Langevin Equation

The Langevin equation is a stochastic ODE describing the velocity $V$ of a particle subject to friction and random thermal forcing:

$$\frac{dV}{dt}=-kV+\xi (t)$$

where $\xi (t)$ is white noise with correlation

$$⟨\xi (t)\xi (t^{\prime })⟩=2\delta (t-t^{\prime })$$

Equivalence to the Ornstein-Uhlenbeck SDE

The Langevin equation is equivalent to the Ornstein-Uhlenbeck process SDE $dV=-kVdt+dW$. The connection is that $\xi (t)$ is the formal derivative of the Wiener process, $\xi (t)=dW/dt$, which does not exist as a classical function but is defined in the distributional sense.

Numerical Discretisation

In a time step $dt$, white noise is represented as $\xi (t_i)\to Z_i/\sqrt{dt}$ where $Z_i\sim N(0,1)$. Substituting into the Langevin equation and rearranging gives the Euler-Maruyama scheme:

$$V(t+dt)=(1-kdt)V(t)+\sqrt{dt}\cdot Z$$

Ornstein-Uhlenbeck process | Wiener process | Stochastic differential equation | Euler-Maruyama scheme | First-passage time