Statistical Properties of Ideal Ensemble of Disordered 1D Steric Spin-Chains

TL;DR

This paper investigates the statistical properties of disordered 1D spin-chains, revealing that interaction constants follow Lévy's alpha-stable distribution and analyzing energy distributions under various conditions.

Contribution

It introduces an analytical and numerical study of 1D disordered spin-chains, establishing the distribution law of interaction constants and exploring energy distribution behaviors.

Findings

Interaction constants follow Lévy's alpha-stable distribution.

Energy distribution exhibits local maxima when spin-chain number is much less than N_x^2.

When spin-chain number is comparable to N_x^2, energy distribution has a single global maximum.

Abstract

The statistical properties of ensemble of disordered 1D steric spin-chains (SSC) of various length are investigated. Using 1D spin-glass type classical Hamiltonian, the recurrent trigonometrical equations for stationary points and corresponding conditions for the construction of stable 1D SSCs are found. The ideal ensemble of spin-chains is analyzed and the latent interconnections between random angles and interaction constants for each set of three nearest-neighboring spins are found. It is analytically proved and by numerical calculation is shown that the interaction constant satisfies L\'{e}vy's alpha-stable distribution law. Energy distribution in ensemble is calculated depending on different conditions of possible polarization of spin-chains. It is specifically shown that the dimensional effects in the form of set of local maximums in the energy distribution arise when the number of spin-chains M << N_x^2 (where N_x is number of spins in a chain) while in the case when M ~ N_x^2 energy distribution has one global maximum and ensemble of spin-chains satisfies Birkhoff's ergodic theorem. Effective algorithm for parallel simulation of problem which includes calculation of different statistic parameters of 1D SSCs ensemble is elaborated.