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Boltzmann's Equation and its solution

Figure 8.1: The phase-space element is six-dimensional and selects
particles that lie in a space element ${d}^{3}x$ and simultaneously in
a velocity element ${d}^{3}v$.

We use a statistical description originally invented by James Clerk
Maxwell called the "distribution function". The distribution
function is a quantity $f(\mathit{v},\mathit{x},t)$ that is a function of
velocity $\mathit{v}$, position $\mathit{x}$, and time $t$. The distribution
function is defined by considering an element in
$f(\mathit{v},\mathit{x},t){d}^{3}v{d}^{3}x.$ | $(8.1)$ |

$f(\mathit{v},\mathit{x},t)=n(\mathit{x},t){\left(\frac{m}{2\mathit{\pi}kT}\right)}^{3/2}\mathrm{exp}(-\frac{{\mathit{mv}}^{2}}{2\mathit{kT}}).$ | $(8.2)$ |

$\mathrm{exp}(-\frac{{\mathit{mv}}^{2}}{2\mathit{kT}})=\mathrm{exp}(-\frac{{\mathit{mv}}_{x}^{2}}{2\mathit{kT}})\mathrm{exp}(-\frac{{\mathit{mv}}_{y}^{2}}{2\mathit{kT}})\mathrm{exp}(-\frac{{\mathit{mv}}_{z}^{2}}{2\mathit{kT}}).$ | $(8.3)$ |

Figure 8.2: A Maxwellian distribution function in two dimensions
displayed as a perspective view of the surface $f({v}_{x},{v}_{y})$, with
velocities $({v}_{x},{v}_{y})$ normalized to the thermal velocity
$\sqrt{2T/m}$. This can be considered to be the proportional to
the distribution at a fixed value of ${v}_{z}$, since the Maxwellian
is separable.

The factor ${\left(\frac{m}{2\mathit{\pi}\mathit{kT}}\right)}^{3/2}$ normalizes the
distributions in the three velocity dimensions. It is equal to the
inverse of the integral of eq. (8.3) over all
velocities. Therefore the leading term $n(\mathit{x},t)$ is just the
density in space (not phase-space). It might vary with
position or time. The Maxwellian distribution is what occurs in
$\mathit{\Gamma}=n\mathit{v}=\int \mathit{v}f(\mathit{v},\mathit{x},t){d}^{3}v.$ | $(8.4)$ |

$E=\frac{3}{2}nkT=\int \frac{1}{2}m{v}^{2}f{d}^{3}v.$ | $(8.5)$ |

${f}_{x}({v}_{x})=\int f(\mathit{v}){\mathit{dv}}_{y}{\mathit{dv}}_{z},$ | $(8.6)$ |

Figure 8.3: In phase-space, velocity $v$ carries a particle in the
$x$-direction, acceleration carries it in the
$v$-direction. Particle flux out of an element $\mathit{dvdx}$ arises from
the divergence of the fluxes $\mathit{fv}$ and $\mathit{fa}$ in the respective
directions

As time passes, particles move in phase-space. The rate of change of
$x$ is the velocity $\mathit{dx}/\mathit{dt}=v$. The rate of change of velocity $v$ is
acceleration $\mathit{dv}/\mathit{dt}=a$. Generally acceleration arises from force (per
particle) divided by particle mass. The force might be gravity, or
(for charged particles) electric or magnetic field. An individual
particle thus moves through the phase-space (in the $\mathit{xv}$ plane of our
diagram). If we therefore consider some phase-space volume we can
write the conservation of particles within it just as we did for the
fluid continuity equation as
$\frac{\partial f}{\partial t}+{\nabla}_{\mathit{ps}}.(f{\mathit{v}}_{\mathit{ps}})=\frac{\partial f}{\partial t}+\frac{\partial}{\partial \mathit{x}}.(f\mathit{v})+\frac{\partial}{\partial \mathit{v}}.(f\mathit{a})=S.$ | $(8.7)$ |

${\mathit{v}}_{\mathit{ps}}=\left(\genfrac{}{}{0ex}{}{\mathit{v}}{\mathit{a}}\right)=\left(\begin{array}{c}\hfill {v}_{x}\hfill \\ \hfill {v}_{y}\hfill \\ \hfill {v}_{z}\hfill \\ \hfill {a}_{x}\hfill \\ \hfill {a}_{y}\hfill \\ \hfill {a}_{z}\hfill \end{array}\right),\hspace{0.5em}\text{and}\hspace{0.5em}{\mathit{\nabla}}_{\mathit{ps}}=\left(\genfrac{}{}{0ex}{}{\mathit{\nabla}}{{\mathit{\nabla}}_{v}}\right)=\left(\genfrac{}{}{0ex}{}{\frac{\partial}{\partial \mathit{x}}}{\frac{\partial}{\partial \mathit{v}}}\right)=\left(\begin{array}{c}\hfill \partial /\partial x\hfill \\ \hfill \partial /\partial y\hfill \\ \hfill \partial /\partial z\hfill \\ \hfill \partial /\partial {v}_{x}\hfill \\ \hfill \partial /\partial {v}_{y}\hfill \\ \hfill \partial /\partial {v}_{z}\hfill \end{array}\right).$ | $(8.8)$ |

