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Next: 2.2 The Maxwell-Boltzmann distribution Up: 2. Elements of Kinetic Previous: 2. Elements of Kinetic

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.

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)
\end{displaymath} (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
\end{displaymath} (2.2)

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

\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)
\end{displaymath} (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}
\end{displaymath} (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
$\displaystyle \frac{\partial f(x,v_{x})}{\partial t}$ $\textstyle =$ $\displaystyle \frac{n_{in}-n_{out}}{dt  dx   dv_{x}}$ (2.6)
  $\textstyle =$ $\displaystyle \frac{f(x-v_{x}dt,v_{x})-f(x,v_{x})}{dt}$ (2.7)
  $\textstyle =$ $\displaystyle \frac{f(x,v_{x})-(v_{x}dt) (\partial f/\partial x) -f(x,v_{x}}{dt}$ (2.9)
  $\textstyle =$ $\displaystyle -v_{x}   \frac{\partial f(x,v_{x})}{\partial x}$ (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}}
\end{displaymath} (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.
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}
\end{displaymath} (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}^{'}\}   .
\end{displaymath} (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...
\left( f_{1}^{'}f_{2}^{'}-f_{1}f_{2} \right)
\end{displaymath} (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})
\end{displaymath} (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
$\textstyle \parbox{360pt}{
{\bf Simulation: The power of Boltzmann's...
...tion densities in r-space and in v-space
{\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
\end{displaymath} (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}) \}
\end{displaymath} (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}  ,
\end{displaymath} (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.
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Next: 2.2 The Maxwell-Boltzmann distribution Up: 2. Elements of Kinetic Previous: 2. Elements of Kinetic
Franz Vesely