Direct Monte Carlo scheme for Helmholtz free energy

Helmholtz free energy is a central quantity in statistical physics, as it fully characterizes the thermodynamic properties of a system. However, apart from a few exactly solvable models, computing the free energy for generic many-body systems typically requires numerical methods, most commonly Markov chain Monte Carlo combined with thermodynamic integration. In this study, we propose an alternative approach that rewrites the partition function as a high-dimensional integral over auxiliary fields and solves it directly, thereby circumventing the need for Markov chains and thermodynamic integration. We formulate the VEGAS algorithm within the framework of stochastic optimization. This improves the achievable integration dimension by an order of magnitude. We benchmark the proposed scheme against Markov chain Monte Carlo simulations, demonstrating favorable accuracy and scalability.