Langevin equation in external potential

[skateboarding]

I am trying to solve the following equation.

$$m\frac{du}{dt}=-\gamma u -\frac{dV}{dx} + A(t)$$
Where u is the momentum, x is position, V is the external potential dependent on position, and A is the random stochastic force dependent on time. There is no initial condition in this problem, but probably limiting conditions to get the constants of integration.

I am basically after the following quantities.

$$<\bigtriangleup u> and <\bigtriangleup u^2>$$
Where the brackets denote the ensemble average.

I tried solving the equation for u(x,t), by Fourier transforming both sides into frequency domain, but didn't know how to get it back to the time domain since the potential functions are not defined. Separation of variables would not work due to the potentials as well.

So my question is, can I get an analytical solution for the momentum in terms of the potentials? If not, will the following work?

Attempt sort of... for $$<\bigtriangleup u>$$

the equation at the top can be written as

$$\lim_{\bigtriangleup t\to 0 } m\frac{\bigtriangleup u}{\bigtriangleup t} = -\gamma u(t +\bigtriangleup t) -\frac{\partial V}{\partial x} + A(t + \bigtriangleup t)$$

expanding the right side around t and multiplying both sides by delta t we get

$$\lim_{\bigtriangleup t\to 0 } m\bigtriangleup u = -\gamma u(t)\bigtriangleup t -\frac{\partial V}{\partial x}\bigtriangleup t + A(t)\bigtriangleup t + O (\bigtriangleup t^2)$$

I can then take the average of this, where the average of A(t) is assumed to be zero.

For the second moment, I can't just take the expression for delta t and square it can I? Thanks.

Homework Equations

The Attempt at a Solution

 

[member 553083]

Yes, you can square the expression for $\Delta t$ to find the second moment. However, it is not possible to get an analytical solution for the momentum in terms of the potentials. In order to determine the momentum given the potentials, one must solve the equation numerically or by simulation.