The spectrum of fractal dimensions D(q) of non-interacting classical a
nd quantum gases in non-equilibrium, whose states are described by the
set of occupation number {n1, n2, ..., n(K)}, has been obtained. With
this point of departure, the fractal inspired entropies of the relate
d gases have been deduced. The generalized non-exponential forms of th
e distribution functions have been found by the application of Boltzma
nn's H-theorem.