Monte Carlo investigations of phase transitions: status and perspectives

Abstract

Using the concept of finite-size scaling, Monte Carlo calculations of various models have become a very useful tool for the study of critical phenomena, with the system linear dimension as a variable. As an example, several recent studies of Ising models are discussed, as well as the extension to models of polymer mixtures and solutions. It is shown that using appropriate cluster algorithms, even the scaling functions describing the crossover from the Ising universality class to the mean-field behavior with increasing interaction range can be described. Additionally, the issue of finitesize scaling in Ising models above the marginal dimension (d * = 4) is discussed.

Fast flat-histogram method for generalized spin models

“…IV; we will concentrate here on precision matters. For q = 10, we obtained T c (L) = 0.70699(5), 0.70491(5), 0.70300(2), 0.70278(1), 0.70164(1), 0.701328(4) and 0.701249(2) for L = 16,20,30,32,64,128, and 256, where T c was determined from the location of peaks of the specific heat. C v was computed directly from the estimated density of states, and then refined from an additional production run of length 10 7 MCS.…”

“…It must be noted that, with standard multicanonical methods based on single-spin updates, such huge sizes are simply intractable, since the largest size of 2 16 investigated in this work would demand several months of computation. As regards dynamic performance, we obtain a substantial reduction in the dynamic exponent, from e.g., z ∼ 1.35(3) to z ∼ 1.05(1) for q = 6 and σ = 0.7.…”

Collective motion of self-propelled particles interacting without cohesion

“…The FSS approach [44,45] involves the numerical estimation of various moments of the order parameter distribution as the linear system size L is systematically varied. Of particular interest are the variance…”

Condensed matter theory by computer simulation: from materials to chemical biology features

“…Subsequently the study of phase transitions (both of lattice models and of offlattice systems) has been a very important task of MC and MD methods, since analytical methods are rarely reliable for such problems [5][6][7]. However, a naive application of sampling techniques to such phenomena will also encounter some problems: at first order transitions, the states corresponding to the two competing phases are normally separated by huge free energy barriers in phase space [5][6][7].…”

“…This requires (with any standard algorithm) the observation of a large number of transitions back and forth between the two states. Conversely, at second order transitions the growth of the order parameter correlation length (which ultimately diverges when one approaches criticality) and the associated slowness of relaxation ("critical slowing down") have been longstanding obstacles to the quest for highly accurate results [5,6].…”