(113d) Simulations of Phase Transitions, Metastability, and Nucleation Via Spatial Updating and Tempering Techniques

Spatial updating grand canonical Monte Carlo algorithms are generalizations of random and sequential updating algorithms for lattice systems to continuum fluid models. The elementary steps, insertions and removals, are constructed by generating points in space, either at random (random updating) or in a definite order (sequential updating). The type of move is deduced by examining the local environment in the neighborhood of the selected point. Simulation results indicate that these algorithms converge faster than standard grand canonical algorithms. The efficiency enhancement of spatial updating may be understood by the following physical reasoning: successful insertion (or removal) of a particle triggers a higher probability of a successful insertion (or removal) of another particle at a neighboring point in space. These neighbor type of effects are rarely observed in standard grand canonical algorithms.

Due to the nature of the updating and the absence of strict detailed balance, spatial updating algorithms based on sequential updates are ideal for parallel implementation via domain or geometric decomposition techniques. Spatial updating algorithms are only applicable in the grand canonical ensemble, and they are distinctly different from sequential updating of particles that is commonly employed in canonical Monte Carlo simulations. Due to particle indistinguishability, sequential updating of particles in canonical Monte Carlo simulations is identical to random updating. However, sequential updating of particles can be used to reduce inter-processor communication time in parallel canonical simulations. Simulation results on Lennard-Jones systems indicate a nearly perfect improvement in parallel efficiency for large systems both for grand canonical and canonical simulations.

Simulations of phase transitions are hampered by long relaxation times associated with the unbounded growth of correlations and fluctuations in the vicinity of the critical region. Although several attempts have been made, there is no universally accepted grand canonical Monte Carlo algorithm capable of reducing the critical slowing down effects associated with phase transitions. In this work, it is shown that a combination of spatial updating with tempering techniques can increase the simulation efficiency by several orders of magnitude. In simulated tempering, several macrostates are combined to form a super-ensemble or an expanded ensemble. Each macrostate comprises a grand canonical system at a given value of the chemical potential and temperature. The elementary steps consist of particle transfers as well as transitions between different macrostates. Particle transfers are implemented according to spatial updating and transitions between different macrostates according to a heat-bath algorithm. The combination of spatial updating with the heat-bath algorithm yields an efficiency enhancement which is 2-4 orders of magnitude higher compared to standard algorithms. The crucial parameters that determine the efficiency gain are the frequency of transitions and the number of different macrostates.

The previous algorithmic advances allow for much bigger system sizes to be simulated and have thus profound ramifications in data analysis via finite-size scaling techniques. These techniques have been applied in the determination of the fluid-fluid and fluid-solid phase diagrams of systems of competing interactions, a case that is relevant in globular protein crystallization. Due to the nature of the updating, spatial updating is efficient even at high, liquid-like densities and can lead to direct determination of fluid-solid coexistence. In addition, the proposed combination of spatial updating with simulated tempering has been used to study metastable fluid-fluid separation via simulations of constrained systems. The increased efficiency of the present methodology also allows it to be used in numerical studies of homogeneous and heterogeneous nucleation to obtain: (i) nucleation barriers, (ii) free energies of cluster formation, and (iii) the critical cluster size beyond which clusters (on the average) grow.