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

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