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Despite their formal simplicity, most lattice spin models cannot be easilysolved, even under the simplifying assumptions of mean field theory. In thismanuscript, we present a method for generating mean field solutions toclassical continuous spins. We focus our attention on systems with non-localinteractions and non-periodic boundaries, which require careful handling withexisting approaches, such as Monte Carlo sampling. Our approach utilizesfunctional optimization to derive a closed-form optimality condition and arriveat self-consistent mean field equations. We show that this approachsignificantly outperforms conventional Monte Carlo sampling in convergencespeed and accuracy. To convey the general concept behind the approach, we firstdemonstrate its application to a simple system - a finite one-dimensionaldipolar chain in an external electric field. We then describe how the approachnaturally extends to more complicated spin systems and to continuum fieldtheories. Furthermore, we numerically illustrate the efficacy of our approachby highlighting its utility on nonperiodic spin models of variousdimensionality.

A directed walk in probability space that locates mean field solutions to spin models