A Monte Carlo Sampling Scheme for the Ising Model

Abstract

In this paper we describe a Monte Carlo sampling scheme for the Ising model and similar discrete state models. The scheme does not involve any particular method of state generation but rather focuses on a new way of measuring and using the Monte Carlo data.We show how to reconstruct the entropy S of the model, from which e.g the free energy can be obtained. Furthermore we discuss how this scheme allows us to more or less completely remove the effects of critical fluctuations near the critical temperature and likewise how it reduces critical slowing down. This makes it possible to use simple state generation methods like the Metropolis algorithm also for large lattices.

Reconstruction of the finite size canonical ensemble from incomplete micro-canonical data

“…Using these data and the exact K -function we can reconstruct the magnetisation and susceptibility at any temperature in that region using Eq. (5) and (6) adding data as prescribed in [5]. The reconstructed curves agree very well with the sampled data.…”

“…This will be demonstrated first in a case where we know the exact partition function, the Ising model on the 256 × 256 square lattice, and then for a case where we have ensemble nonequivalence: the 3-state Potts model on the 3-dimensional cubic lattice. All in all we find that with data collected with the methods of [5] in mind one can get a good picture of the canonical ensemble as well as the micro-canonical. In fact, thanks to knowing the density of states for a full interval of energies we will be able to reconstruct the canonical ensemble for all couplings in some interval rather than just those used in the sampling process.…”

“…The data were generated and collected by using the sampling method described in detail in [5] and we refer to that paper for further details. Since this model is conjectured to have a first order phase transition and cluster methods thus are expected to have exponential mixing time [15] we opted for a highly optimised single spin Metropolis method.…”

“…Such recent methods were given in [2] and [3], in [4] several such methods were given, and in [5] all of the later methods as well as several others were united in a common framework.…”

“…Among other things it governs the behaviour of many sampling algorithms and for systems where we have ensemble nonequivalence its dynamic can be very interesting. In order to reconstruct the canonical ensemble one would in principle need to know the density of states for all values of the energy E. However, using methods as in [5] this is very costly, and also not needed for work in the micro-canonical ensemble.…”