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

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.

arXiv comment7 pages, 6 figures. This is the version that was accepted for publication in Physical Review E Letters

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• Casimir and Helmholtz forces in one-dimensional Ising model with Dirichlet (free) boundary conditions
• A Brief Survey of Fluctuation-induced Interactions in Micro- and Nano-systems and One Exactly Solvable Model as Example
• Constraint correlation functions of the one-dimensional Ising model in the scaling limit