Numerically integrate equations of motion with noise

Consider the equations of motion

$$\begin{cases} \dot{x}(t) & = v(t) \\ \dot{v}(t) & = -\frac{\lambda}{m} v(t) + \frac{1}{m}F^{c}(x(t)) = a(x(t), v(t)) \end{cases},$$

where $x$ is the position, $v$ is the velocity, $m$ is the mass, $\lambda$ is the damping coefficient, $F^c(x(t))$ is a conservative force and $a$ is the net acceleration.

One way to numerically integrate the equations of motion is the Velocity Verlet algorithm (VV). Let $x_i = x(i\Delta t)$ and $v_i = v(i \Delta t)$, where $\Delta t > 0$ is the desired time step. VV steps are the following:

1. $x_{i+1} = x_i + v_i \Delta t + \frac{1}{2}a(x_i, v_i) \Delta t^2$
2. $v_{inter} = v_i + \frac{1}{2}a(x_i, v_i) \Delta t$
3. $v_{i+1} = v_{inter} + \frac{1}{2}a(x_{i+1}, v_{inter}) \Delta t$

How does this scheme change when we consider also the presence of a random force? Formally, the equations of motion are: $$\begin{cases} \dot{x}(t) & = v(t) \\ m\dot{v}(t) & = -\frac{\lambda}{m}v(t) + \frac{1}{m}F^{c}(x(t)) + \frac{1}{m}F^{r} = a(x(t), v(t), F^r) \end{cases},$$

where $F^r$ is a random force with zero mean and variance $\sigma^2$.

Moreover, given a generic numerical integrator

$$\begin{cases} x_{i+1} & = f(x_i, v_i) \\ v_{i+1} & = g(x_i, v_i) \end{cases}$$

how it changes when we consider also the presence of random forces?

• Google 'numerically integrate Langevin equations' – lemon May 6 '16 at 10:41