• Non-Boltzmann ensembles and Landau free energy

• # Fulltext

https://www.ias.ac.in/article/fulltext/conf/002/0008

• # Keywords

Monte Carlo methods; entropy; free energy; ensembles; random sampling; Metropolis algorithm; entropic sampling.

• # Abstract

Boltzmann sampling based on the 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 of microstates generated by Boltzmann Monte Carlo methods. However, a thermalproperty 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 cannot make such an assignment for entropy. Entropy is not a propertyassociated with any single microstate. It is a collective property of all the microstates. Towards calculating entropy and other thermal properties, a non-Boltzmann Monte Carlo technique called Umbrella sampling was proposedin the mid-seventies (of the last century). Umbrella sampling has since undergone several metamorphoses and we have now, multicanonical Monte Carlo, entropic sampling, flat histogram methods, Wang–Landau algorithm etc. This class of methods generates non-Boltzmann ensembles which are unphysical. However, physical quantities can be calculated by un-weighting the microstates of the entropic ensemble, followed by re-weighting to the desired physical ensemble.In this review we shall tell you of the Metropolis algorithm for estimating the mechanical properties and of the Wang–Landau algorithm for estimating both mechanical and thermal properties of an equilibrium system. We shall demonstrate the utility of non-Boltzmann Monte Carlo methods by calculating Landau free energy in a model system consisting of $q$-state Potts spins on a two-dimensional square lattice. The model exhibits a first-order phase transition for $q > 4$ and a second-order phase transition for $q \leq 4$. We report results on the Potts spin model for $q = 8$ (first-order phase transition) and for $q = 2$ (second-order phase transition). We also present the results on a more realistic problem of temperature-induced unbinding or denaturation of a hairpin DNA. The transition from a closed phase to an open phase is found to be first order.We shall attempt to make this review as pedagogical and self-contained as possible.

• # Author Affiliations

1. Department of Physical Sciences, Indian Institute of Science Education and Research Mohali, Sector 81, Sahibzada Ajit Singh Nagar, Manauli 140 306, India
2. Department of Physics, School of Physical Sciences, Central University of Rajasthan, Bandarsindri, N. H. 8 (Jaipur-Ajmer Highway), Kishangargh 305 817, Ajmer, India