Subsections

• Markov Systems
  • Time-dependent Distribution Function
  • Transfer Probabilities
  • Property of the Transfer Function
• Chapman-Kolmogorov Equation
• Expansion
• Focker-Planck Equation
• Calculation of Coefficients
• Large Viscosity. Smoluchowski Equation

Diffusion Approach. Focker-Planck Equation

Markov Systems

Time-dependent Distribution Function

In equilibrium thermodynamics we introduced distribution function $\rho(\tilde p ,\tilde q)$: the probability to have momenta around $\tilde p$ and coordinates around $\tilde q$ is

$$\Prob(\tilde p, \tilde q) = \rho(\tilde p ,\tilde q)\,d\tilde p \, d\tilde q$$

• We will study one Brownian particle--one coordinate $\mathbf{r}$ and one velocity $\mathbf{v}$
• We want a time-dependent solution--we must introduce t

We introduce a time-dependent distribution function $\rho(\mathbf{r},\mathbf{v},t)$--the probability to be in certain place with certain velocity at certain time!

Transfer Probabilities

If the Brownian motion is the slowest mode in the system, there exists such time $\Delta t$ that:

1.

$\Delta t$ is large enough, so there are many molecular collisions during this time

2.

$\Delta t$ is small enough, so $\mathbf{v}$ and $\mathbf{r}$ do not change ``much''

We know the state of our particle at time t: $\mathbf{r}$ and $\mathbf{v}$. What is its state at the time $t+\Delta t$?

Assumption:

The state of the system at the moment $t+\Delta t$depends only on the state at the moment t. The system has short memory.

Definition:

A system satisfying this assumption is called Markov system. We assume Brownian motion to be Markovian.

We introduce $\psi(\mathbf{r},\mathbf{v},\Delta\mathbf{r},\Delta\mathbf{v},\Delta t)$--the probability to go from state 1 to state 2:

$$\mathbf{r},\mathbf{v}\xrightarrow[\Delta t]{\psi(\mathbf{r}... ...lta t)} \mathbf{r}+\Delta\mathbf{r},\mathbf{v}+\Delta\mathbf{v}$$

or, introducing the variable $u=(\mathbf{r},\mathbf{v})$

$$u\xrightarrow[\Delta t]{\psi(u,\Delta u,\Delta t)} u+\Delta u$$

The function $\psi$ is called transfer function. The evolution of Markov system depends only on transfer function and initial state.

Property of the Transfer Function

Whatever is the state you started from, you will finish at some state with probability 1:  

$$\int \psi(u,\Delta u,\Delta t)\,d\Delta u=1$$ (8)

Chapman-Kolmogorov Equation

What is the probability to find particle at the point u at the time $t+\Delta t$? If it was at the point $u-\Delta u$ at time t, the probability is $\psi(u-\Delta u,\Delta u,\Delta t)$. The probability to be at $u-\Delta u$was $\rho(u-\Delta u)$. We obtain:  

$$\rho(u,t+\Delta t)=\int \rho(u-\Delta u,t)\psi(u-\Delta u,\Delta u,\Delta t)\,d\Delta u$$ (9)

This is called Chapman-Kolmogorov (or master) equation.

Expansion

Now we will use the fact that $\Delta t$ is small. In the left hand side of (9) we have:

$$\rho(u,t+\Delta t) = \rho(u,t) + \frac{\partial\rho}{\partial t}\Delta t +\ldots$$

In the right hand side (do not forget that u is a vector!):

$$\begin{split} &\rho(u-\Delta u,t)\psi(u-\Delta u,\Delta u,\... ...igl(\rho(u)\psi(u,\Delta u,\Delta t)\Bigr) +\ldots \end{split}$$

Integration:

1.

First term gives

$$\begin{split} &\int \rho(u,t)\psi(u,\Delta u,\Delta t)\,d\D... ... \psi(u,\Delta u,\Delta t)\,d\Delta u = \rho(u,t) \end{split}$$

2.

Second term gives:

$$\begin{split} &\int\Delta u_i \frac{\partial}{\partial u_i... ...rho(u)\left\langle \Delta u_i\right\rangle\Bigr) \end{split}$$

3.

Third term gives:

$$\begin{split} &\int\Delta u_i\Delta u_j \frac{\partial^2}{\... ...t\langle \Delta u_i\Delta u_j\right\rangle\Bigr) \end{split}$$

Focker-Planck Equation

We obtained:  

$$\frac{\partial\rho}{\partial t} = -\sum_i \frac{\partial}{\... ...{\partial^2}{\partial u_i\partial u_j}\Bigl(D_{i,j}\rho\Bigr)$$ (10)

with drift coefficients

$$A_i = \frac{\left\langle \Delta u_i\right\rangle}{\Delta t}$$

and diffusion coefficients

$$D_{i,j} = \frac{\left\langle \Delta u_i\Delta u_j\right\rangle}{\Delta t}$$

This is called Focker-Planck equation. It describes diffusion in phase space.

The evolution is described by the moments $\left\langle \left(\Delta u\right)^n\right\rangle$. In the Focker-Planck equation we use only the first and second moments. In principle, we can construct an approximation based on higher moments (Kramers, 1940).

Calculation of Coefficients

For Brownian motion we have:

1.

Linear terms are

$$\frac{\left\langle \Delta\mathbf{r}\right\rangle}{\Delta t} ... ...\Delta\mathbf{v}\right\rangle}{\Delta t} = -\lambda\mathbf{v}$$

2.

Cross terms are zero:

$$\left\langle r_{\alpha}v_{\beta}\right\rangle=0$$

3.

Quadratic in $\mathbf{r}$ term is:

$$\left\langle (\Delta\mathbf{r})^2\right\rangle=O((\Delta t)^2)$$

4.

Quadratic in $\mathbf{v}$ term is

$$\left\langle (\Delta\mathbf{v})^2\right\rangle=\int_0^{\Delt... ...\mathbf{f}(\tau')\right\rangle = \frac{6kT\lambda}{m}\Delta t$$

Result (summation over $\alpha$ & $\beta$ is implied):

$$\frac{\partial\rho}{\partial t} = - \frac{\partial (v_{\alp... ...ial^2}{\partial v_{\alpha}\partial v_{\beta}}\Bigl(D\rho\Bigr)$$

with

$$D = \frac{kT\lambda}{m}$$

Large Viscosity. Smoluchowski Equation

If viscosity is large, we can assume that $\mathbf{v}$ quickly relaxes to equilibrium value:

$$\rho(\mathbf{r},\mathbf{v})=\phi(\mathbf{r})e^{-m\mathbf{v}^2/2kT}$$

Integrating Focker-Planck equation, we can obtain diffusion equation

$$\frac{\partial\phi}{\partial t} = D\nabla^2\phi$$