  We consider a generic system composed of a fixed number of particles
distributed over a finite number of energy levels. We make only general
assumptions about system's properties and the entropy. System's constraints
other than fixed number of particles can be included by appropriate reduction
of system's state space. For the entropy we consider three generic cases. It
can have a maximum in the interior of system's state space or on the boundary.
On the boundary we can have another two cases. There the entropy can increase
linearly with increase of the number of particles and in the another case grows
slower than linearly. The main results are approximations of system's sum of
states using Laplace's method. Estimates of the error terms are also included.
As an application, we prove the law of large numbers which yields the most
probable state of the system. This state is the one with the maximal entropy.
We also find limiting laws for the fluctuations. These laws are different for
the considered cases of the entropy. They can be mixtures of Normal,
Exponential and Discrete distributions. Explicit rates of convergence are
provided for all the theorems.