$\frac{\partial}{\partial \mathit{x}}.(f\mathit{v})=\frac{\partial (f{v}_{x})}{\partial x}+\frac{\partial (f{v}_{y})}{\partial y}+\frac{\partial (f{v}_{z})}{\partial z}$ | $(8.9)$ |

$\frac{\partial}{\partial \mathit{v}}.(f\mathit{a})=\frac{\partial (f{a}_{x})}{\partial {v}_{x}}+\frac{\partial (f{a}_{y})}{\partial {v}_{y}}+\frac{\partial (f{a}_{z})}{\partial {v}_{z}}.$ | $(8.10)$ |

$\frac{\partial}{\partial \mathit{x}}.(f\mathit{v})=\mathit{v}.\frac{\partial f}{\partial \mathit{x}}={v}_{x}\frac{\partial f}{\partial x}+{v}_{y}\frac{\partial f}{\partial y}+{v}_{z}\frac{\partial f}{\partial z}.$ | $(8.11)$ |

$\frac{\partial f}{\partial t}+\mathit{v}.\frac{\partial f}{\partial \mathit{x}}+\mathit{a}.\frac{\partial f}{\partial \mathit{v}}=S=C.$ | $(8.12)$ |

$\frac{d\mathit{x}}{dt}=\mathit{v},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\frac{d\mathit{v}}{dt}=\mathit{a};\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\text{i.e.}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\frac{d}{dt}\left(\genfrac{}{}{0ex}{}{\mathit{x}}{\mathit{v}}\right)=\left(\genfrac{}{}{0ex}{}{\mathit{v}}{\mathit{a}}\right)$ | $(8.13)$ |

Figure 8.4: A phase-space orbit is determined by a first order ordinary
differential equation. The Boltzmann equation states that the rate
of change of the distribution function along phase-space orbits is
equal to the collision term.

First let's convince ourselves that is true. If, during our ride, we
measure the difference in $f$ between times different by a small
interval $\mathit{dt}$. The $f$ will be different because (1) it may have
intrinsic variation with time, resulting in a change $\mathit{dt}\hspace{0.5em}\partial f/\partial t$; (2) it may have variation with space so our motion has
carried us a distance $\mathit{dt}\hspace{0.5em}\mathit{v}$, to a place where $f$ is different
by $\mathit{dt}\hspace{0.5em}\mathit{v}.\partial f/\partial \mathit{x}$; or (3) it may have
variation with velocity so our motion in velocity space (acceleration)
has carried us a velocity-"distance", $\mathit{dt}\hspace{0.5em}\mathit{a}$, to where $f$ is
different by $\mathit{dt}\hspace{0.5em}\mathit{a}.\partial f/\partial \mathit{v}$. The total of
these three, divided by $\mathit{dt}$, is the rate of change of $f$ along the
orbit. That's the left hand side of eq. (8.12).
Second, why is this identity with the total derivative the case? It is
because the flow in phase-space is
$\frac{\partial f}{\partial t}+\mathit{v}.\frac{\partial f}{\partial \mathit{x}}+\mathit{a}.\frac{\partial f}{\partial \mathit{v}}=\frac{\mathit{Df}}{\mathit{Dt}}=C.$ | $(8.14)$ |

$f({\mathit{v}}_{1},{\mathit{x}}_{1},{t}_{1})-f({\mathit{v}}_{0},{\mathit{x}}_{0},{t}_{0})={\int}_{0}^{1}C\mathit{dt}.$ | $(8.15)$ |

Figure 8.5: In the collisionless Boltzmann equation the distribution is
constant along orbits. The distribution (a) is different at the top
of a potential hill (b) because the speed on an orbit is smaller
(conserving energy). The distribution values ${f}_{0}=f({x}_{0})$ and
${f}_{1}=f({x}_{1})$ are the same but at different velocities. Orbits have
moved the distribution along the horizontal dotted lines in (a). The
lowest velocity orbits of distribution ${f}_{0}$ (upper dashed part) can't
reach the top of the hill where ${f}_{1}$ is, and do not contribute to
it.

