the configuration should be independent of the central density and 2 the central density of the configuration should be independent of the surface density obviously the condition 1 will be satisfied by the configurations pertaining to an infinite value of the central density that is the singular solutions andor by the two density or multiple density distributions corresponding to a surface density which turns out to be independent of the central density because the regular configurations governed by a single exact solution or eos pertaining to this category are not possible whereas the condition 2 will be fulfilled by the configurations corresponding to a surface density which vanishes together with pressure the configurations in this category will include the density variation governed by a single exact solution or eos as well as the two density or multiple density distributions however the point to be emphasized here is that a two density distribution in any of the two categories mentioned here will fulfill only a necessary condition for hydrostatic equilibrium unless the compatibility criterion 20 is satisfied by them which also assure a necessary and sufficient condition for any structure in the state of hydrostatic equilibrium as mentioned above the criterion obtained in 20 can be summarized in the following manner for an assigned value of the ratio of central pressure to central energy density xmath60 the compaction parameter of homogeneous density distribution xmath14 should always be larger than or equal to the compaction parameter xmath61 of any static and spherical solution compatible with the structure of general relativity that is xmath62 in the light of eq 15 let us assign the same value xmath55 for the total mass corresponding to various static configurations in hydrostatic equilibrium if we denote the density of the homogeneous sphere by xmath63 we can write xmath64 where xmath65 denotes the radius of the homogeneous density sphere if xmath66 represents the radius of any other regular sphere for the same mass xmath55 the average density xmath67 of this configuration would correspond to xmath68 eq 15 indicates that xmath69 by the use of eqs 16 and 17 we find that xmath70 that is for an assign value of xmath13 the average energy density of any static spherical configuration xmath67 should always be less than or equal to the density xmath63 of the homogeneous density sphere for the same mass xmath55 although the regular configurations with finite non vanishing surface densities represented by a single density variation can not exist because for such configurations the necessary condition set up by exterior schwarzschild solution can not be satisfied however we can construct regular configurations composed of core envelope models corresponding to a finite central with vanishing and non vanishing surface densities such that the necessary conditions imposed by the schwarzschild s exterior solution at the surface of the configuration are appropriately satisfied however it should be noted that the necessary conditions satisfied by such core envelope models at the surface may not always turn out to be sufficient for describing the state of hydrostatic equilibrium because for an assigned value of xmath13 the average density of such configurations may not always turn out to be less than or equal to the density of the homogeneous density sphere for the same mass as indicated by eqs16 and 17 respectively it would depend upon the types of the density variations considered for the core and envelope regions and the the matching conditions at the core envelope boundary thus it follows that the criterion obtained in 20 is able to provide a necessary and sufficient condition for any regular configuration to be consistent with the state of hydrostatic equilibrium the future study of such core envelope models see for example the models described in 22 and 23 based upon the criterion obtained in 20 could be interesting regarding two density structures of neutron stars and other stellar objects compatible with the structure of gr we have considered the following exact solutions expressed in units of compaction parameter xmath71 mass to size ratio in geometrized units and radial coordinate measured in units of configuration size xmath72 for convenience the other parameters which will appear in these solutions are defined at the relevant places in these equations xmath24 andxmath0 represent respectively the pressure and energy density inside the configuration the surface density is denoted by xmath73 and the central pressure and central energy density are denoted by xmath74 and xmath75 respectively the regular exact solutions which pertain to a non vanishing value of the surface density are given under the sub sections a d while those correspond to a vanishing value of the surface density are described under the sub sections e and f respectively sub sections g and h represent the singular solutions having non vanishing values of the surface densities a tolmans iv solution xmath76 xmath77 by the use of eq20 we can obtain the relation connecting central and surface densities in the following form xmath78 eq21 shows that the surface density is dependent upon the central density and vice versa by using eqs 19 and 20 we obtain xmath79 this solution finds application it can be seen from eq 19 for the values of xmath80 b adler 6 adams and cohen 7 and kuchowicz 8 solution xmath81 xmath82 where xmath83 and xmath84 eq24 gives the relation connecting central and surface densities of the configuration in the following form xmath85 thus the surface density depends upon the central density and vice versa equations 23 and 24 give xmath86 it is seen from eq23 that this solution finds application for values of xmath87 c vaidya and tikekar 10 and durgapal and bannerji 11 solution xmath88 xmath89 where the variable xmath90 and the constants xmath91 and xmath92 are given by xmath93 and xmath94 by the use of eq28 we find that the surface and central densities are connected by the following relation xmath95 it is evident from eq29 that the surface density is dependent upon the central density and vice versa by the use of eqs27 and 28 we obtain xmath96 this solution finds application for the values of xmath97 as shown by eq27 d durgapal and fuloria 13 solution xmath98 