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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

Solar Wind Dispersion Relation Neutron Star Helmholtz Free Energy Virial Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1980

Authors and Affiliations

  • Kenneth R. Lang
    • 1
  1. 1.Department of PhysicsTufts UniversityMedfordUSA

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