Theses and Dissertations

Katja Biswas

Issuing Body

Mississippi State University

Advisor

Novotny, Mark A.

Committee Member

Lacy, Thomas E., Jr.

Committee Member

Gwaltney, Steven

Committee Member

Kim, Seong-Gon

Committee Member

Pierce, Donna M.

Date of Degree

12-9-2011

Document Type

Dissertation - Open Access

Major

Applied Physics

Degree Name

Doctor of Philosophy

College

James Worth Bagley College of Engineering

Department

Applied Physics Program

Abstract

This dissertation presents a theoretical study of arbitrary discretizations of general nonequilibrium and non-steady-state systems. It will be shown that, without requiring the partitions of the phase-space to fulfill certain assumptions, such as culminating in Markovian partitions, a Markov chain can be constructed which has the same macro-change of probability of the occupation of the states as the original process. This is true for any classical and semiclassical system under any discrete or continuous, deterministic or stochastic, Markovian or non-Markovian dynamics. Restricted to classical and semi-classical systems, a formalism is developed which treats the projection of arbitrary (multidimensional) complex systems onto a discrete set of states of an abstract state-space using time and ensemble sampled transitions between the states of the trajectories of the original process. This formalism is then used to develop expressions for the mean first passage time and (in the case of projections resulting in pseudo-one-dimensional motion) for the individual residence times of the states using just the time and ensemble sampled transition rates. The theoretical work is illustrated by several numerical examples of non-linear diffusion processes. Those include the escape over a Kramers potential and a rough energy barrier, the escape from an entropic barrier, the folding process of a toy model of a linear polymer chain and the escape over a fluctuating barrier. The latter is an example of a non- Markovian dynamics of the original process. The results for the mean first passage time and the residence times (using both physically meaningful and non-meaningful partitions of the phase-space) confirms the theory. With an accuracy restricted only by the resolution of the measurement and/or the finite sampling size, the values of the mean first passage time of the projected process agree with those of a direct measurement on the original dynamics and with any available semi-analytical solution.

URI

https://hdl.handle.net/11668/19097

Comments

mean first passage time||absorbing Markov chain||diffusion process||stochastic processes||master equation||probability||partitioning||non-ergodic systems||non-equilibrium and non-steady-state dynamics

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