Probing subglass relaxation in polymers via a geometric representation of probabilities, observables, and relaxation modes for discrete stochastic systems

Accessible Molecular System Creator: Building Molecular Configurations Based on the Inaccessible Molecular Volume and Accessible Molecular Surface via Static Monte Carlo Sampling

Monte Carlo (MC) stochastic sampling is a powerful tool in classical molecular simulations that directly connects the observable macroscopic properties of matter and the underlying atomistic interactions. This connection operates within the framework of the statistical mechanics proposed by Gibbs. Most MC simulations are "dynamic," creating statistical ensembles of microstates via a Markovian chain, where each microstate in the ensemble depends only on its previous microstate. Herein, we re-examine an alternative form of MC that generates ensemble members through a "static" approach, building molecular systems stepwise. The basic theory for such an approach traces back to Rosenbluth and Rosenbluth, who proposed "static" stepwise sampling of a polymeric chain. It is almost as old as the Metropolis importance sampling approach used in dynamic MC, although the latter has been considerably more popular than the former. Herein, we address the main obstacle in static MC that has hindered the widespread adoption of Rosenbluth-based approaches in atomistic simulations. The obstacle lies in mapping the molecular accessible volume for adding a molecule in a Rosenbluth-like static sampling of atomistic configurations. We demonstrate a breakthrough by leveraging the ability to analytically map the inaccessible molecular volume and the accessible molecular surface owing to interatomically excluded volume interactions. This advance substantially enhances the ability to create molecular samples using a Rosenbluth-like static building process. The proposed approach can be used as a tool for creating initial configurations in MC or molecular dynamics simulations─a field where Rosenbluth-like static building has been applied. Additionally, this approach can be used as the first step in a perturbation scheme that accurately estimates free energy differences by estimating the chemical work related to molecule addition, removal, or reinsertion within the context of free energy perturbation schemes employed in molecular simulations.

Temporal disconnectivity of the energy landscape in glassy systems

An alternative graphical representation of the potential energy landscape (PEL) has been developed and applied to a binary Lennard-Jones glassy system, providing insight into the unique topology of the system's potential energy hypersurface. With the help of this representation one is able to monitor the different explored basins of the PEL, as well as how--and mainly when--subsets of basins communicate with each other via transitions in such a way that details of the prior temporal history have been erased, i.e., local equilibration between the basins in each subset has been achieved. In this way, apart from detailed information about the structure of the PEL, the system's temporal evolution on the PEL is described. In order to gather all necessary information about the identities of two or more basins that are connected with each other, we consider two different approaches. The first one is based on consideration of the time needed for two basins to mutually equilibrate their populations according to the transition rate between them, in the absence of any effect induced by the rest of the landscape. The second approach is based on an analytical solution of the master equation that explicitly takes into account the entire explored landscape. It is shown that both approaches lead to the same result concerning the topology of the PEL and dynamical evolution on it. Moreover, a "temporal disconnectivity graph" is introduced to represent a lumped system stemming from the initial one. The lumped system is obtained via a specially designed algorithm [N. Lempesis, D. G. Tsalikis, G. C. Boulougouris, and D. N. Theodorou, J. Chem. Phys. 135, 204507 (2011)]. The temporal disconnectivity graph provides useful information about both the lumped and the initial systems, including the definition of "metabasins" as collections of basins that communicate with each other via transitions that are fast relative to the observation time. Finally, the two examined approaches are compared to an "on the fly" molecular dynamics-based algorithm [D. G. Tsalikis, N. Lempesis, G. C. Boulougouris, and D. N. Theodorou, J. Chem. Theory Comput. 6, 1307 (2010)].

On solving the master equation in spatially periodic systems

We present a new method for solving the master equation for a system evolving on a spatially periodic network of states. The network contains 2(ν) images of a "unit cell" of n states, arranged along one direction with periodic boundary conditions at the ends. We analyze the structure of the symmetrized (2(ν)n) × (2(ν)n) rate constant matrix for this system and derive a recursive scheme for determining its eigenvalues and eigenvectors, and therefore analytically expressing the time-dependent probabilities of all states in the network, based on diagonalizations of n × n matrices formed by consideration of a single unit cell. We apply our new method to the problem of low-temperature, low-occupancy diffusion of xenon in the zeolite silicalite-1 using the states, interstate transitions, and transition state theory-based rate constants previously derived by June et al. [J. Phys. Chem. 95, 8866 (1991)]. The new method yields a diffusion tensor for this system which differs by less than 3% from the values derived previously via kinetic Monte Carlo (KMC) simulations and confirmed by new KMC simulations conducted in the present work. The computational requirements of the new method are compared against those of KMC, numerical solution of the master equation by the Euler method, and direct molecular dynamics. In the problem of diffusion of xenon in silicalite-1, the new method is shown to be faster than these alternative methods by factors of about 3.177 × 10(4), 4.237 × 10(3), and 1.75 × 10(7), respectively. The computational savings and ease of setting up calculations afforded by the new method of master equation solution by recursive reduction of dimensionality in diagonalizing the rate constant matrix make it attractive as a means of predicting long-time dynamical phenomena in spatially periodic systems from atomic-level information.