Advances in Mathematical Physics

Volume 2017 (2017), Article ID 6970870, 5 pages

https://doi.org/10.1155/2017/6970870

## Exact Partition Function for the Random Walk of an Electrostatic Field

^{1}Cátedras CONACYT, Universidad Autónoma de San Luis Potosí, 78000 San Luis Potosí, SLP, Mexico^{2}Coordinación para la Innovación y la Aplicación de la Ciencia y la Tecnología, Universidad Autónoma de San Luis Potosí, 78000 San Luis Potosí, SLP, Mexico

Correspondence should be addressed to Gabriel González; xm.plsau@zelaznog.leirbag

Received 7 April 2017; Revised 6 June 2017; Accepted 14 June 2017; Published 13 July 2017

Academic Editor: Jacopo Bellazzini

Copyright © 2017 Gabriel González. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The partition function for the random walk of an electrostatic field produced by several static parallel infinite charged planes in which the charge distribution could be either is obtained. We find the electrostatic energy of the system and show that it can be analyzed through generalized Dyck paths. The relation between the electrostatic field and generalized Dyck paths allows us to sum overall possible electrostatic field configurations and is used for obtaining the partition function of the system. We illustrate our results with one example.

#### 1. Introduction

In 1961 Lenard considered the problem of a system of infinite charged planes free to move in one direction without any inhibition of free crossing over each other [1]. If all the charged planes carry a surface mass density of magnitude unity and the th one carries an electric surface charge density , the Hamiltonian of Lenard’s problem is given bywhere the system is overall neutral. With the Hamiltonian function at hand Lenard was able to obtain the exact statistical mechanics of the system with the technique of generating functions [1, 2].

From a physical point of view, Lenard’s model can be used in the study of a one-dimensional Coulomb gas [3]. The one-dimensional Coulomb gas is a statistical mechanical problem where particles of equal or opposite charges interact through the Coulomb potential [4]. The model has been extensively studied in the past and forms one of the classical exactly soluble problems in one dimension [5]. Also, the Coulomb gas model has also been used to describe the main features of ionic liquids [6, 7].

More recently, a very similar problem was proposed in [8] with the constraint that each of the charged planes has a fixed position in space and that the surface charge distribution in each plane could be either . This modified model gives rise to a random walk behavior of the electrostatic field that can be analyzed as a Markovian stochastic process.

The purpose of this paper is to give a statistical description of the electrostatic field generated by several static parallel infinite charged planes in which the surface charge distribution could be either . The key step in our formulation consists of the summation of all possible trajectories of the electrostatic field for every different charge configuration. We do this by showing that there is a one to one correspondence between every electrostatic field trajectory and a generalized Dyck path.

The article is organized as follows. In Section 2 we describe the model and derive a system of equations for obtaining the electrostatic energy of the system or Hamiltonian. In Section 3 we make a one to one correspondence between electrostatic field trajectories and generalized Dyck paths. In Section 4 the explicit expression for the partition function is given. In Section 5 a simple numerical analysis is given to show some features of the partition function. In the last section we summarize our conclusions.

#### 2. Model of the System

Suppose that we are given a collection of infinite charged planes, half of them have a constant charge distribution , and the other half have a constant charge distribution ; that is, the system is overall neutral. If we randomly place the infinite charged planes parallel to each other along the -axis at position where , then the electrostatic field would evolve along the -axis making random jumps each time it crosses an infinite charged sheet.

For example, consider a system of four charged planes. We know that equals zero at the left and right side of the configuration due to the neutrality of the system [9]. After crossing the first charged plane, would increase or decrease in an amount of depending on whether the first charged plane had a positive or negative surface charge density. In Figure 1 we show all the possible electrostatic field configurations for the case when the charge density is not known.