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

Certain fluctuations in particle number, n, at fixed total energy, E, lead exactly to a cut-power law distribution in the one-particle energy, ω, via the induced fluctuations in the phase-space volume ratio, Ωn(E − ω)/Ωn(E) = (1 − ω/E). The only parameters are 1/T = 〈β〉 = 〈n〉/E and q = 1−1/〈n〉+∆n2/〈n〉2. For the binomial distribution of n one obtains q = 1−1/k, for the negative binomial q = 1+1/(k+1). These results also represent an approximation for general particle number distributions in the reservoir up to second order in the canonical expansion ω ≪ E. For general systems the average phase-space volume ratio 〈eS(E−ω)/eS(E)〉 to second order delivers q = 1−1/C+∆β2/〈β〉2 with β = S (E) and C = dE/dT heat capacity. However, q 6= 1 leads to non-additivity of the Boltzmann–Gibbs entropy, S. We demonstrate that a deformed entropy, K(S), can be constructed and used for demanding additivity, i.e., qK = 1. This requirement leads to a second order differential equation for K(S). Finally, the generalized q-entropy formula, K(S) = ∑ piK(− ln pi), contains the Tsallis, Rényi and Boltzmann–Gibbs–Shannon expressions as particular cases. For diverging variance, ∆β we obtain a novel entropy formula.