A simple nonparametric procedure is devised for constructing Boltzmann entropy functions from statistically weighted entropy differences calculated from overlapping histograms. The method is noniterative, avoids numerical problems associated with large state densities, and accommodates variable bin widths for reducing systematic and statistical errors inherent to histogram techniques. Results show that the procedure can yield thermodynamic functions for an Ising spin lattice model that have average errors comparable to ones obtained from a conventional approach. Analysis of thermofunctions computed for a polyalanine peptide simulated by hybrid Monte Carlo replica exchange indicates that method performance can be enhanced through the use of nonuniform state space discretization schemes. An extension of the reweighting procedure for multidimensional applications is presented through calculations of vapor-liquid equilibrium densities of a model fluid simulated by grand canonical replica exchange.
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For this system, and , where and are spin variables equal to and signifies a sum over all nearest-neighbor spins. Periodic boundary conditions were used to remove edge effects of the finite lattice.
Replica exchange moves are not critical for equilibrating this Ising system. They are employed mainly to provide a consistent methodological framework. In addition, they provide exchange probabilities, which are a useful measure of histogram overlap.
, where and (the brackets ‘’ denote ensemble averages).
Two minor changes were made to the CHARMM program (Ref. 60). Net global rotation and translation elimination, which is carried out after Gaussian velocity assignment at the beginning of dynamics trajectories, was removed for proper implementation of the hybrid MC moves. Scalar arrays were enlarged for storage of current replica coordinates.
For simulations in the grand canonical ensemble, the probability distribution of the number of particles and the total potential energy may be written as , where is the system volume, is the partition function, and is a function of the chemical potential (Ref. 79). The particle pair potential describing the Lennard-Jones system is , where is the interparticle separation distance, is the potential well-depth, and is the separation corresponding to zero potential. Results reported for , , , and are expressed in units of , , , and , respectively.
It is noted that and in the sum denote and values of the same curve for constant whereas and in Eq. (6) correspond to curves for and .