Probability distribution function of dipolar field in two-dimensional spin ensemble

TL;DR

This paper derives the probability distribution function of the net dipolar field in a two-dimensional ensemble of polarized particles, revealing a transition from special functions to Gaussian behavior as surface concentration increases.

Contribution

It introduces a method based on cumulant expansion to determine the distribution function across different concentration regimes, including the transitional region.

Findings

At low concentrations, the distribution is expressed with special functions.

At high concentrations (~0.6), the distribution follows a Gaussian law.

The distribution functions are asymmetric at low and moderate concentrations.

Abstract

We theoretically determine the probability distribution function of the net field of the random planar structure of dipoles which represent polarized particles. At small surface concentrations c of the point dipoles this distribution is expressed in terms of special functions. At the surface concentrations of the dipoles as high as 0.6 the dipolar field obey the Gaussian law. To obtain the distribution function within transitional region c<0.6, we propose the method based on the cumulant expansion. We calculate the parameters of the distributions for some specific configurations of the dipoles. The distribution functions of the ordered ensembles of the dipoles at the low and moderate surface concentrations have asymmetric shape with respect to distribution medians. The distribution functions allow to calculate various physical parameters of two-dimensional interacting nanoparticle ensembles.