Langevin equation for systems with a preferred spatial direction

In this paper, we generalize the theory of Brownian motion and the Onsager–Machlup theory of fluctuations for spatially symmetric systems to equilibrium and nonequilibrium steady-state systems with a preferred spatial direction, due to an external force. To do this, we extend the Langevin equation to include a bias, which is introduced by an external force and alters the Gaussian structure of the system's fluctuations. In addition, by solving this extended equation, we provide a physical interpretation for the statistical properties of the fluctuations in these systems. Connections of the extended Langevin equation with the theory of active Brownian motion are discussed as well.

Figure 1

Histograms of two random walk simulations: (a) $N=25$, (b) $N=1000$. In panel (b) the Gaussian approximation is not plotted, because it is indistinguishable from the graph of the Gamma distribution when $N$ is so large. The Poisson distribution provides a poor approximation of the histogram data, since the probability of the forward step is not sufficiently small.