  Starting with the additivity condition for Lyapunov functions of master
equation, we derive a one-parametric family of entropy functions which may be
appropriate for a description of certain effects of finiteness of statistical
systems, in particular, distribution functions with long tails. This
one-parametric family is different from Tsallis entropies, and is essentially a
convex combination of the Boltzmann-Gibbs-Shannon entropy and the entropy
function introduced by Burg. An example of how longer tails are described
within the present approach is worked out for the canonical ensemble. In
addition, we discuss a possible origin of a hidden statistical dependence, and
give explicit recipes how to construct corresponding generalizations of master
equation.