bigr xmath99 where xmath92 is a constant and xmath100 the variables xmath101 and xmath102 are given by xmath103 xmath104 where the arbitrary constant xmath105 is given by xmath106 and xmath107 eq32 gives the relation connecting central and surface densities as xmath108 eq33 indicates that the surface density is dependent upon the central density and vice versa by the use of equations 31 and 32 we get xmath109 bigr where xmath110 and xmath111 are given by xmath112 xmath113 and xmath114 as seen from eq31 this solution is applicable for the values of xmath115 e buchdahl s gaseous solution 9 xmath116 xmath117 where xmath118 n 21 u and xmath119 0 leq z leq pi eq36 shows that the surface density vanishes together with pressure thus the central density will become independent of the surface density given by the equation xmath120 by using equations 35 and 36 we obtain xmath121 it is evident from eq35 that this solution is applicable for the values of xmath87 f tolman s vii solution with vanishing surface density xmath122 xmath123 where xmath75 is the central energy density given by xmath124 and xmath125 xmath126 xmath12712 na2 u5 3x xmath1283u12 xmath12912 xmath130 by using eqs 39 and 40 we get xmath131 where xmath132 is given by xmath133 it follows from eq40 that the surface density is always zero hence the central density is always independent of the surface density eq39 indicates that his solution is applicable for the values of xmath134 g tolman s v solution xmath135 and xmath136yq bigr 2n 1 n2y2 where xmath137 is given by xmath138 and xmath139 is defined as xmath140 eq44 shows that the central density is always infinite for xmath141 together with central pressure eq43 however their ratio xmath142 is finite at all points inside the configuration and at the centre yields in the following form xmath143 the consequence of the infinite central density is that the surface density will become independent of the central density given by the equation xmath144 it is evident from eq45 that this solution is applicable for a value of xmath145 i e for a value of xmath87 h tolman s vi solution xmath147 xmath148 eqs47 and 48 indicate that the central pressure and central density are always infinite however their ratio xmath142 is finite at all points inside the structure and at the centre reduces into the following form xmath149 and the surface density obviously independent of the central density would be given by the equation xmath150 where xmath139 is defined as xmath151 eq49 indicates that this solution is applicable for a value of xmath152 let us denote the compaction parameter of the homogeneous density configuration by xmath15 and for the exact solutions corresponding to the sub sections a d by xmath153 and xmath154 respectively the compaction parameters of the exact solutions described under sub section e and f are denoted by xmath155 and xmath156 respectively and those discussed under sub sections g and h are denoted by xmath157 and xmath158 respectively solving these analytic solutions for various assigned values of the ratio of central pressure to central energy density xmath159 we obtain the corresponding values of the compaction parameters as shown in table 1 and table 2 respectively it is seen that for each and every assigned value of xmath13 the values represented by xmath153 and xmath154 respectively table 1 turn out to be higher than xmath15 that is xmath153 and xmath160 while those represented by xmath161 and xmath158 respectively table 2 correspond to a value which always remains less than xmath15 that is xmath162 and xmath163 thus we conclude that the configurations defined by xmath153 and xmath154 respectively do not show compatibility with the structure of general relativity while those defined by xmath161 and xmath158 respectively show compatibility with the structure of general relativity however this type of characteristics that is the value of compaction parameter larger than the value of xmath15 for some or all assigned values of xmath13 can be seen for any regular exact solution having a finite non vanishing surface density because such exact solutions having finite central densities with non vanishing surface densities can not possess the actual mass xmath55 required to fulfill the boundary conditions at the surface on the other hand the value of compaction parameter for a regular solution with vanishing surface density and a singular solution with non vanishing surface density will always remain less than the value of xmath15 for all assigned values of xmath13 because such solutions naturally fulfill the definition of the actual mass xmath55 required for the hydrostatic equilibrium therefore it is evident that the findings based upon the compatibility criterion carried out in this section are fully consistent with the definition of the mass xmath55 defined as the type independence property under section 3 of the present study we have investigated the criterion obtained in the reference 20 which states for an assigned value of the ratio of central pressure to central energy density xmath165 the compaction parameter xmath166 of any static and spherically symmetric solution should always be less than or equal to the compaction parameter xmath15 of the homogeneous density distribution we conclude that this criterion is fully consistent with the reasoning discussed under section 3 which states that in order to fulfill the requirement set up by exterior schwarzschild solution that is to ensure the condition of hydrostatic equilibrium the total mass xmath55 of the configuration should depend either upon the surface density that is independent of the central density or upon the central density that is independent of the surface density and in any case not upon both of them an examination based upon this criterion show that among various exact solutions of the field equations available in the literature the regular solutions corresponding to a vanishing surface density together with pressure namely i tolman s vii solution with vanishing surface density and ii buchdahl s gaseous solution and the singular solutions with non vanishing surface density namely tolman s v and vi solutions are compatible with the structure of general relativity the only regular solution with finite non vanishing surface density which could exist in this regard