Laws of thermodynamics, extensivity and stationary equilibrium distributions

Details Laws of thermodynamics, extensivity and stationary equilibrium distributions Laws of thermodynamics, extensivity and stationary equilibrium distributions

2011-01-13

arxiv.org/abs/1101.2607v2

Physics

Condensed Matter

Statistical Mechanics

8 pages

Scientific paper

We first obtain the stationary equilibrium distributions of entropies such as Boltzmann-Gibbs and Tsallis on purely thermodynamic grounds, without appealing to entropy maximization procedure of Jaynes. Although both thermodynamic approach and entropy maximization procedure yield the same stationary equilibrium distribution for Boltzmann-Gibbs entropy, differences are noted in the case of nonadditive Tsallis entropy. This difference between the purely thermodynamic approach and entropy maximization procedure is then traced back to the assumption of extensivity, which is to hold for macroscopic conservation of energy i.e., the first law of thermodynamics. Therefore, an additional constraint due to the requirement of extensivity is needed in the maximization procedure, leading to a novel generalized maximization procedure. Having shown that this generalized maximization procedure is compatible with the purely thermodynamic formalism, we investigate the R\'enyi entropy solely on thermodynamic grounds, since the novel maximization procedure presented herein applies only to trace form entropies unlike the R\'enyi measure. However, even the sole application of thermodynamic laws to R\'enyi entropy is surprising, since it yields an exponential equilibrium stationary distribution instead of commonly presumed inverse power law behavior.

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