Gas Processes

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.

“This notation may be perhaps further explained by conceiving the air near the earth to be such a heap of little bodies, lying one upon another as may be resembled to a fleece of wool. For this consists of many slender and flexible hairs, each of which may indeed, like a little spring, be easily bent or rolled up; but will also, like a spring, be still endeavouring to stretch itself out again.”

R. Boyle, (1660)

“O dark dark dark. They all go into the dark.

The vacant interstellar spaces, the vacant into the vacant.”

T. S. Eliot in East Coker III (1940)