Classical diffusion of a particle in a one dimensional random potential

This thesis examines the topic of classical diffusion of a particle in the presence of disorder. The presence of disorder has the effect of subjecting the classical particle to an additional random potential and it is the form of this random potential that is of interest. We consider two forms of the random potential and calculate several disorder averaged quantities including the particles probability distribution which is described by the Fokker-Planck equation [1, 2] and the transport properties of the particle, including the mean-square displacement and the velocity and diffusion coefficients. The first part of the thesis deals with a random potential that is characterized by shortranged correlations and some constant term known as drift. This is a problem that was first formulated some thirty years ago by Sinai [3], who showed that for a particle with zero drift the mean-square displacement had the form (x\(^2\)(t)) ≈ ln\(^4\)(t). We employ a combination of Green’s functions, distribution functions and asymptotic matching to not only analytically re-produce this result, but also the expectation value of the probability distribution and all transport properties for an arbitrary value of drift, which is an original result. For the second half of the thesis we consider essentially the same problem again but with a random potential that has long-ranged logarithmic correlations. To solve the problem we use the renormalization and functional renormalization group techniques in an attempt to re-create known results in an effort to find a general method that can deal with such one-dimensional systems. We calculate the particles distribution function using a functional renormalization group approach, which we use to partially re-derive the phase transition in the first-passage time distribution.

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