is described by constant homogeneous density distribution this criterion provides a necessary and sufficient condition for any static spherical configuration to be compatible with the structure of general relativity and may be used to construct core envelope models of stellar objects like neutron stars with vanishing and non vanishing surface densities such that for an assigned value of central pressure to central density the average density of the configuration should always remain less than or equal to the density of the homogeneous sphere for the same mass this criterion could provide a convenient and reliable tool for testing equations of state eoss for dense nuclear matter and models of relativistic star clusters and may find application to investigate new analytic solutions and eoss r c tolman phys 55 364 1939 r j adlar j math phys 15 727 1974 r c adams and j m cohen astrophys 198 507 1975 b kuchowicz astrophys space science l13 33 1975 h a buchdahl astrophys j 147 310 1967 p c vaidya and r tikekar j astrophys 3 325 1982 m c durgapal and r bannerji phys d27 328 1983 erratum d28 2695 m c durgapal 15 2637 1982 m c durgapal and r s fuloria gen 17 671 1985 h knutsen astrophys space science 140 385 1988 h knutsen mon not soc 232 163 1988 h knutsen astrophys space science 162 315 1989 a l mehra j aust 6 153 1966 m c durgapal and p s rawat mon not 192 659 1980 p s negi and m c durgapal astrophys space science 245 97 1996 p s negi and m c durgapal gen 31 13 1999 h knutsen gen grav 20 317 1988 p s negi and m c durgapal gravitation cosmology 7 37 2001astro ph0312516 j r oppenheimer and g m volkoff phys rev 55 374 1939 p s negi a k pande and m c durgapal class quantum grav 6 1141 1989 p s negi a k pande and m c durgapal gen rel 22 735 1990 p s negi and m c durgapal gravitation cosmology 5 191 1999 p s negi and m c durgapal astron astrophys 353 641 2000
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we examine various well known exact solutions available in the literature to investigate the recent criterion obtained in ref 20 which should be fulfilled by any static and spherically symmetric solution in the state of hydrostatic equilibrium it is seen that this criterion is fulfilled only by i the regular solutions having a vanishing surface density together with the pressure and ii the singular solutions corresponding to a non vanishing density at the surface of the configuration on the other hand the regular solutions corresponding to a non vanishing surface density do not fulfill this criterion based upon this investigation we point out that the exterior schwarzschild solution itself provides necessary conditions for the types of the density distributions to be considered inside the mass in order to obtain exact solutions or equations of state compatible with the structure of general relativity the regular solutions with finite centre and non zero surface densities which do not fulfill the criterion 20 in fact can not meet the requirement of the actual mass set up by exterior schwarzschild solution the only regular solution which could be possible in this regard is represented by uniform homogeneous density distribution the criterion 20 provides a necessary and sufficient condition for any static and spherical configuration including core envelope models to be compatible with the structure of general relativity thus it may find application to construct the appropriate core envelope models of stellar objects like neutron stars and may be used to test various equations of state for dense nuclear matter and the models of relativistic stellar structures like star clusters pacs nos 0420jd 0440dg 9760jd exact solutions of einstein s field equations 25 in p s negi department of physics kumaun university nainital 263 002 india
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introduction
field equations and exact solutions
boundary conditions: the valid and invalid assumptions for mass distribution
criterion for static spherical configurations to be consistent with the structure of general relativity
examination of the compatibility criterion for various well known exact solutions available in the literature
results and conclusions
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a great deal of progress was recently achieved in our understanding of the multifragmentation phenomenon xcite when an exact analytical solution of a simplified version of the statistical multifragmentation model smm xcite was found in refs an invention of a new powerful mathematical method xcite the laplace fourier transform allowed us not only to solve this version of smm analytically for finite volumes xcite but to find the surface partition and surface entropy of large clusters for a variety of statistical ensembles xcite it was shown xcite that for finite volumes the analysis of the grand canonical partition gcp of the simplified smm is reduced to the analysis of the simple poles of the corresponding isobaric partition obtained as a laplace fourier transform of the gcp this method opens a principally new possibility to study the nuclear liquid gas phase transition directly from the partition of finite system and without taking its thermodynamic limit exactly solvable models with phase transitions play a special role in the statistical physics they are the benchmarks of our understanding of critical phenomena that occur in more complicated substances they are our theoretical laboratories where we can study the most fundamental problems of critical phenomena which can not be studied elsewhere note that these questions in principle can not be clarified either within the widely used mean filed approach or numerically despite this success the application of the exact solution xcite to the description of experimental data is limited because this solution corresponds to an infinite system volume therefore from a practical point of view it is necessary to extend the formalism for finite volumes such an extension is also necessary because despite a general success in the understanding the nuclear multifragmentation there is a lack of a systematic and rigorous theoretical approach to study the phase transition phenomena in finite systems for instance even the best formulation of the statistical mechanics and thermodynamics of finite systems by hill xcite is not