Derivation of an equation of pair correlation function from BBGKY hierarchy in a weakly coupled self gravitating system Anirban Bose_2

2025-05-06 0 0 187.28KB 9 页 10玖币

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Derivation of an equation of pair correlation function from BBGKY hierarchy

in a weakly coupled self gravitating system

Anirban Bose

Serampore College, Serampore, Hooghly 712201, India

An equation of pair correlation function has been derived from the ﬁrst two members

of BBGKY hierarchy in a weakly coupled inhomogeneous self gravitating system in quasi

thermal equilibrium. This work may be useful to study the thermodynamic properties of the

central region of a star cluster which is older than a few or more central relaxation time.

arXiv:2210.15442v2 [cond-mat.stat-mech] 9 Nov 2022

I. INTRODUCTION

Finite, bound self gravitating system has negative heat capacity. The thermodynamics [1–8]

of such system diﬀers signiﬁcantly from that of normal laboratory system and there is a question

that if self gravitating system, where the constituent particles are interacting with each other via

long range gravitational force, may at all attain the thermal equilibrium state.

In fact, absolute thermal equilibrium is not possible. Since, in contrast to the laboratory system,

there is no physical container to conﬁne the system to a particular place. Therefore, evaporation of

particles from the high energy tail of the distribution function does not allow the system to reach

thermal equilibrium.

This conﬁnement problem may be theoretically resolved by the introduction of artiﬁcial bound-

ary. Thermodynamic properties of such system have been extensively investigated [9–11] by con-

ﬁning N point particles each with mass m in a spherical container. The system is assumed to

be thermally conducting. Therefore, the equilibrium state of this system is expected to be the

isothermal sphere. But that is not the actual fact. For both canonical and microcanonical ensem-

bles instabilities occur if the density contrast between the centre and edge of an isothermal gas

exceeds certain values. For microcanonical ensembles this phenomenon is recognised as gravother-

mal catastrophe [9–11] and such system can increase entropy by settling down to a state which is

not isothermal.

However, self gravitating system like globular cluster whose relaxation time is less than the age

of the cluster has a chance to reach close to the thermal equilibrium state. At least, the central

part of a cluster older than a few central relaxation time is close to thermal equilibrium[2]. These

globular clusters are approximately spherical in shape and may be modelled by the N particle

system conﬁned in a spherical container.

There are diﬀerent theoretical approaches [1, 2, 12–15]to study the self gravitating system.

Probably, the most common way is to explore the Fokker-Planck equation [1, 2] of the concerned

system.

There are other ways. For example, integrating the N particle Liouville equation over the

phase space of (N-1) particles we obtain the ﬁrst member of BBGKY hierarchy equation which, in

the limit of N→ ∞, becomes the collisionless Vlasov equation. This equation deals successfully

with the self gravitating system where the inﬂuence of encounters are neglected. Integrating the

Liouville equation once again over the phase space of (N-2) particles we obtain the second member

of BBGKY hierarchy. These two equations, in the limit of large but ﬁnite values of N, are capable

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DerivationofanequationofpaircorrelationfunctionfromBBGKYhierarchyinaweaklycoupledselfgravitatingsystemAnirbanBoseSeramporeCollege,Serampore,Hooghly712201,IndiaAnequationofpaircorrelationfunctionhasbeenderivedfromthersttwomembersofBBGKYhierarchyinaweaklycoupledinhomogeneousselfgravitatingsysteminqua...

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Derivation of an equation of pair correlation function from BBGKY hierarchy in a weakly coupled self gravitating system Anirban Bose_2

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