Noise-induced drift in two-dimensional anisotropic systems

We study the isothermal Brownian dynamics of a particle in a system with spatially varying diffusivity. Due to the heterogeneity of the system, the particle's mean displacement does not vanish even if it does not experience any physical force. This phenomenon has been termed “noise-induced drift,” and has been extensively studied for one-dimensional systems. Here, we examine the noise-induced drift in a two-dimensional anisotropic system, characterized by a symmetric diffusion tensor with unequal diagonal elements. A general expression for the mean displacement vector is derived and presented as a sum of two vectors, depicting two distinct drifting effects. The first vector describes the tendency of the particle to drift toward the high diffusivity side in each orthogonal principal diffusion direction. This is a generalization of the well-known expression for the noise-induced drift in one-dimensional systems. The second vector represents a novel drifting effect, not found in one-dimensional systems, originating from the spatial rotation in the directions of the principal axes. The validity of the derived expressions is verified by using Langevin dynamics simulations. As a specific example, we consider the relative diffusion of two transmembrane proteins, and demonstrate that the average distance between them increases at a surprisingly fast rate of several tens of micrometers per second.

Figure 1

The local principal directions define an orthogonal curvilinear coordinate system with unit vectors ${\widehat{x}}_i(i=1,2)$ that are tangent to the coordinate curves $x_i=x_i\left(0\right)=\mathrm{const}$.

Figure 2

The mean displacements $\langle \mathrm{\Delta }{x}_1\rangle$ (solid line) and $\langle \mathrm{\Delta }{x}_2\rangle$ (dashed line) as a function of time, computed from Langevin dynamics simulations of $5\times {10}^5$ stochastic trajectories of a particle moving in an anisotropic system with $D_1=10x_1/{\left(x_2\right)}^2$ and $D_2=10/x_2$, and starting at $x_1=x_2=100$.

Figure 3

The mean displacement in the $x$ direction, $\langle \mathrm{\Delta }x\rangle$, as a function of time. The monotonically increasing (decreasing) curve depicts results for a particle starting at $(x,y)=(1,0)$ and moving in an anisotropic system with $D_r=1/25$, and $D_{\theta }=1/125$ ($D_r=1/125$ and $D_{\theta }=1/25$). The inset shows a magnification of the initial region $\mathrm{\Delta }t\le 15$, where the tangent dashed lines depict the asymptotic behavior $\langle \mathrm{\Delta }x\rangle \sim \pm (4/125)\mathrm{\Delta }t$, expected from Eq. (13).