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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ellis, R., Rosen, J.: Asymptotc analysis of Gaussian integrals II. Manifold of minimum points. Commun. Math. Phys. 82, 153–181 (1981)
Ellis, R., Rosen, J.: Asymptotc analysis of Gaussian integrals I. Isolated minimum points. Trans. Am. Math. Soc. 273(2), 447–481 (1982)
Ellis, R., Rosen, J.: Laplace’s method for gaussian integrals with an application to statistical mechanics. Ann. Probab. 10(1), 47–66 (1982)
Hwang, C.-R.: Laplace’s method revisited: Weak convergence of probability measures. Ann. Probab. 8(6), 1177–1182 (1980)
Kolokoltsov, V.N.: Semiclassical Analysis for Diffusions and Strochastic processes. Springer (2000)
Łapiński, T.M.: Law of large numbers unifying Maxwell-Boltzmann. Bose-Einstein and Zipf-Mandelbort distributions, and related fluctuations. arXiv:1501.06718 (2015)
Łapiński, T.M.: Multivariate Laplace’s approximation with estimates and application to limit theorems. Journal of Approximation Theory, 248. https://doi.org/10.1016/j.jat.2019.105305 (2019)
Pathria, R., Beale, P.: Statistical Mechanics, 3rd edn. Elsevier (2011)
Reif, F.: Fundamentals of Statistical and Thermal Physics. Waveland Press (2013)
Werpachowska, A.: Exact and approximate methods of calculating the sum of states for noninteracting classical and quantum particles occupying a finite number of modes. Phys. Rev. E 84(4), 041125 (2011)
Acknowledgments
Author dedicates special thanks to Professor Vassili Kolokoltsov from University of Warwick, for hints in developing the results of this work. That was, using the Taylor’s Theorem in the proofs and, as outlined in the Remark 1, obtaining the solution for some cases by reducing the limit for natural numbers to the limit for specific subsequence of the natural numbers.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Łapiński, T.M. 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. Math Phys Anal Geom 23, 9 (2020). https://doi.org/10.1007/s11040-020-9330-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11040-020-9330-8