Boltzmann Equation - The Force and Diffusion Terms

The Force and Diffusion Terms

Consider particles described by f, each experiencing an external force F not due to other particles (see the collision term for the latter treatment).

Suppose at time t some number of particles all have position r within element d3r and momentum p within d3p. If a force F instantly acts on each particle, then at time t + Δt their position will be r + Δr = r + pΔt/m and momentum p + Δp = p + FΔt. Then, in the absence of collisions, f must satisfy

f \left (\mathbf{r}+\frac{\mathbf{p}}{m} \Delta t,\mathbf{p}+\mathbf{F}\Delta t,t+\Delta t \right )\,d^3\mathbf{r}\,d^3\mathbf{p} = f(\mathbf{r},\mathbf{p},t)\,d^3\mathbf{r}\,d^3\mathbf{p}

Note that we have used the fact that the phase space volume element d3rd3p is constant, which can be shown using Hamilton's equations (see the discussion under Liouville's theorem). However, since collisions do occur, the particle density in the phase-space volume d3rd3p changes, so

\begin{align} dN_\mathrm{coll} & = \left(\frac{\partial f}{\partial t} \right)_\mathrm{coll}\Delta td^3\mathbf{r} d^3\mathbf{p} \\ & = f \left (\mathbf{r}+\frac{\mathbf{p}}{m}\Delta t,\mathbf{p} + \mathbf{F}\Delta t,t+\Delta t \right)d^3\mathbf{r}d^3\mathbf{p} - f(\mathbf{r},\mathbf{p},t)d^3\mathbf{r}d^3\mathbf{p} \\ & = \Delta f d^3\mathbf{r}d^3\mathbf{p} \end{align}

(1)

where Δf is the total change in f. Dividing (1) by d3rd3pΔt and taking the limits Δt → 0 and Δf → 0, we have

(2)

The total differential of f is:

\begin{align} d f & = \frac{\partial f}{\partial t}dt +\left(\frac{\partial f}{\partial x}dx +\frac{\partial f}{\partial y}dy +\frac{\partial f}{\partial z}dz \right) +\left(\frac{\partial f}{\partial p_x}dp_x +\frac{\partial f}{\partial p_y}dp_y +\frac{\partial f}{\partial p_z}dp_z \right)\\ & = \frac{\partial f}{\partial t}dt +\nabla f \cdot d\mathbf{r} + \frac{\partial f}{\partial \mathbf{p}}\cdot d\mathbf{p} \\ & = \frac{\partial f}{\partial t}dt +\nabla f \cdot \frac{\mathbf{p}dt}{m} + \frac{\partial f}{\partial \mathbf{p}}\cdot \mathbf{F}dt \end{align}

(3)

where ∇ is the gradient operator, · is the dot product,

\frac{\partial f}{\partial \mathbf{p}} = \mathbf{\hat{e}}_x\frac{\partial f}{\partial p_x} + \mathbf{\hat{e}}_y\frac{\partial f}{\partial p_y}+\mathbf{\hat{e}}_z\frac{\partial f}{\partial p_z}= \nabla_\mathbf{p}

is a shorthand for the momentum analogue of ∇, and êx, êy, êz are cartesian unit vectors.

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