Brownian motion with velocity-dependent friction in a periodic potential

Abstract

This thesis investigates the stochastic dynamics of a Brownian particle in a one-dimensional periodic potential under velocity-dependent friction. Motivated by physical systems such as atomic diffusion on crystal surfaces and biological transport in structured environments, we explore how different forms of damping influence the particle’s motion, particularly its flight-length statistics and timedependent diffusion behavior. We consider three friction models: Coulomb (α = 0.5), Lorentzian (α = 1), and Gaussian damping. Each model introduces a distinct velocity dependence in the friction coefficient, which in turn affects the particle’s ability to escape potential wells and traverse the periodic landscape. The Langevin equation governing the particle’s dynamics is solved numerically using a second-order accurate Milsteintype integration scheme, adapted for multiplicative Gaussian white noise and implemented with spline-based interpolation for computational efficiency. Flight-length distributions are extracted from coarse-grained trajectories and analyzed across a range of temperatures. We find that the distributions follow a modified power-law with an exponential cutoff: P(l) = Al⁻ᴮ exp[−C(T) l], where the exponent 𝐵 is largely insensitive to temperature but increases with the steepness of the damping function, and the cutoff parameter 𝐷(𝑡) decreases with temperature following a power-law scaling. Gaussian damping yields the longest flights and the heaviest tails, indicating reduced energy dissipation at high velocities. The time-dependent diffusion coefficient 𝐷(𝑡) is computed by ensemble averaging over stochastic realizations. For Coulomb and Lorentzian friction, 𝐷(𝑡) saturates at long times, consistent with normal diffusion. In contrast, Gaussian damping leads to persistent growth in 𝐷(𝑡), revealing superdiffusive behavior even in the presence of a confining potential. Overall, this work provides new insights into how velocity-dependent damping shapes transport in periodic systems. The numerical framework developed here offers a tool for simulating non-equilibrium dynamics and can be extended to higher dimensions, interacting particles, and experimentally relevant conditions.