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Theoretical studies in stochastic processes

University of British Columbia

A general method of analysis of a variety of stochastic processes in terms of probability density functions (PDFs) is developed and applied to several model as well as physically realistic systems. A model for diffusion in a bistable potential is the first system considered. The time dependence of the PDF for this system is described by a Fokker-Planck equation with non-linear coefficients. A numerical procedure is developed for finding the solution of this class of Fokker-Planck equations. The solution of the Fokker-Planck equation is obtained in terms of an eigenfunction expansion. The numerical procedure provides an efficient method of determining the eigenfunctions and eigenvalues of Fokker-Planck operators. The methods developed in the study of the model system are then applied to the trans-gauche isomerization of n-butane in CC1₄. This system is studied with the use of Kramers equation to describe the time evolution of the PDF. It is found that at room temperature the isomerization rate obeys a first order rate law. The rate constant for this system is sensitive to the collision frequency between the the n-butane and CC1₄ as has been previously suggested. It is also found that transition state theory underestimates the rate constant at all collision frequencies. However, the activation energy given by transition state theory is consistent with the activation energy obtained in this work. The problem of the escape of light constituents from planetary atmospheres is also considered. Here, the primary objective is to construct a collisional kinetic theory of planetary exospheres based on a rigorous solution of the Boltzmann equation. It is shown that this problem has many physical and mathematical similarities with the problems previously considered. The temperature and density profiles of light gases in the exosphere as well as their escape fluxes are calculated. In the present work, only a thermal escape mechanism was considered, although it is shown how non-thermal escape mechanisms may be included. In addition, these results are compared with various Monte-Carlo calculations of escape fluxes.

Text

2010-06-11

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http://hdl.handle.net/2429/25554

Chemistry

University of British Columbia

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