A Partition Function Approximation Using Elementary Symmetric Functions

Abstract

In statistical mechanics, the canonical partition function Z can be used to compute equilibrium properties of a physical system. Calculating Z however, is in general computationally intractable, since the computation scales exponentially with the number of particles N in the system. A commonly used method for approximating equilibrium properties, is the Monte Carlo (MC) method. For some problems the MC method converges slowly, requiring a very large number of MC steps. For such problems the computational cost of the Monte Carlo method can be prohibitive. Presented here is a deterministic algorithm - the direct interaction algorithm (DIA) - for approximating the canonical partition function Z in ~N2 operations. The DIA approximates the partition function as a combinatorial sum of products known as elementary symmetric functions (ESFs), which can be computed in ~N2 operations. The DIA was used to compute equilibrium properties for the isotropic 2D Ising model, and the accuracy of the DIA was compared to that of the basic Metropolis Monte Carlo method. Our results show that the DIA may be a practical alternative for some problems where the Monte Carlo method converge slowly, and computational speed is a critical constraint, such as for very large systems or web-based applications. © 2012 Ramu Anandakrishnan.