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 ±σ 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.
Highlights
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]
The onedimensional Coulomb gas is a statistical mechanical problem where particles of equal or opposite charges interact through the Coulomb potential [4]
Let us denote by Γ the ensemble of generalized Dyck paths of length 2N and consider γ ∈ Γ; the partition function of the electrostatic field will be given by [14]
Summary
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]. 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 ±σ. This modified model gives rise to a random walk behavior of the electrostatic field that can be analyzed as a Markovian stochastic process. The key step in our formulation consists of the summation of all possible trajectories of the electrostatic field for every different charge configuration.
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