# Brownian Motion¶

The equation for Brownian Motion is

where \(R(t)\) is random force which requires

This equation is the linear Langevin equation.

The formal solution, which seems to be useless, is

Knowing that \(R(t)\) is random, or stochastic, we imediately realize that \(v(t)\) is stochastic. The thing to do is to find out the statistics.

This quantity squared is,

Average of the solution is

where these magenta colored terms are zero.

Hint

Note that the average here is ensemble average. Define probability density \(P(v,t)\) for velocity at time t. Recall that in velocity space, Liouville equation is

It’s very important to realize that this current density is the current density in velocity space. Applying \(d_t v = -\gamma vt R(t)\),

Warning

Now I am curious what ENSEMBLE average is. It’s not this probability density because there is time dependence.

Warning

What about Fokker-Plank equation?

By using the definition of random motion, the ensemble average shows us very simple results.

In the limit that time goes to infinity, we have

The legacy of Einstein is to measure Boltzmann constant using Brownian motion. The regorous derivation can be found in Physical Mathematics . The results, however, is that \(\avg{v(t)^2}\) is related to thermal energy of the particle and also related to the temperature according to equipartition theorem,

## Other Forms of Random Motion¶

### Wiener Process¶

Equation for this process is

Derived from this,

### Ornstein-Uhlenbeck Process¶

Change the velocity in Brownian motion to displacement,

Suppose we have \(\avg{v(0)}=0\), \(\avg{x^2}\) becomes constant.

## Harmonic Oscillators The Skeleton Key¶

A damped, driven harmonic oscillator is

Divided by \(\alpha\),

We understand immediately that this reduces to a Ornstein-Uhlenbeck equation if mass is very small.

Warning

Does it ring a bell of the WKB approximation?

.