Exact Partition Function for the Random Walk of an Electrostatic Field
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Exact Partition Function for the Random Walk of an Electrostatic Field
Aug 7, 2016 - 1Cátedras CONACYT, Universidad Autónoma de San Luis Potosı, San Luis Potosı, ... know that E(z) equals zero at the left and right side of the.
Exact partition function for the random walk of an electrostatic field Gabriel Gonz´alez1, 2, ∗
arXiv:1608.02179v1 [cond-mat.stat-mech] 7 Aug 2016
1
C´atedras CONACYT, Universidad Aut´onoma de San Luis Potos´ı, San Luis Potos´ı, 78000 MEXICO 2 Coordinaci´on para la Innovaci´on y la Aplicaci´on de la Ciencia y la Tecnolog´ıa, Universidad Aut´onoma de San Luis Potos´ı,San Luis Potos´ı, 78000 MEXICO
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 over all possible electrostatic field configurations and is used for obtaining the partition function of the system. We illustrate our results with one example. PACS numbers: Keywords:
I. INTRODUCTION
In 1961 A. Lenard consider the problem of a system of infinite charged planes freely 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 ith one carries an electric surface charge density σi , the Hamiltonian of Lenard’s problem is given by H(q1 , q2 , . . . ; p1 , p2 , . . .) =