$\mathbf{v}.\frac{\partial}{\partial \mathbf{x}}\mathit{\psi}=S,$ | $(8.16)$ |

$\mathbf{v}.\frac{\partial}{\partial \mathbf{x}}\mathit{\psi}=\frac{d\mathbf{x}}{\mathit{dt}}.\frac{\partial}{\partial \mathbf{x}}\mathit{\psi}=\sum _{j=1}^{N}\frac{d{x}_{j}}{\mathit{dt}}.\frac{\partial}{\partial {x}_{j}}\mathit{\psi}={\frac{d\mathit{\psi}}{\mathit{dt}}|}_{\mathrm{orbit}}=S.$ | $(8.17)$ |

$\frac{\partial f}{\partial t}+\mathit{v}.\frac{\partial f}{\partial \mathit{x}}+\mathit{a}.\frac{\partial f}{\partial \mathit{v}}=0.$ | $(8.18)$ |

$C(f)=-\mathit{\nu}f(\mathit{v})+\mathit{\nu}{f}_{2}(\mathit{v}).$ | $(8.19)$ |

Figure 8.6: Charge exchange collisions, where an electron is
transferred from a neutral to an ion, give rise to a simple
collision term. If they occur at a constant rate, $\mathit{\nu}$, eq. 8.19 applies.

This is sometimes called the BGK collision
form. It represents depletion of the original ions at rate $\mathit{\nu}$
giving the term $-\mathit{\nu}f(\mathit{v})$, and their direct replacement at the same
rate by new ions. The newly born ions, before their collision, were
neutrals. They retain the velocity distribution ${f}_{2}(\mathit{v})$ they had
before the collision, because the collision just transfers an electron
from one to the other.
Another idealized example (Fig. 8.7) is when collisions
are with heavy stationary targets (which
therefore acquire negligible recoil energy) which happen to scatter equally,
isotropically, in all directions.
Figure 8.7: Isotropic scattering (an idealized approximation) gives
particles emerging equally in all directions $\mathit{\Omega}$. With heavy
targets, $v$ is not changed in magnitude, only in direction. Eq. (8.20) is the result.

In a collision, a particle
just changes the direction of its velocity, not its magnitude.
If the density of targets is ${n}_{2}$ and the collision cross-section is
$\mathit{\sigma}$, then
$C(f)=-{n}_{2}\mathit{\sigma}v(f(\mathit{v})-\int f(\mathit{v}){d}^{2}\mathit{\Omega}/4\mathit{\pi}).$ | $(8.20)$ |

${\mathit{\Sigma}}_{t}=\sum _{j}{n}_{j}{\mathit{\sigma}}_{j};$ | $(8.21)$ |

The steady collisionless one-dimensional Boltzmann (Vlasov) equation is

$0=\frac{Df}{Dt}=v.\frac{\partial f}{\partial x}+a.\frac{\partial f}{\partial v}$ | $(8.22)$ |

$\frac{dx}{dt}=v;\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\frac{dv}{dt}=a=-\frac{1}{m}\frac{d\mathit{\phi}}{dx}.$ | $(8.23)$ |

$v\frac{dv}{dt}+\frac{1}{m}\frac{d\mathit{\phi}}{dx}\frac{dx}{dt}=0,$ | $(8.24)$ |

$\frac{1}{2}m{v}^{2}+\mathit{\phi}=\mathit{const}.$ | $(8.25)$ |

$f(v,x)={f}_{\mathit{\infty}}({v}_{\mathit{\infty}})=\mathrm{exp}(-m{v}_{\mathit{\infty}}^{2}/2T)=\mathrm{exp}(-[m{v}^{2}+2\mathit{\phi}(x)]/2T).$ | $(8.26)$ |

$f(v,x)=\mathrm{exp}(-[m{v}^{2}+2{\mathit{\phi}}_{0}{e}^{-{x}^{2}/{w}^{2}}]/2T)=\mathrm{exp}(-\frac{m{v}^{2}}{2T})\mathrm{exp}(-\frac{{\mathit{\phi}}_{0}}{T}{e}^{-{x}^{2}/{w}^{2}}).$ | $(8.27)$ |

Figure 8.8: Contours of constant $f(v,x)$ are also orbits. Therefore the
orbits can be plotted simply by contouring $f$,
whose value is determined by the total (kinetic plus potential)
energy at any point in phase-space. When the potential has a hill
(a), all orbits extend to $x\to \mathit{\infty}$ and $f$ is determined by
boundary conditions. When the potential has a well (b), the value of
$f$ on trapped orbits (shaded) is undetermined. [The parameters used
in these plots are ${\mathit{\phi}}_{0}/m=\pm 1$, $w=1$, $T/m=1$.]

1. Divergence of acceleration in phase-space. (a) Prove that particles of charge $q$ moving in a magnetic field $\mathit{B}$ and hence subject to a force $q\mathit{v}\times \mathit{B}$, nevertheless have ${\nabla}_{v}.\mathit{a}=0$. (b) Consider a frictional force that slows particles down in accordance with $\mathit{a}=-K\mathit{v}$, where $K$ is a constant. What is the "velocity-divergence", of this acceleration, ${\nabla}_{v}.\mathit{a}$? Does this cause the distribution function $f$ to increase or decrease as a function of time?

2. Write down the Boltzmann equation governing the distribution function of neutral particles of mass $m$ in a gravitational field $g\hat{\mathit{x}}$, moving through matter that consists of two different species of density ${n}_{a}$, ${n}_{b}$ whose

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