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.

中文翻译：

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有限能级粒子系统的拉普拉斯方法对状态和的近似及其在极限定理中的应用

我们考虑一个由分布在有限数量的能级上的固定数量的粒子组成的通用系统。我们只对系统的属性和熵做出一般假设。通过适当减少系统的状态空间，可以包括固定数量粒子以外的系统约束。对于熵，我们考虑三种一般情况。它可以在系统状态空间的内部或边界上具有最大值。在边界上，我们可以有另外两种情况。在那里，熵可以随着粒子数量的增加而线性增加，在另一种情况下，比线性增长更慢。主要结果是使用拉普拉斯方法对系统状态总和的近似值。误差项的估计也包括在内。作为应用程序，我们证明了产生系统最可能状态的大数定律。这种状态是具有最大熵的状态。我们还发现了波动的限制规律。对于所考虑的熵情况，这些定律是不同的。它们可以是正态分布、指数分布和离散分布的混合。为所有定理提供了明确的收敛速度。