rigorous while discussing the phase transitions exactly solvable models of phase transitions applied to finite systems may provide us with the first principle results unspoiled by the additional simplifying assumptions herewe present a finite volume extension of the smm to have a more realistic model for finite volumes we would like to account for the finite size and geometrical shape of the largest fragments when they are comparable with the system volume for this we will abandon the arbitrary size of largest fragment and consider the constrained smm csmm in which the largest fragment size is explicitly related to the volume xmath0 of the system a similar model but with the fixed size of the largest fragment was recently analyzed in ref xcite in this workwe will solve the csmm analytically at finite volumes using a new powerful method consider how the first order phase transition develops from the singularities of the smm isobaric partition xcite in thermodynamic limit study the finite volume analogs of phases and discuss the finite size effects for large fragments the system states in the smm are specified by the multiplicity sets xmath1 xmath2 of xmath3nucleon fragments the partition function of a single fragment with xmath3 nucleons is xcite xmath4 where xmath5 xmath6 is the total number of nucleons in the system xmath0 and xmath7 are respectively the volume and the temperature of the system xmath8 is the nucleon mass the first two factors on the right hand side rhs of the single fragment partition originate from the non relativistic thermal motion and the last factor xmath9 represents the intrinsic partition function of the xmath3nucleon fragment therefore the function xmath10 is a phase space density of the k nucleon fragment for nucleon we take xmath11 4 internal spin isospin states and for fragments with xmath12 we use the expression motivated by the liquid drop model see details in xmath13 with fragment free energy xmath14k sigma t k23 tau 32 tln k with xmath15 heremev is the bulk binding energy per nucleon xmath17 is the contribution of the excited states taken in the fermi gas approximation xmath18 mev xmath19 is the temperature dependent surface tension parameterized in the following relation xmath2054 with xmath21 mev and xmath22 mev xmath23 at xmath24 the last contribution in eq one involves the famous fisher s term with dimensionless parameter xmath25 the canonical partition function cpf of nuclear fragments in the smm has the following form xmath26nknk biggr textstyle deltaasumk knk in eq two the nuclear fragments are treated as point like objects however these fragments have non zero proper volumes and they should not overlap in the coordinate space in the excluded volume van der waals approximation this is achieved by substituting the total volume xmath0 in eq two by the free available volume xmath27 where xmath28 xmath29 xmath30 is the normal nuclear density therefore the corrected cpf becomes xmath31 the smm defined by eq two was studied numerically in refs this is a simplified version of the smm eg the symmetry and coulomb contributions are neglected however its investigation appears to be of principal importance for studies of the liquid gas phase transition the calculation of xmath32 is difficult due to the constraint xmath33 this difficulty can be partly avoided by evaluating the grand canonical partition gcp xmath34 where xmath35 denotes a chemical potential the calculation of xmath36 is still rather difficult the summation over xmath1 sets in xmath37 can not be performed analytically because of additional xmath6dependence in the free volume xmath38 and the restriction xmath39 this problem was resolved xcite by the laplace transformation method to the so called isobaric ensemble xcite in this workwe would like to consider a more strict constraint xmath40 where the size of the largest fragment xmath41 can not exceed the total volume of the system the parameter xmath42 is introduced for convenience the case xmath43 is also included in our treatment a similar restriction should be also applied to the upper limit of the product in all partitions xmath44 xmath32 and xmath45 introduced above how to deal with the real values of xmath46 see later then the model with this constraint the csmm can not be solved by the laplace transform method because the volume integrals can not be evaluated due to a complicated functional xmath0dependence however the csmm can be solved analytically with the help of the following identity xmath47 which is based on the fourier representation of the dirac xmath48function the representation four allows us to decouple the additional volume dependence and reduce it to the exponential one which can be dealt by the usual laplace transformation in the following sequence of steps xmath49 thetavprime nonumber hspace01cmint0inftyhspace02cmdvprime intlimitsinftyinfty d xi intlimitsinftyinfty fracd eta2 pi textstyle e i eta vprime xi lambda vprime vprime cal fxi lambda i eta textstyle e vprime cal fxi lambda i eta endaligned after changing the integration variable xmath50 the constraint of xmath51function has disappeared then all xmath52 were summed independently leading to the exponential function now the integration over xmath53 in eq five can be straightforwardly done resulting in xmath54 where the function xmath55 is defined as follows xmath56 endaligned as usual in order to find the gcp by the inverse laplace transformation it is necessary to study the structure of singularities of the isobaric partition seven the isobaric partition seven of the csmm is of course more complicated than its smm analog xcite because for finite volumes the structure of singularities in the csmm is much richer than in the smm and they match in the limit xmath57 only to see thislet us first make the inverse laplace transform xmath581 endaligned where the contour xmath59integral is reduced to the sum over the residues of all singular points xmath60 with xmath61 since this contour in the complex xmath59plane obeys the inequality xmath62 now both remaining integrations in eight can be done and the gcp becomes xmath631 ie the double integral in eight simply reduces to the substitution xmath64 in the sum over singularities this is a remarkable result which can be formulated as the following the simple poles in eight are defined