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

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.