Gradientdrift

Gradientdrift is a term used in the study of stochastic processes to describe a drift component that can be written as the gradient of a scalar potential function. In a diffusion process described by an Itô equation dx_t = μ(x_t) dt + σ dW_t, gradientdrift refers to a drift μ(x) that takes the form μ(x) = ∇Φ(x) for some smooth potential Φ. A common convention in physics and chemistry uses a drift of the form μ(x) = -∇V(x), yielding overdamped Langevin dynamics dx_t = -∇V(x_t) dt + √(2D) dW_t, where D is a diffusion coefficient.

The gradient form is conservative: in a simply connected region the curl of μ vanishes, ∇ × μ = 0. This structure implies that the process is attracted to minima of the potential, and the stationary distribution, when diffusion is constant, is proportional to exp(-V(x)/D) (or exp(Φ/D) with the opposite sign convention). Gradientdrift thus often leads to detailed balance and reversible dynamics.

If the drift is not a gradient field, the curl is nonzero and the system can exhibit non-equilibrium behavior with circulating probability currents and broken detailed balance, even in steady state. This distinction helps explain when a stochastic model will converge to a predictable equilibrium distribution versus when it sustains ongoing fluxes.

Applications of gradientdrift appear across physics, chemistry, and related fields, including diffusion in energy landscapes, reaction coordinate modeling, and optimization-inspired stochastic processes. The concept also connects to broader ideas about gradient flows in probability spaces and the role of potential landscapes in dynamical systems.