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