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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

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 [

From a physical point of view, Lenard’s model can be used in the study of a one-dimensional Coulomb gas [

More recently, a very similar problem was proposed in [

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 article is organized as follows. In Section

Suppose that we are given a collection of

For example, consider a system of four charged planes. We know that

Possible electrostatic field configurations inside a four-charged-plane system where the charge distribution is not known.

Since we are concern with the statistical properties of the electrostatic field we have first to obtain the Hamiltonian of the field or the electrostatic energy of the system. We know from basic electrodynamics courses that the electrostatic energy is given by [

We can write down the boundary condition given in (

In this section we present the relation between the electrostatic field and generalized Dyck paths. Let us consider for the sake of simplicity that all the charged planes carry a surface charge density of

It is convenient to investigate the nature of all the possible electrostatic field configurations by plotting

Generalized Dyck paths representing a possible electrostatic field configuration of

Note that every possible electrostatic field configuration consists of a closed path of length

Let us denote by

In order to calculate the sum of (

For example, consider the case of 4 parallel static randomly charged planes with surface charge distribution

We can calculate the expectation value for the energy of the system when

We can perform a direct evaluation of the partition function given in (

We have shown that it is possible to obtain the exact partition function for the electrostatic field generated by means of several static parallel infinite charged planes in which the surface charge distribution is not explicitly known. We have worked out the special case where the charged planes have a constant surface charge distribution given by

In this appendix we present a simple Mathematica© code which evaluates the partition function given in (

The author declares that they have no conflicts of interest.

This work was supported by the program “Cátedras CONACYT.”