On solving the master equation in spatially periodic systems

Abstract

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.

KoBra: A rate constant method for prediction of the diffusion of sorbates inside nanoporous materials at different loadings

“…In all these applications, diffusion of adsorbates (benzene, p‐xylene, xenon, isobutene, methane, ethane, propane and generally hydrocarbons) inside nanoporous materials takes place, and a detailed description of this process is of a large interest. There are different methods to estimate the diffusion of adsorbates inside these materials, and a large number of experiments have been conducted for this purpose.…”

“…In the case of MD simulations, if the porous diameter is comparable with the kinetic diameter of the sorbate, the sorbate is trapped inside the pore, and diffusion jumps become extremely rare within the computational margins of a conventional MD simulation, which are usually of the order of nanoseconds. In these cases, MD simulation has to be used together with the transition state theory (TST) . TST is based on the calculation of transition rate constants, which describe the probability of a transition from one state to another state inside the porous network per one time unit .…”

“…The corresponding algorithm and its proof are presented in the Supporting Information (see part I). Finally, Kolokathis and Theodorou developed the master equation solution by recursive reduction dimensionality (MESoRReD) method, which is much faster than the KMC method. The disadvantage of this method is that it can be applied successfully for the orthogonal crystal systems at infinite dilution only, in contrast to the KMC method, which can be used for all crystal systems at different loadings.…”

“…Our program is developed with the hope to replace the KMC method for the case of diffusion calculation at infinite dilution, because, as we will show later, it is much faster than the KMC, and simultaneously the user can use it like a black box, without the need to understand the heavy mathematics of the method. In developing of this program, the formalism proposed by Kolokathis and Theodorou has been used.…”

Multidimensional master equation and its Monte-Carlo simulation

“…[2][3][4][5][6][7][8] This equation is often written artificially in terms of continue viewpoint and thus the transition function appearing in the ME needs to be determined by either the model or the experiment. 9, 10 For a given Markovian ME, Hänggi 11 presented the description of stochastically equivalent Langevin equation (LE).…”