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,

$$ Nf\left( {r,v,t} \right)dxdydzd{v_x}d{v_y}d{v_z} = Nf\left( {r,v,t} \right)d{V_r}d{V_r} $$

((3–2))

gives the probable number of particles in the six dimensional phase space dV _r dV _dV around position, r, and velocity, v. Boltzmann’s equation for f( r,p, t) may be written as (Boltzmann, 1872)

$$ \frac{{\partial f}}{{\partial t}} + \frac{p}{m}\;{\nabla _r}\varphi \;{\nabla _p}f = {\left( {\frac{{df}}{{dt}}} \right)_{coll}} $$

((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 p ̇ = -∇_rφ, we may write Eq. (3–3) in Cartesian coordinates as

$$ \frac{{\partial f}}{{\partial t}} + \mathop x\limits^. \frac{{\partial f}}{{\partial x}} + \mathop y\limits^. \frac{{\partial f}}{{\partial y}} + \mathop z\limits^. \frac{{\partial f}}{{\partial z}} + {\mathop p\limits^. _x}\frac{{\partial f}}{{\partial {p_x}}} + {\mathop p\limits^. _y}\frac{{\partial f}}{{\partial {p_y}}} + {\mathop p\limits^. _z}\frac{{\partial f}}{{\partial z}} = {\left( {\frac{{df}}{{dt}}} \right)_{coll}} $$

where denotes the first derivative with respect to time.

“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)