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

Abstract

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, Ω n (E − ω)/Ω n (E) = (1 − ω/E) n . The only parameters are 1/T = β = n /E and q = 1 − 1/ n + ∆n 2 / 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 e S(E−ω) /e S(E)to second order delivers q = 1 − 1/C + ∆β 2 / β 2 with β = S ′ (E) and C = dE/dT heat capacity. However, q = 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., q K = 1. This requirement leads to a second order differential equation for K(S). Finally, the generalized q-entropy formula, K(S) = p i K(− ln p i ), contains the Tsallis, Rényi and Boltzmann-Gibbs-Shannon expressions as particular cases. For diverging variance, ∆β 2 we obtain a novel entropy formula.

Kinetics of low-temperature transitions and a reaction rate theory from non-equilibrium distributions

“…However, the presentation emphasizes that an intermediate step in the derivation of all the three statistical distributions in equations (3.4), (3.6) and (3.7) includes what we refer to as the Euler deformed exponential function, (1 − ( n /N)) N . This step is crucial here as well as to discuss the role of the Tsallis distribution in non-equilibrium statistics for finite heat baths: when finite size effects are negligible, we can establish the connection between the thermodynamic and the microscopic parameters: n = Eβ and d = 1/N [44,[47][48][49]. See also important previous work on statistical distributions in the kinetics of rate processes [50][51][52].…”

“…In microscopic systems where quantum effects are operative and the indistinguishability of particles has to be taken into account separately, another physically significant case applies, wherein no limitation occurs on the number of particles per state (Boson particles): [40,41,44] and appears that 'tertium non datur' [45]. The most probable distributions resulting for equations (3.4), (3.6) and (3.7) lead to Boltzmann, Fermi-Dirac and Bose-Einstein statistics [39,46], respectively, when the thermodynamic limit is taken.…”

Non-Extensive Entropic Distance Based on Diffusion: Restrictions on Parameters in Entropy Formulae

“…Finally, the recently suggested doubly logarithmic entropy formula, designed for extreme large fluctuations in a reservoir by Biro et al [23,24], considers…”

“…emerges with K(S) satisfying a second order differential equation, the additivity restoration condition (ARC) [24]. Connecting to a more traditional notation, the K function is related to the deformed logarithm function as…”

Nonextensive critical effects in the NJL model

“…Following [13] we use the q version of the Einstein's formula [13] for probability of states, W ∼ exp q S q 1 , and expand it to the second order in E [14]:…”