# Gas Processes

• Kenneth R. Lang

## Abstract

The one particle probability distribution function, f(r, p, t), is defined so that
$$f\left( {r,\,p,\,t} \right)dxdydzd{p_x}d{p_y}d{p_z} = f\left( {r,\,p,\,t} \right)d{V_r}d{V_p}$$
(3-1)
, is the probability that, at the time, t, a particle has momentum, p, in the volume element dV p at p and position, r, in the volume element dV r at r. Similarly, the distribution function f(r, v, t) is defined so that for an average particle density, N,
$$N\,f\left( {r,\,v,\,t} \right)dxdydzd{v_x}d{v_y}d{v_z} = N\,f\left( {r,\,v,\,t} \right)d{V_r}d{V_v}$$
(3-2)
, gives the probable number of particles in the six dimensional phase space dV r dV v around position, r, and velocity, v. Boltzmann’s equation for f(r, p, t) may be written as (Boltzmann, 1872)
$$\frac{{\partial f}}{{\partial t}} + \frac{p}{m} \cdot {\nabla _r}f - {\nabla _r}\varphi \cdot {\nabla _r}f = {\left( {\frac{{df}}{{dt}}} \right)_{coll}}$$
(3-3)
, where φ(r) is the potential energy acting on every particle, p is the momentum, m is the particle mass, ∇ r is the gradient in position space, ∇ p is the gradient in momentum space, and (df /dt)coll is the rate of change in f due to collisions. Noting that $$\dot p = - {\nabla _r}\varphi$$, we may write Eq. (3-3) in Cartesian coordinates as
$$\frac{{\partial f}}{{\partial t}} + \dot x\frac{{\partial f}}{{\partial x}} + \dot y\frac{{\partial f}}{{\partial y}} + \dot z\frac{{\partial f}}{{\partial z}} + {\dot p_x}\frac{{\partial f}}{{\partial {p_x}}} + {\dot p_y}\frac{{\partial f}}{{\partial {p_y}}} + {\dot p_z}\frac{{\partial f}}{{\partial {p_z}}} = {\left( {\frac{{df}}{{dt}}} \right)_{coll}}$$
, where · denotes the first derivative with respect to time. The Boltzmann equation for f(r, v,t) is
$$\frac{{\partial f}}{{\partial t}} + v \cdot {\nabla _r}f + \frac{F}{m} \cdot {\nabla _v}f = {\left( {\frac{{df}}{{dt}}} \right)_{coll}}$$
(3-4)
, where v is the velocity, F is the force acting on each particle, m is the particle mass, and ∇ r and ∇ v denote, respectively, gradients in position and velocity space. As an example of astrophysical forces, a particle of charge, q, and mass, m, experiences the force
$$F = q\left( {E + \frac{1}{c}v \times H} \right) - m\,g\,{n_r}$$
, in the presence of an electric field of strength E, a magnetic field of strength H, and a gravitational field of acceleration g. Here n r is a unit vector in the radial direction from the mass, m, to another mass, M, and the acceleration due to gravity is G M/r 2, where the gravitational constant G=6.67 · 10−8 dyn cm2g−2 and r is the distance between the mass, M, and the particle of mass, m.

### Keywords

Permeability Entropy Anisotropy Convection Enthalpy

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