Eric I. Barnes, Liliya L. R. Williams
We present an analysis of two thermodynamic techniques for determining
equilibria of self-gravitating systems. One is the Lynden-Bell entropy
maximization analysis that introduced violent relaxation. Since we do not use
the Stirling approximation which is invalid at small occupation numbers, our
systems have finite mass, unlike Lynden-Bell's isothermal spheres. (Instead of
Stirling, we utilize a very accurate smooth approximation for $\ln{x!}$.) The
second analysis extends entropy production extremization to self-gravitating
systems, also without the use of the Stirling approximation. In addition to the
Lynden-Bell (LB) statistical family characterized by the exclusion principle in
phase-space, and designed to treat collisionless systems, we also apply the two
approaches to the Maxwell-Boltzmann (MB) families, which have no exclusion
principle and hence represent collisional systems. We implicitly assume that
all of the phase-space is equally accessible. We derive entropy production
expressions for both families, and give the extremum conditions for entropy
production. Surprisingly, our analysis indicates that extremizing entropy
production rate results in systems that have maximum entropy, in both LB and MB
statistics. In other words, both thermodynamic approaches lead to the same
equilibrium structures.
View original:
http://arxiv.org/abs/1201.5899
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