Exact Partition Function for the Random Walk of an Electrostatic Field

TL;DR

This paper derives an exact partition function for a random walk model of electrostatic fields generated by multiple charged planes, linking electrostatics with combinatorial Dyck paths to analyze field configurations.

Contribution

It introduces a novel method connecting electrostatic field configurations with generalized Dyck paths to compute the partition function exactly.

Findings

Derived the electrostatic energy for multiple charged planes.

Established a correspondence between electrostatic fields and Dyck paths.

Provided an explicit example illustrating the method.

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 $\pm\sigma$ 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 over all possible electrostatic field configurations and is used for obtaining the partition function of the system. We illustrate our results with one example.