We propose a replica-exchange method (REM) which does not use pseudo random numbers. For this purpose, we first give a conditional probability for Gibbs sampling replica-exchange method (GSREM) based on the heat bath method. In GSREM, replica exchange is performed by conditional probability based on the weight of states using pseudo random numbers. From the conditional probability, we propose a new method called deterministic replica-exchange method (DETREM) that produces thermal equilibrium distribution based on a differential equation instead of using pseudo random numbers. This method satisfies the detailed balance condition using a conditional probability of Gibbs heat bath method and thus results can reproduce the Boltzmann distribution within the condition of the probability. We confirmed that the equivalent results were obtained by REM and DETREM with two-dimensional Ising model. DETREM can avoid problems of choice of seeds in pseudo random numbers for parallel computing of REM and gives analytic method for REM using a differential equation.Keywords: generalized-ensemble algorithm, replica-exchange method (REM), simulated tempering (ST), Monte Carlo (MC) simulation, differential equation, gibbs sampling, heat-bath method, conditional probability, pseudo random numbers, Ising model
INTRODUCTIONThe enhancement of sampling during Monte Carlo (MC) and molecular dynamics (MD) simulations is very important for complex systems. Replica-exchange method (REM) (or parallel tempering) is one of the most popular ways to improve sampling efficiency [1][2][3][4] including biomolecular system in explicit solvent [5,6] or biomembrane [7,8] (for reviews, see, e.g., Refs. [9,10]). To realize a thermal equilibrium distribution, REM uses Metropolis criterion with pseudo random numbers. However, random numbers sometimes give inaccurate results of simulations [11]. Moreover, generation of high quality random numbers is often difficult and does not assure good simulation results [12]. REM and its extension is suited for parallel computing [13][14][15][16]. Most of pseudo random number generators decrease the scalability in parallelization [17]. Hence, the complementary method producing the same results without pseudo random numbers is meaningful.In addition, the analytic approach for temperature selections in REM have been performed [18,19]. For performance and the condition of REM, several works were also performed. For example, Nymeyer [20] showed how efficient REM is than conventional simulations using the number of independent configurations. Abraham and Gready introduced some measurement and compared the results [21]. Rosta and Hummer [22] evaluated the practical efficiency of REM simulation for protein folding with a two-state model. However, the examination of the condition for convergence of REM is difficult partly because the mixing of temperature in REM is determined by pseudo random numbers with Metropolis criteria. As a result, most of analyses estimated the REM performance by simulation results.Recently, Suzuki et...