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Theory, Molecular, Mesoscopic Simulations, and Experimental Techniques of Aqueous Phase Adsorption
Published in Jayant K. Singh, Nishith Verma, Aqueous Phase Adsorption, 2018
Jayant K. Singh, Nishith Verma
Numerous methods have been proposed to unbias and recombine the results from umbrella sampling calculations, a promising one being the WHAM [28–30]. It aims to minimize the statistical error of unbiased distribution of function and this approach is routinely used to calculate the PMF along single coordinate: Piu(ξ)=∑i=1NwniPib(ξ)×[∑i=1Nwnje−β[wi(ξ)−Fi]]−1
Trajectory reweighting for non-equilibrium steady states
Published in Molecular Physics, 2018
Patrick B. Warren, Rosalind J. Allen
The Boltzmann factor, which describes exactly the relative probability of microstates at equilibrium in systems whose dynamics obeys detailed balance, forms the cornerstone of a plethora of simulation methods in the physical sciences. For example, the seminal Metropolis–Hastings algorithm for Monte-Carlo simulation exploits the Boltzmann factor to generate a trajectory of configurations which sample the Gibbs–Boltzmann distribution [1]. Knowledge of the Boltzmann factor also makes possible a host of biased sampling methods, which allow efficient characterisation of rugged free energy landscapes comprising multiple free energy minima separated by barriers. In these methods, information on a target system of interest is obtained by simulating a reference system, whose microstate probabilities are biased to be different from the target system. The results are corrected for the bias by reweighting with, for example, a Boltzmann factor. The reference system is typically easier to sample than the target system. Thus in umbrella sampling [2], an external potential is used to coerce the reference system (or a sequence of such systems) to sample a free energy barrier. The basis of biased sampling schemes is the generic relation where is some quantity of interest (an order parameter for example), the brackets refer to an average over microstates for the reference system, the brackets refer to an average for the target system and W is a reweighting factor Here, the ratio is the relative probability of observing the microstate , where and are the steady-state (superscript ‘∞’) probability distributions for the reference and target systems, respectively. For systems whose dynamics obeys detailed balance this ratio is given analytically by the Boltzmann factor (up to an overall constant of proportionality). Thus, Equation (1) provides a way to compute averages over the target system from a simulation of the reference system: during the simulation, one simply tracks the quantity W and uses it to reweight the average of the quantity of interest Θ. The constant of proportionality does not need to be calculated since it cancels in Equation (1).