Restricted Canonical Ensemble Research Articles

The free energy of the metastable supersaturated vapor via restricted ensemble simulations. III. An extension to the Corti and Debenedetti subcell constraint algorithm.

In order to improve the sampling of restricted microstates in our previous work [C. Nie, J. Geng, and W. H. Marlow, J. Chem. Phys. 127, 154505 (2007); 128, 234310 (2008)] and quantitatively predict thermal properties of supersaturated vapors, an extension is made to the Corti and Debenedetti subcell constraint algorithm [D. S. Corti and P. Debenedetti, Chem. Eng. Sci. 49, 2717 (1994)], which restricts the maximum allowed local density at any point in a simulation box. The maximum allowed local density at a point in a simulation box is defined by the maximum number of particles Nm allowed to appear inside a sphere of radius R, with this point as the center of the sphere. Both Nm and R serve as extra thermodynamic variables for maintaining a certain degree of spatial homogeneity in a supersaturated system. In a restricted canonical ensemble, at a given temperature and an overall density, series of local minima on the Helmholtz free energy surface F(Nm, R) are found subject to different (Nm, R) pairs. The true equilibrium metastable state is identified through the analysis of the formation free energies of Stillinger clusters of various sizes obtained from these restricted states. The simulation results of a supersaturated Lennard-Jones vapor at reduced temperature 0.7 including the vapor pressure isotherm, formation free energies of critical nuclei, and chemical potential differences are presented and analyzed. In addition, with slight modifications, the current algorithm can be applied to computing thermal properties of superheated liquids.

Phase transitions in systems with non-additive long-range interactions

We consider spin systems with long-range interactions in a non-additive regime. When the non-additive scaling limit is employed, the energy and the entropy compete and the system exhibits some phase transitions. Such systems do not satisfy the additivity, which results in some unfamiliar properties related to phase transitions. In this paper, the concept of additivity and its consequences are explained, and recent progress on the statistical mechanics of long-range interacting systems is reviewed. It is shown that parameter space is clearly decomposed into three regions according to the stability of the uniform state predicted by the mean-field (MF) theory. Based on this parameter space decomposition, recent results on the exactness of the MF theory are explained. When the interaction is non-negative (ferromagnetic), the analysis of the MF theory is exact and a typical spin configuration is always uniform in the canonical ensemble. However, in the restricted canonical ensemble, i.e., the canonical ensemble with a restriction of the value of the magnetization, it is shown that the MF theory does not necessarily give the exact description of the system and phase transitions between the MF uniform states (MF phase) and the inhomogeneous states (non-MF phase) occur. A new finding is that when the interaction potential changes its sign depending on the distance, the non-MF phase appears even in the canonical ensemble.

Methods for calculating nonconcave entropies

Five different methods which can be used to analytically calculate entropies that arenonconcave as functions of the energy in the thermodynamic limit are discussed andcompared. The five methods are based on the following ideas and techniques:(i) microcanonical contraction, (ii) metastable branches of the free energy, (iii) generalizedcanonical ensembles with specific illustrations involving the so-called Gaussian and Betragensembles, (iv) the restricted canonical ensemble, and (v) the inverse Laplace transform. Asimple long-range spin model having a nonconcave entropy is used to illustrate eachmethod.

Self-diffusion coefficients of the metastable Lennard-Jones vapor

Self-diffusion coefficients of a metastable Lennard-Jones vapor were obtained using thememory function formalism and the frequency moments of the velocity autocorrelationfunction at reduced temperatures from 0.75 to 1.0. The radial density distribution functionsused to evaluate the second, fourth and sixth frequency moments of the velocityautocorrelation function were obtained from the restricted canonical ensemble Monte Carlosimulation (Corti and Debenedetti 1994 Chem. Eng. Sci. 49 2717). The self-diffusioncoefficients at reduced temperature 0.75 do not vary monotonically as the densityincreases, and for the other three temperatures the self-diffusion coefficients varynormally.

Radial density distribution of the metastable supersaturated vapor via restricted ensemble simulations

Extensive restricted canonical ensemble Monte Carlo simulations [D. S. Corti and P. Debenedetti, Chem. Eng. Sci. 49, 2717 (1994)] for the supersaturated Lennard--Jones (LJ) vapor were performed. These simulations were conducted at different densities and reduced temperatures from 0.7 to 1.0 and the radial density distribution functions were obtained, most of which are unavailable via integral equation theory due to phase separations. Among different constraints imposed on the system studied, the one with the local minimum of the excess free energy was taken to be the one that approximates the equilibrium state of the metastable LJ vapor. For the slightly saturated state points, where integral equation theory does have a solution, compared with our simulations, differences of the radial density distribution functions were found and they are attributed to ignoring density fluctuations in the integral equation theory.

