New entropy formula with fluctuating reservoir

Abstract

h i g h l i g h t s• We present a mathematical procedure to obtain a deformed entropy function.• We describe effects due to finite heat capacity and temperature fluctuations in the heat reservoir.• For the Gaussian fluctuation model the resulting entropy-probability relation recovers the traditional ''log'' formula. • Without temperature fluctuations (but at finite heat capacity) we obtain the Tsallis formula.• For extreme large temperature fluctuations we obtain a new ''log(1 − log)'' formula. a b s t r a c tFinite heat reservoir capacity, C , and temperature fluctuation, ∆T /T , lead to modifications of the well known canonical exponential weight factor. Requiring that the corrections least depend on the one-particle energy, ω, we derive a deformed entropy, K (S). The resulting formula contains the Boltzmann-Gibbs, Rényi, and Tsallis formulas as particular cases.For extreme large fluctuations, in the limit C ∆T 2 /T 2 → ∞, a new parameter-free entropy-probability relation is gained. The corresponding canonical energy distribution is nearly Boltzmannian for high probability, but for low probability approaches the cumulative Gompertz distribution. The latter is met in several phenomena, like earthquakes, demography, tumor growth models, extreme value probability, etc.

On the Uniqueness Theorem for Pseudo-Additive Entropies

“…Since the (pseudo-)extensivity is more a characteristic property of the actual interaction/correlation between X and Y rather than entropy itself, we shall not dwell on this issue here. Let us just mention that the actual form of non-extensivity and the value of parameters can be often connected to some specific physical phenomena [8,22]. In Section 3, we will see that, from a strictly mathematical standpoint, it is logically more satisfactory to deal with dependent, but non-interacting systems.…”

“…The meaning of the chain rule (8) is depicted in Figure 1. By induction, one can generalize the previous relation (8) to:…”

“…Among these one can especially mention: (a) innate general statistics approaches such as the q-deformed thermostatistics of Tsallis [1], the superstatistics of Beck et al [3] and various generalized non-extensive entropies [4][5][6][7][8]; (b) information-theoretic concepts such as the information theory of Rényi [9,10], transfer entropies [11,12], complexity theory [1,13] or information geometry [14]. Particularly the concept of entropy, both in its information-theoretic and combinatorial disguise, plays a central role here.…”

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…”

“…Instead of treating the temperature or its inverse as a fluctuating quantity, one may also consider the number of degrees of freedom, and hence the dimensionality of phase space, as a fluctuating quantity. In [19][20][21][22][23] it has been shown that simple fluctuation patterns of the particle number, N, influencing the phase space volume Ω occupied by an ideal gas, can lead both to an exponential distribution of exp(−ω/T) by a Poisson N distribution, or to cut power-law like Tsallis-Pareto distribution by a negative binomial N distribution. In general, a trace form entropy,…”