by the equation xmath65 in contrast to the usual smm xcite the singularities xmath66 are i are volume dependent functions if xmath46 is not constant and ii they can have a non zero imaginary part but in this case there exist pairs of complex conjugate roots of ten because the gcp is real introducing the real xmath67 and imaginary xmath68 parts of xmath69 we can rewrite eq ten as a system of coupled transcendental equations xmath70 where we have introduced the set of the effective chemical potentials xmath71 with xmath72 and the reduced distributions xmath73 and xmath74 for convenience consider the real root xmath75 first for xmath76 the real root xmath77 exists for any xmath7 and xmath35 comparing xmath77 with the expression for vapor pressure of the analytical smm solution xciteshows that xmath78 is a constrained grand canonical pressure of the gas as usual for finite volumes the total mechanical pressure xcite as we will see in section v differs from xmath78 equation twelve shows that for xmath79 the inequality xmath80 never become the equality for all xmath3values simultaneously then from eq eleven one obtains xmath81 xmath82 where the second inequality thirteen immediately follows from the first one in other words the gas singularity is always the rightmost one this fact plays a decisive role in the thermodynamic limit xmath57 the interpretation of the complex roots xmath83 is less straightforward according to eq nine the gcp is a superposition of the states of different free energies xmath84 strictly speaking xmath84 has a meaning of the change of free energy but we will use the traditional term for it for xmath81 the free energies are complex therefore xmath85 is the density of free energy the real part of the free energy density xmath86 defines the significance of the state s contribution to the partition due to thirteen the largest contribution always comes from the gaseous state and has the smallest real part of free energy density as usual the states which do not have the smallest value of the real part of free energy i e xmath87 are thermodynamically metastable for infinite volumethey should not contribute unless they are infinitesimally close to xmath88 but for finite volumes their contribution to the gcp may be important as one sees from eleven and twelve the states of different free energies have different values of the effective chemical potential xmath89 which is not the case for infinite volume xcite where there exists a single value for the effective chemical potential thus for finite xmath0 the states which contribute to the gcp nine are not in a true chemical equilibrium the meaning of the imaginary part of the free energy density becomes clear from eleven and twelve as one can see from eleven the imaginary part xmath90 effectively changes the number of degrees of freedom of each xmath3nucleon fragment xmath91 contribution to the free energy density xmath87 it is clear that the change of the effective number of degrees of freedom can occur virtually only and if xmath83 state is accompanied by some kind of equilibration process both of these statements become clear if we recall that the statistical operator in statistical mechanics and the quantum mechanical convolution operator are related by the wick rotation xcite in other words the inverse temperature can be considered as an imaginary time therefore depending on the sign the quantity xmath92 that appears in the trigonometric functions of the equations eleven and twelve in front of the imaginary time xmath93 can be regarded as the inverse decay formation time xmath94 of the metastable state which corresponds to the pole xmath83 for more details see next sections as will be shown further for xmath95the inverse chemical potential can be considered as a characteristic equilibration time as well this interpretation of xmath94 naturally explains the thermodynamic metastability of all states except the gaseous one the metastable states can exist in the system only virtually because of their finite decay formation time whereas the gaseous state is stable because it has an infinite decay formation time for xmath96 mev and xmath97 the lhs straight line and rhs of eq twelve all dashed curves are shown as the function of dimensionless parameter xmath98 for the three values of the largest fragment size xmath46 the intersection point at xmath99 corresponds to a real root of eq ten each tangent point with the straight line generates two complex roots of ten width325height226 it is instructive to treat the effective chemical potential xmath100 as an independent variable instead of xmath35 in contrast to the infinite xmath0 where the upper limit xmath101 defines the liquid phase singularity of the isobaric partition and gives the pressure of a liquid phase xmath102 xcite for finite volumes and finite xmath46 the effective chemical potential can be complex with either sign for its real part and its value defines the number and position of the imaginary roots xmath103 in the complex plane positive and negative values of the effective chemical potential for finite systems were considered xcite within the fisher droplet model but to our knowledge its complex values have never been discussed from the definition of the effective chemical potential xmath104it is evident that its complex values for finite systems exist only because of the excluded volume interaction which is not taken into account in the fisher droplet model xcite as it is seen from fig 1 the rhs of eq twelve is the amplitude and frequency modulated sine like function of dimensionless parameter xmath105 therefore depending on xmath7 and xmath106 values there may exist no complex roots xmath107 a finite number of them or an infinite number of them in fig 1 we showed a special case which corresponds to exactly three roots of eq ten for each value of xmath46 the real root xmath108 and two complex conjugate roots xmath109 since the rhs of twelve is monotonously increasing function of xmath106 when the former is positive it is possible to map the xmath110 plane into regions of a fixed number of roots of eq ten each curve in divides the xmath110 plane into three parts for xmath106values below the curve there is only one real root gaseous phase for points on the curve there exist three roots and above the curve there are five