New entropy formula with fluctuating reservoir

•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.

Finite heat reservoir capacity, CC, and temperature fluctuation, ΔT/TΔT/T, lead to modifications of the well known canonical exponential weight factor. Requiring that the corrections least depend on the one-particle energy, ωω, we derive a deformed entropy, K(S)K(S). The resultingformula contains the Boltzmann–Gibbs, Rényi, and Tsallis formulas as particular cases. For extreme large fluctuations, in the limit CΔT2/T2→∞CΔT2/T2→∞, 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.