Exact expressions for the partition function of the one-dimensional Ising model in the fixed-$M$ ensemble

TL;DR

This paper derives exact formulas for the partition function of the 1D Ising model with fixed magnetization under various boundary conditions, enabling analysis of fluctuation-induced Helmholtz forces.

Contribution

It provides the first exact closed-form expressions for the partition function in the fixed-M ensemble for the 1D Ising model with multiple boundary conditions.

Findings

Exact partition functions for periodic, antiperiodic, and Dirichlet boundaries.

Facilitates calculation of fluctuation-induced Helmholtz forces.

Insights into fluctuation forces in new regimes.

Abstract

We obtain exact closed-form expressions for the partition function of the one-dimensional Ising model in the fixed-$M$ ensemble, for three commonly-used boundary conditions: periodic, antiperiodic and Dirichlet. These expressions allow for the determination of fluctuation-induced forces in the canonical ensemble, which we term Helmholtz forces. The thermodynamic expressions and the calculations flowing from them should provide insights into the nature and behavior of fluctuation induced forces in interesting and as-yet unexplored regimes.