Statistical properties of 1D spin glasses from first principles of classical mechanics

TL;DR

This paper investigates the statistical properties of one-dimensional classical Heisenberg spin glasses, deriving recurrence equations from Hamiltonian mechanics, revealing NP-hardness, and proposing a new integral representation for the partition function.

Contribution

It introduces a first-principles approach to 1D spin glasses, deriving recurrence relations and a novel integral form for the partition function, connecting classical mechanics with statistical mechanics.

Findings

Calculations from classical mechanics lead to NP-hard problems.

In the equilibrium limit, the problem can be solved with a P algorithm.

A new integral representation for the partition function is proposed.

Abstract

We study the classical 1D Heisenberg spin glasses. Based on the Hamilton equations we obtained the system of recurrence equations which allows to perform node-by-node calculations of a spin-chain. It is shown that calculations from first principles of classical mechanics lead to NP hard problem, that however in the limit of the statistical equilibrium can be calculated by P algorithm. For the partition function of the ensemble a new representation is offered in the form of one-dimensional integral of spin-chains' energy distribution.