Boundary-Induced Drift and Negative Mobility in Constrained Stochastic Systems

Abstract

We study overdamped stochastic dynamics confined by hard reflecting boundaries and show that the combination of boundary geometry and an anisotropic diffusion tensor generically generates directed motion. At the level of individual trajectories, the no-flux condition enforces an oblique reflection at the boundary, which produces a systematic drift parallel to the surface. The resulting local velocity takes the general form $v_B(\mathbf{x})=\mathbf{t}(\mathbf{x})^{\!\top}\mathbf{D}\,\mathbf{n}(\mathbf{x})$, determined by the diffusion tensor $\mathbf{D}$ and the local boundary geometry encoded in the normal $\mathbf{n}$ and tangent $\mathbf{t}$. While this boundary-induced drift is local, it can accumulate into a macroscopic response, depending on the statistics of boundary encounters. We illustrate how this local boundary-induced drift gives rise to macroscopic transport using a minimal one-dimensional dimer composed of two particles with unequal diffusion coefficients. The repeated collisions act as reflections in configuration space and lead to sustained center-of-mass motion, including regimes of absolute negative mobility under constant forcing.

Figures (1)

• Figure 1: Time evolution of the dimer center-of-mass position $X_{\mathrm{cm}}(t)$ under a fixed negative total generalized force $F_A+F_B<0$. For certain diffusion-coefficient ratios, the center of mass moves against the applied force, $X_{\mathrm{cm}}(t)$ increasing with time, demonstrating absolute negative mobility ($\textcolor{red}{\bigcirc}$, $\textcolor{blue}{\bigtriangledown}$). Other choices of $D_A$ and $D_B$ lead to conventional drift in the force direction ($\Box$,$\textcolor{BurntOrange}{\Diamond}$,$\textcolor{ForestGreen}{\bigtriangleup}$). The horizontal black line marks the initial center-of-mass position and highlights motion against the applied force. Symbols show numerical simulations, while thick solid lines are analytical predictions obtained from Eq.\ref{['eq:vcmAveragegl']} (with $c(t)$ given in Eq.(A14) of the SM). Specifically $\textcolor{red}{\bigcirc}$: ($D_A=0.5$, $D_B=0.25$), $\textcolor{blue}{\bigtriangledown}$: ($D_A=0.013$, $D_B=0.0075$), $\Box$: ($D_A=D_B=0.5$), $\textcolor{BurntOrange}{\Diamond}$: ($D_A=0.01$, $D_B=0.013$), and $\textcolor{ForestGreen}{\bigtriangleup}$: ($D_A=0.25$, $D_B=0.5$). For all cases $F_A=-1$ and $F_B=0.6$. The dashed line indicates the short-time behavior, which persists only in the isotropic case where $D_A=D_B$ ($\Box$). Results of numerical simulations were averaged over $10^5$ realizations and $\Delta t = 10^{-3}$ (additional details are in SM).