A methodology is presented for Monte Carlo simulation of fluids in a new ensemble that can be used to obtain phase coexistence properties of multicomponent systems from a single computer experiment. The method is based on performing a simulation simultaneously in two distinct physical regions of generally different densities and compositions. Three types of perturbations are performed, a random displacement of molecules that ensures equilibrium within each region, an equal and opposite change in the volume of the two regions that results in equality of pressures, and random transfers of molecules that equalize the chemical potentials of each component in the two regions. The method is applied to the calculation of the liquid-gas coexistence envelope for the pure Lennard-Jones (6, 12) fluid for several reduced temperatures from the vicinity of the triple point to close to the critical point (T* = 0·75 to T* = 1·30). Good overall agreement with previously available literature results is obtained, with some ...

/pdf/direct-determination-of-phase-coexistence-properties-of-2fy6jes3hr.pdf

Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble

We present a new estimator for computing free energy differences and thermodynamic expectations as well as their uncertainties from samples obtained from multiple equilibrium states via either simulation or experiment. The estimator, which we call the multistate Bennett acceptance ratio estimator (MBAR) because it reduces to the Bennett acceptance ratio estimator (BAR) when only two states are considered, has significant advantages over multiple histogram reweighting methods for combining data from multiple states. It does not require the sampled energy range to be discretized to produce histograms, eliminating bias due to energy binning and significantly reducing the time complexity of computing a solution to the estimating equations in many cases. Additionally, an estimate of the statistical uncertainty is provided for all estimated quantities. In the large sample limit, MBAR is unbiased and has the lowest variance of any known estimator for making use of equilibrium data collected from multiple states. We illustrate this method by producing a highly precise estimate of the potential of mean force for a DNA hairpin system, combining data from multiple optical tweezer measurements under constant force bias.

/pdf/statistically-optimal-analysis-of-samples-from-multiple-1is5zcxbzi.pdf

Statistically optimal analysis of samples from multiple equilibrium states

We present a new estimator for computing free energy differences and thermodynamic expectations as well as their uncertainties from samples obtained from multiple equilibrium states via either simulation or experiment. The estimator, which we term the multistate Bennett acceptance ratio (MBAR) estimator because it reduces to the Bennett acceptance ratio when only two states are considered, has significant advantages over multiple histogram reweighting methods for combining data from multiple states. It does not require the sampled energy range to be discretized to produce histograms, eliminating bias due to energy binning and significantly reducing the time complexity of computing a solution to the estimating equations in many cases. Additionally, an estimate of the statistical uncertainty is provided for all estimated quantities. In the large sample limit, MBAR is unbiased and has the lowest variance of any known estimator for making use of equilibrium data collected from multiple states. We illustrate this method by producing a highly precise estimate of the potential of mean force for a DNA hairpin system, combining data from multiple optical tweezer measurements under constant force bias.

/pdf/statistically-optimal-analysis-of-samples-from-multiple-3h4h5z0m8c.pdf

The Gibbs-ensemble Monte Carlo simulation methodology for phase equilibrium calculations proposed by Panagiotopoulos [1] is generalized and applied to mixture and membrane equilibria. An alternative derivation of the Gibbs simulation criteria based on the limiting distributions for the appropriate statistical mechanical ensembles is presented. The method is then generalized for the calculation of phase equilibria of mixtures by simulation in a constantpressure Gibbs ensemble and the calculation of equilibria across semipermeable membranes with an imposed osmotic pressure difference. The method is used to calculate phase equilibria for binary mixtures of Lennard-Jones molecules. Good agreement is found with published results obtained using other simulation techniques. The computer time required for the Gibbs method is only a small fraction of the corresponding requirement for previously available simulation techniques. Calculations for simple osmotic systems are performed for the first time by simulation, ...

Phase equilibria by simulation in the Gibbs ensemble

We propose a new effective Monte Carlo (MC) procedure for direct calculation of the free energy in a single MC run. The partition function of the expanded ensemble is introduced including a sum of canonical partition functions with a set of temperatures and additive factors (modification). Random walk in the space of both particle coordinates and temperatures provides calculation of free energy in a wide range of T. The method was applied to a primitive model of electrolyte including the region of low temperatures. In similar way other variants of expanded ensembles are constructed (e.g., over the number of particles N or volume V). Its facilities in quantum statistics (path integral Monte Carlo) and some other applications are also discussed.

New approach to Monte Carlo calculation of the free energy: Method of expanded ensembles

It is shown how certain thermodynamic functions, and also the radial distribution function, can be expressed in terms of the potential energy distribution in a fluid. A miscellany of results is derived from this unified point of view. (i) With g(r) the radial distribution function and Φ(r) the pair potential, it is shown that g exp (Φ/kT) may be written as a Fourier integral, or as a power series in r2 the terms of which alternate in sign. (ii) A potential‐energy distribution which is independent of the temperature implies an equation of state which is a generalization of a number of well‐known approximations. (iii) The grand partition function of the one‐dimensional lattice gas is obtained from thermodynamic arguments without evaluating a sum over states. (iv) If in a two‐dimensional honeycomb (three‐coordinates) lattice gas fr(r=0, 1, 2, 3) is the fraction of all the empty sites which at equilibrium are neighbored by exactly r filled sites, then at the critical density the values of all four of the f's ...

B. Widom

Papers

Some Topics in the Theory of Fluids