Random walks in a trapping environment

Achille Giacometti, Purdue University

Abstract

We discuss the statistics of random walks in the presence of a trapping environment, that is, when the environment can act as a trap. This model is also referred as ideal chain or static random walks. Peculiar features arise for diffusion in this model due to the presence of rare events which, however, play a fundamental role in the determination of the asymptotic behaviour. Using a combination of analytical and numerical techniques, we show the origin and effect of this pathological behaviour which is a common feature in many models with disorder and its difference from the kinetic random walks on disorder without trapping. New results for many quantities of interest are derived and physically interpreted. An alternative approach, based on the properties of the spectrum of the transition (or diffusion) matrix is also shown to give consistent results. An exact calculation for an analytical solvable hierarchical model is shown to give different behaviour for the quantities of interest. Finally an improved and self-consistent real space renormalization group approach for random walks is presented. This approach allows the study of the complete phase diagram for the problem of a random walk in the presence of a surface or defects but without disorder. The approach is valid in any dimension and allows the determination of the phase diagram with an arbitrary degree of accuracy. Problems arising from the extension of this approach to disordered systems are also discussed.

Degree

Ph.D.

Advisors

Nakanishi, Purdue University.

Subject Area

Condensation|Mathematics

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