or more roots of eq ten for constant values of xmath111the number of terms in the rhs of twelve does not depend on the volume and consequently in thermodynamic limit xmath57 only the farthest right simple pole in the complex xmath59plane survives out of a finite number of simple poles according to the inequality thirteen the real root xmath112 is the farthest right singularity of isobaric partition six however there is a possibility that the real parts of other roots xmath113 become infinitesimally close to xmath77 when there is an infinite number of terms which contribute to the gcp nine region of one real root of eq ten below the curve three complex roots at the curve and five and more roots above the curve for three values of xmath46 and the same parameters as in fig 1 width325height226 let us show now that even for an infinite number of simple poles in nine only the real root xmath112 survives in the limit xmath57 for this purpose consider the limit xmath114 in thislimit the distance between the imaginary parts of the nearest roots remains finite even for infinite volume indeed for xmath115 the leading contribution to the rhs of twelve corresponds to the harmonic with xmath116 and consequently an exponentially large amplitude of this term can be only compensated by a vanishing value of xmath117 ie xmath118 with xmath119 hereafter we will analyze only the branch xmath120 and therefore the corresponding decay formation time xmath1211 is volume independent keeping the leading term on the rhs of twelve and solving for xmath122 one finds xmath123 where in the last step we used eq eleven and condition xmath119 since for xmath57all negative values of xmath67 can not contribute to the gcp nine it is sufficient to analyze even values of xmath124 which according to msixteen generate xmath125 since the inequality thirteen can not be broken a single possibility when xmath126 pole can contribute to the partition nine corresponds to the case xmath127 for some finite xmath124 assuming this we find xmath128 for the same value of xmath35 substituting these results into equation eleven one gets xmath129ll r0 the inequality mseventeen follows from the equation for xmath77 and the fact that even for equal leading terms in the sums above with xmath130 and even xmath124 the difference between xmath77 and xmath67 is large due to the next to leading term xmath131 which is proportional to xmath132 thus we arrive at a contradiction with our assumption xmath133 and consequently it can not be true therefore for large volumes the real root xmath112 always gives the main contribution to the gcp nine and this is the only root that survives in the limit xmath57 thus we showed that the model with the fixed size of the largest fragment has no phase transition because there is a single singularity of the isobaric partition six which exists in thermodynamic limit if xmath46 monotonically grows with the volume the situation is different in this case for positive value of xmath134 the leading exponent in the rhs of twelve also corresponds to a largest fragment ie to xmath135 therefore we can apply the same arguments which were used above for the case xmath136 and derive similarly equations mfourteenmsixteen for xmath68 and xmath67 from xmath137 it follows that when xmath0 increases the number of simple poles in eight also increases and the imaginary part of the closest to the real xmath59axis poles becomes very small ie xmath138 for xmath139 and consequently the associated decay formation time xmath1401 grows with the volume of the system due to xmath141 the inequality mseventeen can not be established for the poles with xmath139 therefore in contrast to the previous case for large xmath46 the simple poles with xmath139 will be infinitesimally close to the real axis of the complex xmath59plane from eq msixteen it follows that xmath142 for xmath143 and xmath144 thus we proved that for infinite volume the infinite number of simple poles moves toward the real xmath59axis to the vicinity of liquid phase singularity xmath145 of the isobaric partition xcite and generates an essential singularity of function xmath146 in seven irrespective to the sign of chemical potential xmath35 as we showed above the states with xmath147become stable because they acquire infinitely large decay formation time xmath94 in the limit xmath57 therefore these states should be identified as a liquid phase for finite volumes as well such a conclusion can be easily understood if we recall that the partial pressure xmath148 of meighteen corresponds to a single fragment of the largest possible size now it is clear that each curve in fig 2 is the finite volume analog of the phase boundary xmath149 for a given value of xmath46 below the phase boundary there exists a gaseous phase but at and above each curve there are states which can be identified with a finite volume analog of the mixed phase and finally at xmath150 there exists a liquid phase when there is no phase transition ie xmath151 the structure of simple poles is similar but first the line which separates the gaseous states from the metastable states does not change with the volume and second as shown above the metastable states will never become stable therefore a systematic study of the volume dependence of free energy or pressure for very large xmath0 along with the formation and decay times may be of a crucial importance for experimental studies of the nuclear liquid gas phase transition the above results demonstrate that in contrast to hill s expectations xcite the finite volume analog of the mixed phase does not consist just of two pure phases the mixed phase for finite volumes consists of a stable gaseous phase and the set of metastable states which differ by the free energy moreover the difference between the free energies of these states is not surface like as hill assumed in his treatment xcite but volume like furthermore according to eqs eleven and twelve each of these states consists of the same fragments but with different weights as seen above for the case xmath150 some fragments that belong to the states in which the largest fragment is dominant may in principle have negative weights effective number of degrees of freedom in the expression for xmath152 eleven this can be understood easily because higher concentrations of large fragments