Approximations of the Sum of States by Laplace's Method for a System of Particles with a Finite Number of Energy Levels and Application to Limit Theorems

TL;DR

This paper develops Laplace's method to approximate the sum of states in a finite-particle system with various entropy behaviors, deriving limit theorems and fluctuation laws with explicit convergence rates.

Contribution

It introduces a general framework for approximating the sum of states using Laplace's method for systems with different entropy profiles, including error estimates and limit laws.

Findings

Approximate sum of states with Laplace's method for various entropy cases.

Proved law of large numbers identifying the most probable state as maximum entropy.

Derived limiting laws for fluctuations, including mixtures of distributions.

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.