Brownian motion in one-dimensional models: scaling, universality, and dispersionless transport

Kaffashnia, Amir (2023) Brownian motion in one-dimensional models: scaling, universality, and dispersionless transport. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

In the studies of Brownian motion, one-dimensional (1D) models play a special role in view of their relative simplicity and also because diffusion in a higher-dimensional space can often be decomposed into independent random motions in the orthogonal directions. In this thesis, we investigate the diffusion behavior of a Brownian particle (BP) within three 1D models: (i) diffusion along a stochastic harmonic oscillator chain (SHOC), (ii) diffusion of a damped BP in a tilted periodic potential, and (iii) free diffusion described by the Langevin equation with velocity-dependent damping. To address the first problem, we invent moving stochastic boundary condition approach, which allows us to simulate a small subset of oscillators in close proximity to the BP, effectively capturing the relevant dynamics. Our investigations have revealed a power law relation between the diffusion coefficient D and temperature T and the existence of dispersionless undamped phases in the BP motion at high temperatures. In the context of the second model, we explore the phenomenon of dispersionless transport of a BP in a tilted periodic potential, which is described in the literature as a broad time interval during which the particle’s dispersion appears to be constant. Our findings demonstrate that the dispersion fluctuations within the dispersion plateau hinder accurate determination of D, but these challenges can be remedied by employing an alternative measurement procedure. Moreover, it is evident that the conventional Langevin equation with velocity-independent damping coefficient as used in model (ii) cannot reproduce the nearly undamped ballistic flights observed in various physical systems. Lastly, we focus on establishing an analytical relation between D and T for a BP subject to velocity-dependent damping γ(v) and apply it to the case of a monotonically decreasing function γ(v) > 0. We show that at low temperatures, the D vs. T relation is linear; however, at high temperatures, a non-Einsteinian behavior emerges, similar to the one found within the SHOC diffusion model.

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