  The Boltzmann entropy $S^{(B)}$ is true in the case of equal probability of
all microstates of a system. In the opposite case it should be averaged over
all microstates that gives rise to the Boltzmann--Shannon entropy (BSE).
Maximum entropy principle (MEP) for the BSE leads to the Gibbs canonical
distribution that is incompatible with power--low distributions typical for
complex system. This brings up the question: Does the maximum of BSE correspond
to an equilibrium (or steady) state of the complex system? Indeed, the
equilibrium state of a thermodynamic system which exchange heat with a
thermostat corresponds to maximum of Helmholtz free energy rather than to
maximum of average energy, that is internal energy $U$. Following derivation of
Helmholtz free energy the Renyi entropy is derived as a cumulant average of the
Boltzmann entropy for systems which exchange an entropy with the thermostat.
The application of MEP to the Renyi entropy gives rise to the Renyi
distribution for an isolated system. It is investigated for a particular case
of a power--law Hamiltonian. Both Lagrange parameters, $\alpha$ and $\beta$ can
be eliminated. It is found that $\beta$ does not depend on a Renyi parameter
$q$ and can be expressed in terms of an exponent $\kappa$ of the power--law
Hamiltonian and $U$. The Renyi entropy for the resulting Renyi distribution
reaches its maximal value at $q=1/(1+\kappa)$ that can be considered as the
most probable value of $q$ when we have no additional information on behavior
of the stochastic process. The Renyi distribution for such $q$ becomes a
power--law distribution with the exponent $-(\kappa +1)$. Such a picture
corresponds to some observed phenomena in complex systems.
