Abstract
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.
Highlights
We consider a system which is composed of N particles distributed over m+1 energy levels
Let us define a set D ⊂ Rm composed of vectors x ∈ Rm such that the following constraints are valid x1 + x2 + . . . + xm 1, xi 0, for i = 1, . . . , m, and define a set
The main results of this paper are approximations of the sum of states (N)
Summary
A single system’s state is represented by the vector of the occupation numbers of the levels, denoted by (N1, N2, . . . , Nm, Nm+1). An example of additional constraint is bounded maximal average energy per particle, i.e. E m+1 i=1 εi Ni /N When such exemplary system has a large number of particles, majority of its states are in the range (NE − , NE), for some small. Particular cases of the considered generic system might include features such as energy level degeneracy and indistinguishability of the particles. Such features might be reflected in the specific form of the functions f and h, as shows the example in the end of introduction. Our results yields the distributions of the fluctuations from the most probable state They are different for two cases of the entropy maximum. The functions f , h and the point of maximum of the entropy were derived from the system’s properties and are different for each case of G(N)
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