Domain statistics in a finite Ising chain

TL;DR

This paper analyzes the statistical properties of domain structures in finite Ising chains, providing probability functions, averages, variances, and criteria for ferromagnetic behavior relevant to nanoscale magnetic systems.

Contribution

It offers a comprehensive combinatorial analysis of domain variables in finite Ising chains, including probability functions and criteria for ferromagnetic-like behavior, extending understanding to finite systems.

Findings

Derived probability functions for domain walls, up domains, and spins in up domains.

Calculated averages and variances for these domain variables.

Introduced a criterion for ferromagnetic-like behavior in finite chains.

Abstract

We present a comprehensive study for the statistical properties of random variables that describe the domain structure of a finite Ising chain with nearest-neighbor exchange interactions and free boundary conditions. By use of extensive combinatorics we succeed in obtaining the one-variable probability functions for (i) the number of domain walls, (ii) the number of up domains, and (iii) the number of spins in an up domain. The corresponding averages and variances of these probability distributions are calculated and the limiting case of an infinite chain is considered. Analyzing the averages and the transition time between differing chain states at low temperatures, we also introduce a criterion of the ferromagnetic-like behavior of a finite Ising chain. The results can be used to characterize magnetism in monatomic metal wires and atomic-scale memory devices.