Multicanonical Simulations
Raúl Toral, Pere Colet · 2014
Multicanonical SimulationsSo far, our sampling method has allowed us to obtain good estimates for some quantities of interest, such as the magnetization m, the internal energy U, the specific heat C V , and so on.All these quantities can be expressed as averages with respect to the Boltzmann factor.However, the entropy cannot be written as an average and it cannot be obtained readily with the methods explained so far.This is where the multicanonical simulations appear.The general methodology is to sample distributions other than the Boltzmann factor and then, besides allowing for the calculation of the entropy, can be used to speed up or to extend the usual canonical simulations.There is a vast bibliography in this topic and we cannot even aim to summarize in a comprehensive way all recent developments in this area.Some topics that we cannot cover but are somewhat related to multicanonical simulations include the field of simulated tempering, where the temperature becomes a dynamical variable changing during the updates [77]; parallel tempering, also known under the names of replica exchange, exchange Monte Carlo, and multiple Markov chain Monte Carlo [78-80]; umbrella sampling [81]; broad histograms [82]; transition matrix Monte Carlo [83], and so on, for which we refer the reader to more specialized bibliography.Imagine we perform a Monte Carlo sampling and generate representative configurations X (1) , … , X (M) distributed according to the Boltzmann pdf f X(X ) = -1 e -𝛽(X) .In the process, we measure the energy of these configurationsThe values E (i) are distributed according to the pdf f (E) = -1 Ω(E)e -𝛽E , with Ω(E) being the number of configurations that have energy E. 1) Using the fact that the density of states is related to the entropy by Boltzmann's relation E) .With our simulation, we can produce a histogram H T (E) of the energy values.This histogram is an approximation (within the unavoidable statistical errors) to the true pdf f (E).We can use this histogram H T (E) ≈ -1 e -𝛽E+S(E) to estimate the entropy as S(E) ≈ S 0 + log H T (E) + 𝛽E (F.1) 1) We use, for simplicity, the case in which there is a numerable set of possible energies: E 0 , E 1 , ….If the energy can adopt any real value, Ω(E)dE is the number of configurations with energy in the interval (E, E + dE).Although in the main text we use the discrete notation, we might refer to Ω(E) as the ''density'' of states.2) More precisely, S(E) = k log Ω(E), with k the Boltzmann's constant.We adopt henceforth k = 1.