2.1 Boltzmann's Transport Equation

With his ``Kinetic Theory of Gases'' Boltzmann undertook to explain the properties of dilute gases by analysing the elementary collision processes between pairs of molecules.

The evolution of the distribution density in $\mu$ space, $f\left( \vec{r}, \vec{v}; t \right)$, is described by Boltzmann's transport equation. A thorough treatment of this beautiful achievement is beyond the scope of our discussion. But we may sketch the basic ideas used in its derivation.

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If there were no collisions at all, the swarm of particles in $\mu$ space would flow according to

$$f\left( \vec{r} + \vec{v} dt, \vec{v}+\frac{\vec{K}}{m} dt ; t+dt \right) =f\left( \vec{r}, \vec{v} ; t\right)$$ (2.1)

where $\vec{K}$ denotes an eventual external force acting on particles at point $( \vec{r}, \vec{v})$. The time derivative of $f$ is therefore, in the collisionless case,

$$\left( \frac{\partial}{\partial t} + \vec{v} \cdot \nabla_{\... ...{m} \cdot \nabla_{\vec{v}} \right) f( \vec{r}, \vec{v};t) = 0$$ (2.2)

where

$$\vec{v} \cdot \nabla_{\vec{r}} f \equiv v_{x}\frac{\partial ... ...c{\partial f}{\partial y} + v_{z}\frac{\partial f}{\partial z}$$ (2.3)

and

$$\frac{\vec{K}}{m} \cdot \nabla_{\vec{v}} f \equiv \frac{1}{m... ...artial v_{y}} +K_{z} \frac{\partial f}{\partial v_{z}} \right)$$ (2.4)

To gather the meaning of equation 2.2 for free flow, consider the collisionless, free flow of gas particles through a thin pipe: there is no force (i. e. no change of velocities), and $\mu$-space has only two dimensions, $x$ and $v_{x}$ (see Figure 2.2).

A very simple mu-space

At time $t$ a differential ``volume element'' at $(x,v_{x})$ contains, on the average, $f(x,v_{x})dx dv_{x}$ particles. The temporal change of $f(x,v_{x})$ is then given by

$$\frac{\partial f(x,v_{x})}{\partial t} = -v_{x} \frac{\partial f(x,v_{x})}{\partial x}$$ (2.5)

To see this, count the particles entering during the time span $dt$ from the left (assuming $v_{x}>0$, $n_{in}=f(x-v_{x}dt,v_{x})dx dv_{x}$ and those leaving towards the right, $n_{out}=f(x,v_{x})dx dv_{x}$. The local change per unit time is then

$$\frac{\partial f(x,v_{x})}{\partial t} = \frac{n_{in}-n_{out}}{dt dx dv_{x}}$$ (2.6)

$$= \frac{f(x-v_{x}dt,v_{x})-f(x,v_{x})}{dt}$$ (2.7) (2.8)

$$= \frac{f(x,v_{x})-(v_{x}dt) (\partial f/\partial x) -f(x,v_{x}}{dt}$$ (2.9) (2.10)

$$= -v_{x} \frac{\partial f(x,v_{x})}{\partial x}$$ (2.11)

The relation $\frac{\partial f}{\partial t}+v_{x} \frac{\partial f}{\partial x} =0$ is then easily generalized to the case of a non-vanishing force,

$$\frac{\partial f}{\partial t}+v_{x} \frac{\partial f}{\partial x} +\frac{K_{x}}{m} \frac{\partial f}{\partial v_{x}} =0$$ (2.12)

(this would engender an additional vertical flow in the figure), and to six instead of two dimensions (see equ. 2.2).

All this is for collisionless flow only.

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In order to account for collisions a term $(\partial f / \partial t)_{coll} dt$ is added on the right hand side:

$$\left( \frac{\partial}{\partial t} + \vec{v} \cdot \nabla_{\... ...vec{v};t) = \left( \frac{\partial f}{\partial t}\right)_{coll}$$ (2.13)

The essential step then is to find an explicit expression for $(\partial f / \partial t)_{coll}$. Boltzmann solved this problem under the simplifying assumptions that

- only binary collisions need be considered (dilute gas);

- the influence of container walls may be neglected;

- the influence of the external force $\vec{K}$ (if any) on the rate of collisions is negligible;

- velocity and position of a molecule are uncorrelated (assumption of molecular chaos).