can be achieved at the expense of the smaller fragments and is reflected in the corresponding change of the real part of the free energy xmath153 therefore the actual structure of the mixed phase at finite volumes is more complicated than was expected in earlier works a similar situation occurs for the real values of xmath46 in this caseall sums in eqs tenthirteen should be expressed via the euler maclaurin formula xmath154 xmath155 here xmath156 are the bernoulli numbers the representation fourteen allows one to study the effect of finite volume fv on the gcp nine the above results are valid for any xmath46 dependence however the linear one ie xmath157 with xmath158 is the most natural with the help of the parameter xmath42 it is possible to describe a difference between the geometrical shape of the volume under consideration and that one of the largest fragment for instance by fixing xmath159 it is possible to account for the fact that the largest spherical fragment can not fill completely the cube with the side equal to its two radii while there is enough space available for small fragments due to the xmath160 dependence in the csmmthere are two different ways of how the finite volume affects thermodynamical functions for finite xmath0 and xmath161 there is always a finite number of simple poles in nine but their number and positions in the complex xmath59plane depend on xmath0 to see this let us study the mechanical pressure which corresponds to the gcp nine xmath162 2 biggl b2frac partial lambda npartial v sumlimitsk1kv tildephik k2 textstyle efracnunkt tildephikv times nonumber hspace07 cm textstyle efracnunkvt textstyle kv left 1 alpha fracnunt left frac12 alpha right right okvbiggr biggr hspace005 cm right endaligned where we give the main term for each xmath163 and leading fv corrections explicitly for xmath164 whereas xmath165 accumulates the higher order corrections due to the euler maclaurin eq fourteen in evaluation of fifteen we used an explicit representation of the derivative xmath166 which can be found from eqs ten and fourteen the first term in the rhs of fifteen describes the constrained grand canonical cgc complex pressure generated by the simple pole xmath167 due to its free energy density xmath168 weighted with the probability xmath169 whereas the second and third terms appear due to the volume dependence of xmath46 note that instead of the fv corrections the usage of natural values for xmath46 would generate the artificial delta function terms in fifteen for the volume derivatives now it is clear that in case xmath151 the corrections to the main term will not appear and the number of poles and their positions will be defined by values of xmath7 and xmath35 only as one can see from fifteen for finite volumes the corrections can give a non negligible contribution to the pressure because in this case xmath170 can be positive the real parts of the partial cgc pressures xmath171 may have either sign therefore if the fv corrections to the pressure fifteen are small then according to thirteen the positive cgc pressures xmath172 are mechanically metastable and the negative ones xmath173 are mechanically unstable compared to the gas pressure xmath78 the fv corrections should be accounted for to find the mechanically meta and unstable states in the general case however it is clear that the contribution of the states with xmath173 into partition and its derivatives is exponentially small even for finite volumes as we showed earlier in this section when xmath0 increases the number of simple poles in eight also increases and imaginary part of the closest to the real xmath59axis poles becomes very small therefore for infinite volume the infinite number of simple poles moves toward the real xmath59axis to the vicinity of liquid phase singularity xmath174 and thus generates an essential singularity of function xmath146 in seven in this casethe contribution of any of remote poles from the real xmath59axis to the gcp vanishes then it can be shown that the fv corrections in fifteen become negligible because of the inequality xmath175 and consequently the reduced distribution of largest fragment xmath176 and the derivatives xmath166 vanish for all xmath7values and we obtain the usual smm solution xcite its thermodynamics as we discussed is governed by the farthest right singularity in the complex xmath59plane in this work we discussed a powerful mathematical method which allowed us to solve analytically the csmm at finite volumes it is shown that for finite volumes the gcp function can be identically rewritten in terms of the simple poles of the isobaric partition six the real pole xmath112 exists always and the quantity xmath177 is the cgc pressure of the gaseous phase the complex roots xmath126 appear as pairs of complex conjugate solutions of equation ten as we discussed their most straightforward interpretation is as follows xmath178 has a meaning of the free energy density whereas xmath179 depending on sign gives the inverse decay formation time of such a state the gaseous state is always stable because its decay formation time is infinite and because it has the smallest value of free energy the complex poles describe the metastable states for xmath180 and mechanically unstable states for xmath181 we studied the volume dependence of the simple poles and found a dramatic difference in their behavior in case of phase transition and without it for the formerthis representation allows one also to define the finite volume analogs of phases unambiguously and to establish the finite volume analog of the xmath149 phase diagram see fig 2 at finite volumesthe gaseous phase exists if there is a single simple pole the mixed phase corresponds to three and more simple poles whereas the liquid is represented by an infinite amount of simple poles at highest possible particle density or xmath95 as we showed for given xmath7 and xmath35 the states of the mixed phase which have different xmath182 are not in a true chemical equilibrium for finite volumes this feature can not be obtained within the fisher droplet model due to lack of the hard core repulsion between fragments this fact also demonstrates clearly that in contrast to hill s expectations xcite the mixed phase is not just a composition of two states which are the pure phases as we