The free energy of the metastable supersaturated vapor via restricted ensemble simulations. II. Effects of constraints and comparison with molecular dynamics simulations

Extensive restricted canonical ensemble Monte Carlo simulations [D. S. Corti and P. Debenedetti, Chem. Eng. Sci. 49, 2717 (1994)] were performed. Pressure, excess chemical potential, and excess free energy with respect to ideal gas data were obtained at different densities of the supersaturated Lennard-Jones (LJ) vapor at reduced temperatures from 0.7 to 1.0. Among different constraints imposed on the system studied, the one with the local minimum of the excess free energy was taken to be the approximated equilibrium state of the metastable LJ vapor. Also, a comparison of our results with molecular dynamic simulations [A. Linhart et al., J. Chem. Phys. 122, 144506 (2005)] was made.

Study of thermal properties of the metastable supersaturated vapor with the restricted ensemble

The pressure and internal energy data at different densities of the supersaturated Argon vapor at the reduced temperatures 0.7 and 0.8 are obtained by the restricted canonical ensemble Monte Carlo simulation method [D.S. Corti, P. Debenedetti, Chem. Eng. Sci. 49 (1994) 2717]. In order to maintain the system in its one-phase state, different constraints on the density fluctuations have been imposed, varying with densities approaching and beyond the spinodal. The results are comparable with a molecular dynamics simulation study [A. Linhart, C.C. Chen, J. Vrabec, H. Hasse, J. Chem. Phys. 122 (2005) 144506] on the same system. The current study verifies the conclusion drawn by the simulation work that clustering of Argon atoms exists even in the vicinity of the spinodal. Compared with molecular dynamics simulations, our method can give the equilibrium properties of a metastable fluid, for example internal energies.

The free energy of the metastable supersaturated vapor via restricted ensemble simulations

Pressure, excess chemical potential, and excess free energy, with respect to ideal gas data at different densities of the supersaturated Lennard-Jones particle vapor at the reduced temperature 0.7 are obtained by the restricted canonical ensemble Monte Carlo simulation method [D. S. Corti and P. Debenedetti, Chem. Eng. Sci. 49, 2717 (1994)]. The excess free energy values depend upon the constraints imposed on the system with local minima exhibited for densities below the spinodal density and monotonic variation for densities larger than the spinodal density. The results are compared with a molecular dynamics simulation study [A. Linharton et al., J. Chem. Phys. 122, 144506 (2005)] on the same system. The current study verifies the conclusion drawn by the simulation work that clustering of Lennard-Jones atoms exists even in the vicinity of spinodal. Our method gives an alternative to molecular dynamic simulations for the determination of equilibrium properties of a metastable fluid, especially close to the spinodal, and does not require a very large system to carry out the simulation.

Thermal boson expansions and dynamical symmetry

Abstract In this paper we study collective excitations of many-particle systems at finite temperature. We use three distinct ensembles: a restricted canonical ensemble, the normal canonical ensemble and the grand canonical ensemble. We apply the method of thermo-field dynamics which replaces the computation of thermal averages by quantum mechanics in a doubled Hubert space. We show how the use of dynamical symmetries may simplify matters; this is illustrated for the Lipkin model. For each of the ensembles the thermal hamiltonian has dynamical symmetry SU(2) × SU(2). This symmetry is violated by the thermal vacuum, which does not lie within a single irrep of this dynamical symmetry group. We find that for the restricted canonical ensemble we can classify the thermal state by a single irrep of a larger group (SU(4)), of which SU(2) × SU(2) is a subgroup. In order to be able to discuss the collective excitations we construct a boson mapping for the generators of the algebra SU(4). This is used to evaluate the boson expansion of the thermal hamiltonian, including the thermal RPA, for both the normal and deformed phase. Since the hamiltonian has a more restricted symmetry than the irreps we find Goldstone modes corresponding to the extra symmetry and two bosons corresponding to the SU(2) × SU(2) dynamical symmetry. For the canonical ensemble we find that a classification according to SU(4) is also convenient, even though the thermal state has components in several irreps. In the case of the grand canonical ensemble we show that the relevant algebra is U(4), where the thermal vacuum is completely contained within one irrep of mixed type (particle-hole or covariant-contragradient). We also construct the boson mapping for this irrep and derive the thermal boson expansion.