Langevin forces and white-noise models

For a one-dimensional inertial particle, write $m\,dv=(-\zeta v+F)dt+\sqrt{2\zeta k_BT}\,dW_t$. Here zeta is a friction coefficient in kilograms per second, F is a resolved force, and $W_t$ is a standard Wiener process with independent increments of zero mean and variance dt. The random force itself is a generalized process, formally proportional to the time derivative of W. Its pointwise value and variance are not ordinary finite numbers. Measurements always average its effect over a nonzero interval.

The ideal delta correlation is appropriate when the bath correlation time is short compared with resolved times. At very short times molecular memory and particle inertia can matter; a colored-noise model or generalized friction kernel may be needed. The thermal amplitude above is tied to equilibrium bath temperature and friction. Choosing it independently of friction describes a different model and generally loses the thermal equilibrium distribution. Zero mean random force does not imply zero accumulated variance: independent kicks cancel in average displacement while their squared effects add.

At times long compared with $m/\zeta$, a suitable overdamped description is $dx=\mu Fdt+\sqrt{2D}\,dW_t$, with mobility $\mu=1/\zeta$. Mobility has units of velocity divided by force and D has units of length squared per time. For an isothermal equilibrium thermal bath and constant mobility, $D=\mu k_BT$. This Einstein relation is an assumption justified by thermal consistency, not a definition valid for every fluctuating environment. An actively driven bath can violate it even when its position trajectories appear diffusive.

For constant force and diffusion, the mean displacement grows linearly with time, while the centered variance is twice D times time in one dimension. The uncentered mean-square displacement also includes the square of the mean drift. Confusing these quantities leads to an apparent diffusion coefficient that increases with observation time. In several independent Cartesian directions each contributes its own variance; isotropic three-dimensional diffusion therefore has mean-square displacement six D times time after drift is subtracted. The factor two is already present in each coordinate's noise normalization.

Unless stated otherwise this module writes stochastic differential equations in the Itô convention. For $dx=a_I(x)dt+b(x)dW_t$, the coefficient of each increment is evaluated at its left endpoint. A Stratonovich equation uses a midpoint prescription and is marked with a circle on its stochastic differential. For constant b both conventions give the same paths and drift. For varying b they describe the same process only after the drift is changed by $a_I=a_S+bb'/2$, with the derivative taken with respect to x.

This drift conversion is not an optional numerical correction. Keeping the same displayed drift while switching interpretations generally changes the stochastic process and its stationary density. The ordinary chain rule holds in Stratonovich form; Itô's formula instead includes the quadratic-variation term. The correct interpretation of a reduced physical model follows from how fast degrees of freedom were eliminated, including the noise and friction timescales. Neither notation is universally more physical. A complete model declares the interpretation and any required noise-induced drift before computing predictions.