Statistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle

TL;DR

This paper explores how reservoir fluctuations induce power law distributions in particle energies, linking the q-parameter to fluctuations and heat capacity, and introduces a generalized entropy framework encompassing Tsallis, Renyi, and Boltzmann-Gibbs-Shannon forms.

Contribution

It derives a universal relation between reservoir fluctuations and power law distributions, and develops a generalized entropy formalism that ensures additivity for non-extensive systems.

Findings

Power law distributions arise from reservoir fluctuations.

The q-parameter relates to variance and heat capacity.

A new entropy formula is proposed for diverging temperature variance.

Abstract

Certain fluctuations in particle number at fixed total energy lead exactly to a cut-power law distribution in the one-particle energy, via the induced fluctuations in the phase-space volume ratio. The temperature parameter is expressed automatically by an equipartition relation, while the q-parameter is related to the scaled variance and to the expectation value of the particle number. For the binomial distribution q is smaller, for the negative binomial q is larger than one. These results also represent an approximation for general particle number distributions in the reservoir up to second order in the canonical expansion. For general systems the average phase-space volume ratio expanded to second order delivers a q parameter related to the heat capacity and to the variance of the temperature. However, q differing from one leads to non-additivity of the Boltzmann-Gibbs entropy. We demonstrate that a deformed entropy, K(S), can be constructed and used for demanding additivity. This requirement leads to a second order differential equation for K(S). Finally, the generalized q-entropy formula contains the Tsallis, Renyi and Boltzmann-Gibbs-Shannon expressions as particular cases. For diverging temperature variance we obtain a novel entropy formula.