The Equation of Motion

Consider a Brownian particle moving, under the influence of a uniform driving force $\vec{F}_0$, through the force field $\vec{F}(\vec{r})$ due to an inhomogeneous medium or landscape. Its trajectory is described by the Langevin equation (7,1)

$$\xi \, \frac{d \ensuremath{{\mathbf{r}}}\xspace }{dt} = \vec{F}(\ensuremath{{\mathbf{r}}}\xspace ) + \vec{F}_0 + \vec{\Gamma}(t),$$ (1)

where $\xi$ is the particle's viscous drag coefficient, and $\vec{\Gamma}$ describes random thermal fluctuations. This Langevin force satisfies $\langle \vec{\Gamma}(t) \rangle = 0$ and $\langle \vec{\Gamma}(t) \cdot \vec{\Gamma}(t + \tau) \rangle = \xi \, k_B T \, \delta(\tau)$ at temperature $T$, where $\delta(\tau)$ is the Dirac delta function. A sphere of radius $a$ immersed in an unbounded fluid of viscosity $\eta$, for example, has $\xi = 6 \pi \eta a$.

In the limit that $\vec{F}_0$ and $\vec{F}$ both greatly exceed the scale of thermal forces, $\vec{\Gamma}$, the Langevin equation, Eq. (1), reduces to a first-order deterministic equation of motion. This article focuses on two-dimensional systems, the simplest case exhibiting novel behavior. Even this deceptively simple system yields surprising results, as we will see.