A Partition Function Approximation Using Elementary Symmetric Functions

Ramu Anandakrishnan,Juan P. Garrahan

doi:10.1371/journal.pone.0051352

Abstract

In statistical mechanics, the canonical partition function can be used to compute equilibrium properties of a physical system. Calculating however, is in general computationally intractable, since the computation scales exponentially with the number of particles 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 in operations. The DIA approximates the partition function as a combinatorial sum of products known as elementary symmetric functions (ESFs), which can be computed in 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.

Highlights

In statistical mechanics, the partition function Z, for a canonical ensemble of particles is given by XZ~ e{EX =kBT ð1Þ where kB is the Boltzmann constant, T is the temperature, EX is the energy of microstate X, and the sum is over all accessible microstates [1]
Computation time was measured by CPU time
A goal of the following analysis is to determine if and when the direct interaction algorithm (DIA) may be a practical alternative to the Monte Carlo (MC) method, for applications with computation time constraints, such as web services for computing protonation states in biomolecules [18,19]

Summary

Introduction

The partition function Z, for a canonical ensemble of particles is given by XZ~ e{EX =kBT ð1Þ where kB is the Boltzmann constant, T is the temperature, EX is the energy of microstate X , and the sum is over all accessible microstates [1]. The partition function can be used to calculate macroscopic thermodynamic properties of systems in equilibrium [2]. For applications where energy is a function of non-uniform interactions between particles, [3,4,5,6,7] such as the 3D Ising model for magnetic phase transition, the calculation of the partition function Z in Eq (1), has been shown to be computationally intractable, i.e. NP complete [8,9,10,11]. The DIA computes the exact contribution of direct interactions to the partition function, while using an average value for indirect interactions. In the hypothetical case where all the indirect interactions are equal, the partition function calculated by the DIA is exact to within numerical precision

Methods

Results

Conclusion