On solving the master equation in spatially periodic systems

Kolokathis, PD; Theodorou, DN

dc.contributor.author	Kolokathis, PD	en
dc.contributor.author	Theodorou, DN	en
dc.date.accessioned	2014-03-01T02:11:49Z
dc.date.available	2014-03-01T02:11:49Z
dc.date.issued	2012	en
dc.identifier.issn	00219606	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/29948
dc.subject.other	Alternative methods	en
dc.subject.other	Atomic levels	en
dc.subject.other	Computational requirements	en
dc.subject.other	Computational savings	en
dc.subject.other	Constant matrix	en
dc.subject.other	Diagonalizations	en
dc.subject.other	Diffusion tensor	en
dc.subject.other	Direct molecular dynamics	en
dc.subject.other	Dynamical phenomena	en
dc.subject.other	Eigenvalues and eigenvectors	en
dc.subject.other	Euler method	en
dc.subject.other	Kinetic monte carlo simulation	en
dc.subject.other	Kmc simulations	en
dc.subject.other	Low temperatures	en
dc.subject.other	Master equations	en
dc.subject.other	Numerical solution	en
dc.subject.other	Periodic boundary conditions	en
dc.subject.other	Recursive scheme	en
dc.subject.other	Reduction of dimensionality	en
dc.subject.other	Silicalite-1	en
dc.subject.other	Spatially periodic systems	en
dc.subject.other	Time-dependent	en
dc.subject.other	Transition state	en
dc.subject.other	Unit cells	en
dc.subject.other	Computational chemistry	en
dc.subject.other	Eigenvalues and eigenfunctions	en
dc.subject.other	Molecular dynamics	en
dc.subject.other	Silicate minerals	en
dc.subject.other	Xenon	en
dc.subject.other	Rate constants	en
dc.title	On solving the master equation in spatially periodic systems	en
heal.type	journalArticle	en
heal.identifier.primary	10.1063/1.4733291	en
heal.identifier.secondary	http://dx.doi.org/10.1063/1.4733291	en
heal.identifier.secondary	034112	en
heal.publicationDate	2012	en
heal.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 2ν 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 (2νn) × (2νn) 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 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 × 104, 4.237 × 103, and 1.75 × 107, 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. © 2012 American Institute of Physics.	en
heal.journalName	Journal of Chemical Physics	en
dc.identifier.doi	10.1063/1.4733291	en
dc.identifier.volume	137	en
dc.identifier.issue	3	en