The effect of the binary collisions is expressed in terms of a ``differential scattering cross section'' $\sigma(\Omega)$ which describes the probability density for a certain change of velocities,

$$\{ \vec{v}_{1},\vec{v}_{2}\} \rightarrow \{ \vec{v}_{1}^{'},\vec{v}_{2}^{'}\} .$$ (2.14)

($\Omega$ thus denotes the relative orientation of the vectors $( \vec{v}_{2}^{'}-\vec{v}_{1}^{'})$ and $(\vec{v}_{2}-\vec{v}_{1})$). The function $\sigma(\Omega)$ depends on the intermolecular potential and may be either calculated or measured.

Under all these assumptions, and by a linear expansion of the left hand side of equ. 2.1 with respect to time, the Boltzmann equation takes on the following form:

$$\left( \frac{\partial}{\partial t} + \vec{v}_{1} \cdot \nabl... ...v}_{2}\right\vert \left( f_{1}^{'}f_{2}^{'}-f_{1}f_{2} \right)$$ (2.15)

where $f_{1} \equiv f(\vec{r},\vec{v}_{1}; t)$, $f_{1}^{'} \equiv f(\vec{r},\vec{v}_{1}^{'}; t)$ etc. This integrodifferential equation describes, under the given assumptions, the spatio-temporal behaviour of a dilute gas. Given some initial density $f(\vec{r},\vec{v};t=0)$ in $\mu$-space the solution function $f(\vec{r},\vec{v};t)$ tells us how this density changes over time. Since $f$ has up to six arguments it is difficult to visualize; but there are certain moments of $f$ which represent measurable averages such as the local particle density in 3D space, whose temporal change can thus be computed.

Chapman and Enskog developed a general procedure for the approximate solution of Boltzmann's equation. For certain simple model systems such as hard spheres their method produces predictions for $f\left( \vec{r}, \vec{v}; t \right)$ (or its moments) which may be tested in computer simulations. Another more modern approach to the numerical solution of the transport equation is the ``Lattice Boltzmann'' method in which the continuous variables $\vec{r}$ and $\vec{v}$ are restricted to a set of discrete values; the time change of these values is then described by a modified transport equation which lends itself to fast computation.

The initial distribution density $f(\vec{r}, \vec{v}; 0)$ may be of arbitrary shape. To consider a simple example, we may have all molecules assembled in the left half of a container - think of a removable shutter - and at time $t=0$ make the rest of the volume accessible to the gas particles:

$$f(\vec{r}, \vec{v}, 0) = A \Theta (x_{0}-x) f_{0}(\vec{v})$$ (2.16)

where $f_{0}(\vec{v})$ is the (Maxwell-Boltzmann) distribution density of particle velocities, and $\Theta(x_{0}-x)$ denotes the Heaviside function. The subsequent expansion of the gas into the entire accessible volume, and thus the approach to the stationary final state (= equilibrium state) in which the particles are evenly distributed over the volume may be seen in the solution $f(\vec{r},\vec{v};t)$ of Boltzmann's equation. Thus the greatest importance of this equation is its ability to describe also non-equilibrium processes.

Applet BM: Start

$\parbox{360pt}{ \mbox{}\\ {\bf Simulation: The power of Boltzmann's... ...tion densities in r-space and in v-space \\ \vspace{2ex} {\small [Code: BM]} }$

The Equilibrium distribution $f_{0}(\vec{r},\vec{v})$ is that solution of Boltzmann's equation which is stationary, meaning that

$$\frac{\partial f(\vec{r},\vec{v};t)}{\partial t} = 0$$ (2.17)

It is also the limiting distribution for long times, $t \rightarrow \infty$.

It may be shown that this equilibrium distribution is given by

$$f_{0}(\vec{r},\vec{v}) = \rho(\vec{r}) \left[ \frac{m}{2 ... ...\left[ \vec{v}-\vec{v}_{0}(\vec{r})\right]^{2}/2kT(\vec{r}) \}$$ (2.18)

where $\rho(\vec{r})$ and $T(\vec{r})$ are the local density and temperature, respectively.

If there are no external forces such as gravity or electrostatic interactions we have $\rho(\vec{r})=\rho_{0}=N/V$. In case the temperature is also independent of position, and if the gas as a whole is not moving ($\vec{v}_{0}=0$), then $f(\vec{r},\vec{v})$ $= \rho_{0}f_{0}(\vec{v})$, with

$$f_{0}(\vec{v}) = \left[ \frac{m}{2 \pi k T} \right]^{3/2} e^{-mv^{2}/2kT} ,$$ (2.19)

This is the famous Boltzmann distribution; it may be derived also in different ways, without requiring the explicit solution of the transport equation. $\Longrightarrow$See next section, equ.2.30.