Non-Boltzmann Ensembles and Monte Carlo Simulation

TL;DR

This paper discusses non-Boltzmann Monte Carlo methods, such as umbrella sampling and Wang-Landau algorithm, which generate unphysical ensembles to compute thermal properties like entropy that are difficult to access with traditional Boltzmann sampling.

Contribution

It reviews recent developments in non-Boltzmann Monte Carlo techniques, highlighting their ability to calculate entropy and other thermal properties.

Findings

Non-Boltzmann methods generate unphysical ensembles.

Un-weighting and re-weighting enable calculation of physical quantities.

Recent algorithms improve sampling efficiency and accuracy.

Abstract

Boltzmann sampling based on Metropolis algorithm has been extensively used for simulating a canonical ensemble. An estimate of a mechanical property, like energy, of an equilibrium system, can be made by averaging over a large number microstates generated by Boltzmann Monte Carlo methods. However, a thermal property like entropy is not easily accessible to these methods. The reason is simple. We can assign a numerical value for energy to each microstate. But we can not assign a numerical value for entropy, to a microstate. Entropy is not a property associated with any single microstate.It is a collective property of allthe microstates. Toward calculating entropy and other thermal properties, a non-Boltzmann Monte Carlo technique called Umbrella sampling was proposed in the mid-seventies (of the last century). Umbrella sampling has since undergone several metamorphoses and we have now, multi-canonical Monte Carlo, entropic sampling, flat histogram methods, Wang-Landau algorithm {\it etc.} This class of methods generates non-Boltzmann ensembles which are un-physical. However, physical quantities can be calculated by un-weighting of the microstates of the entropic ensemble, followed by re-weighting to the desired physical ensemble.In this talk I shall tellyou of a few non-Boltzmann Monte Carlo methods with emphasis on recent developments.