Renormalized Multicanonical Sampling in Two-Dimensional Systems

Wolfhard Janke E

Yong Hwan Lee, David O. Yevick

This paper extends to two and three dimensions the recently proposed renormalized multicanonical sampling procedure [D. Yevick, Int. J. Mod. Phys. C 27, 1650033 (2016)IJMPEO0129-183110.1142/S0129183116500339], which generates the density of states of a system from that of a smaller system. The accuracy and efficiency of the method is additionally enhanced through a transition matrix Monte Carlo method that incorporates multicanonical sampling. The performance of the resulting procedure is investigated through calculations of the specific heat of the standard Ising model for square lattices with up to 128^{2} spins and for cubic lattices with up to 24^{3} spins. A comparison of the scaling properties to those of other algorithms based on the transition matrix Monte Carlo method demonstrates that the relative advantage of the renormalized procedure increases rapidly with system size.

Robert H. Swendsen

David O. Yevick

For a homogeneous system divisible into identical, weakly interacting subsystems, the multicanonical procedure can be accelerated if it is first applied to determine the density of states for a single subsystem. This result is then employed to approximate the state density of a subsystem with twice the size that forms the starting point of a new multicanonical iteration. Since this compound subsystem interacts less on average with its environment, iterating this sequence of steps rapidly generates the state density of the full system.

Robert H. Swendsen

M. L. GUERRA, Jose Daniel Muñoz

This work measures the time to equilibrium for the multicanonical method on the 2D-Ising system by using a new criterion, proposed here, to find the time to equilibrium, t eq , of any sampling procedure based on a Markov process. Our new procedure gives the same results that the usual one, based on the magnetization, for the canonical Metropolis sampling on a 2D-Ising model at several temperatures. For the multicanonical method we found a power-law relationship with the system size, L, of t eq =0.27(15) L2.80(13) and with the number of energy levels to explore, k E , of [Formula: see text], in perfect agreement with the above result. In addition, a kind of critical slowing down was observed around the critical energy. Our new procedure is completely general, and can be applied to any sampling method based on Markov processes.