showed the mixed phase is a superposition of three and more collective states and each of them is characterized by its own value of xmath183 because of that the difference between the free energies of these states is not a surface like as hill argued xcite but volume like for the case with phase transition ie for xmath184 we analyzed what happens in thermodynamic limit when xmath0 grows the number of simple poles in eight also increases and imaginary part of the closest to the real xmath59axis poles becomes vanishing for infinite volumethe infinite number of simple poles moves toward the real xmath59axis and forms an essential singularity of function xmath146 in seven which defines the liquid phase singularity xmath174 thus we showed how the phase transition develops in thermodynamic limit also we analyzed the finite volume corrections to the mechanical pressure fifteen the corrections of a similar kind should appear in the entropy particle number and energy density because of the xmath7 and xmath35 dependence of xmath163 due to ten xcite therefore these corrections should be taking into account while analyzing the experimental yields of fragments then the phase diagram of the nuclear liquid gas phase transition can be recovered from the experiments on finite systems nuclei with more confidence a detailed analysis of the isobaric partition singularities in the xmath185 plane allowed us to define the finite volume analogs of phases and study the behavior of these singularities in the limit xmath57 such an analysis opens a possibility to study rigorously the nuclear liquid gas phase transition directly from the finite volume partition this may help to extract the phase diagram of the nuclear liquid gas phase transition from the experiments on finite systems nuclei with more confidence s das gupta a majumder s pratt and a mekjian arxiv nucl th9903007 1999 k a bugaev m i gorenstein i n mishustin and w greiner phys rev c62 2000 044320 arxiv nucl th0007062 2000 k a bugaev m i gorenstein i n mishustin and w greiner phys lett b 498 2001 144 arxiv nucl th0103075 2001 p t reuter and k a bugaev phys b 517 2001 233 k a bugaev arxiv nucl th0406033 2004 k a bugaev l phair and j b elliott arxiv nucl th0406034 2004 k a bugaev and j b elliott arxiv nucl th0501080 2005
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we discuss an exact analytical solution of a simplified version of the statistical multifragmentation model with the restriction that the largest fragment size can not exceed the finite volume of the system a complete analysis of the isobaric partition singularities of this model is done for finite volumes it is shown that the real part of any simple pole of the isobaric partition defines the free energy of the corresponding state whereas its imaginary part depending on the sign defines the inverse decay formation time of this state the developed formalism allows us for the first time to exactly define the finite volume analogs of gaseous liquid and mixed phases of this model from the first principles of statistical mechanics and demonstrate the pitfalls of earlier works the finite size effects for large fragments and the role of metastable unstable states are discussed numbers 2570 pq 2165f 2410 pa
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introduction
laplace-fourier transformation
isobaric partition singularities
no phase transition case
finite volume analogs of phases
conclusions
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"we present a detailed analysis of the regularity and decay properties of linear scalar waves near t(...TRUNCATED)
| " we show that linear scalar waves are bounded and continuous up to the cauchy horizon of reissner (...TRUNCATED)
| "introduction\nreissnernordstrmde sitter space\nkerrde sitter space\nvariable order b-sobolev spaces(...TRUNCATED)
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"electronically tuned microwave oscillators are key components used in a wide variety of microwave c(...TRUNCATED)
| " we have developed a new methodology and a time domain software package for the estimation of the o(...TRUNCATED)
| "introduction\ntheoretical analysis\ntime responses of non-linear oscillators and resonance frequenc(...TRUNCATED)
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"loop quantum gravity had never been considered a candidate of the unification of matter and gravity(...TRUNCATED)
| " we propose a new notation for the states in some models of quantum gravity namely 4valent spin net(...TRUNCATED)
| "introduction\nnotation\nbraids\nequivalence moves\nclassification of braids\nconclusions & perspect(...TRUNCATED)
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"quantum fluctuation can suppress chaotic motion of wave packet in the phase space due to the quantu(...TRUNCATED)
| " we numerically study influence of a polychromatic perturbation on wave packet dynamics in one dime(...TRUNCATED)
| "introduction\nmodel\nnumerical results\nsummary and discussion\neffect of polychromatic perturbatio(...TRUNCATED)
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"the total energy formula obtained by sun is xmath1 labeluv where xmath2 is the total energy per ato(...TRUNCATED)
| " the cohesive energies of solids calculated using mglj eos proposed by sun jiuxun sun jiuxun j phys(...TRUNCATED)
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cohesive energy
mglj potential
conclusion
acknowledgements
references
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"in the past decades atom based metrology has had an enormous impact on science technology and every(...TRUNCATED)
| " we investigate atom based electric field calibration and polarization measurement of a 100mhz line(...TRUNCATED)
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introduction
experimental setup
rydberg-atom-based characterization of an rf field
conclusion
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"recently there has been a lot of interest in understanding the scaling behavior in submonolayer isl(...TRUNCATED)
| " the effects of cluster diffusion on the submonolayer island density xmath0 and island size distrib(...TRUNCATED)
| "introduction\nmodel and simulations\nsimulation methods\ngeneralized scaling form for the island-si(...TRUNCATED)
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