Brownian motion near a wall: the dilemma of Itô or Stratonovich

Abstract

Coordinate or state-dependent diffusion manifests under various non-equilibrium as well as equilibrium conditions. In the equilibrium scenario, a simple case occurs in the vicinity of walls confining to the liquid in which a Brownian particle (BP) undergoes thermal excursions. Coordinate-dependent diffusion is observed in such cases at constant ambient temperature. The present topical review focuses on issues with modelling this case of coordinate-dependent diffusion of a BP near a wall in the presence of constant temperature. The phenomenon is important to understand structural transitions and transport in various complex systems across disciplines. Important kinematic consequences of such coordinate-dependent diffusion exist, which cannot be observed in the homogeneous case. In this context, the present review attempts to compare stochastic methods based on the conventions of Itô and Stratonovich. In this Review, a well-known construction of the Langevin equation on the molecular kinetic basis of Brownian motion is discussed, followed by an evaluation of competing stochastic conventions in relation to the fluctuation–dissipation theorem, which is the physical basis of the local Langevin equation. Important experimental results are discussed, and distributions of Itô and anti-Itô processes are compared with the results of an experiment. An exact result of the Itô process resulting in spontaneous filtering of Brownian motion in the presence of coordinate-dependent diffusion is discussed.