As a final characterization of a statistically varying population, we consider what is often called transport theory or a kinetic description. A compact derivation which leads to a defining equation for a particle density distribution function and which also includes collisional effects as well as source effects-that is, comprehensive transport of particles and their properties-is suggested in the following.

Consider an ensemble of N particles each characterized by a distinguishable velocity vector Vi and coordinate vector Гі at time t. For that we introduce a distribution function fN represented by

f n = f N{ri>vi;r2>v2;~-;ri, Vi;—;rN>VN;t)> (6.27)

which is able to specify the probability that at time t, the particle 1 is found about the coordinate Г] with the velocity vector V] while the position and velocity of particle 2 are about r2 and v2, and the remaining (N-2) particles are similarly found at distinct coordinate intervals in the position-velocity space. With N as the number of particles of interest, each characterized by three Cartesian space coordinates and three Cartesian velocity components, this phase-space is evidently a 6N-dimensional phase-space.

A fundamental principle in the description of particle distributions is known
as the Liouville Theorem; this theorem states that a volume element in the 6N- dimensional phase-space remains constant for the motion of particles obeying the Hamiltonian equation of motion. For the N-particle distribution function fN, Eq.(6.27), this theorem leads to the Liouville Equation

(6.28)

if particle creation and/or particle destruction does not occur. Performing the differentiation of Eq.(6.28), the explicit differentiation which fN must satisfy is evidently

(6.29)

This differential equation is generally the starting point for an analysis of a many particle system. Note that the acceleration term dv/dt can be replaced by F,/m, via Newton’s Law with Fj representing the total force acting on the i-th particle.

Equation (6.29) is now subject to integration over a suitable large set of coordinates Гі and velocities Vj in order to substantially reduce the dimensionality of the phase-space distribution function from fN to, say fr. Thus, for r « N, we obtain

(6.30)

with Ar a normalization constant generally chosen by convention, e. g. Ar = 1/(N — r)!, found via permutational calculus.

We will find it useful to separate the total force F, acting on the i-th particle

j *i, r= 1, 2, …, N — 1 ,

where we replaced Э/Эг and Э/Эу by the gradient symbols

Э, Э. Э

Bulk Particle Transport and

Vv = 3—* + 3—j + 3—-^ • (6.31c)

dyx a у y a vz

Equation (6.31a) is known as the Bogolyubov-Bom-Green-Kirkwood-Yvon hierarchy and closes only with the Liouville Equation, Eq.(6.28), since the equation for fr contains fr+i. To accommodate the cutting off of this sequence at a reasonably small r, a physical approximation for fr+i is therefore needed in terms of f i,…,f r in order to arrive at a solvable set of equations. From this set of r equations the r-1 equations may be used to eliminate f2,…,fr, leaving a single kinetic equation for the one-particle distribution function fb We now consider the case r = 1 which is described by Э f F6*’ 1 r

-3^ + V, • V, / ; + — • Vv, / ; =—- V,2-V, J2d3r2d3V2. (6.32)

dt m, m, J

Here, fi(ri, Vi, t) is the one-particle distribution function used to determine the probability of finding particle 1 at time t about the position ri with velocity Vi; note however, that herein we have lost the corresponding information about the other particles. Further, Eq.(6.32) contains the two-particle distribution

f2(ri, vi, r2,v2,t) which specifies the probability that at time t particle 1 moves with Vi about r, and, simultaneously, particle 2 will be found about r2 with velocity v2, with no such individual information about the remaining N-2 particles.

Writing the one-particle distributions

f,(ri, Mi, t)=f(l) (6.33a)

for particle 1,

f i(r2,v2,t) = f(2) (6.33b)

for particle 2, and expressing the two-particle distribution by cluster expansion as f2(rj, vi, r2,v2,t) = film + C(l,2) (6.33c)

with C representing the so-called pair correlation function relating kinetic variations of particles 1 and 2, most significantly within the close range of their interaction, we find upon substitution of these expansions the following kinetic equation:

+ v, • V, fil) + ^■ Vv, fil) + —( f F12 f(2)d3 r2 d3 v2 at m, m,

= -—[F12 — W, C(l,2)d3r2d3y2. m, J

The expression within brackets in the fourth term on the left-hand side represents the field force experienced by particle one due to the presence of the other particles and may be written as

j¥12f(2)d3r2d3v2 = F{ield • (6.35)

Actually, this is the particle contribution to the self-consistent field (for example:

an ion distribution generates an electromagnetic field which in turn maintains this distribution). The right-hand side of Eq.(6.34) is the collision term for which some approximation must be applied to expand it also as a function of f(l) and f(2). Since various physical models exist for approximating the collision term, we choose to represent it here simply by (5f(l)/5t)c. Thus the final kinetic equation for a one-particle distribution function is

Іr dt Jc

(6.36)

where F includes all external and field forces. Herein and subsequently we omit the subscript and f(r, v,t) will refer to the one-particle distribution function which is representative of any one of the considered N particles of the same species, however does not provide information which distinguishes the individual kinetic behaviour of the particles. For charged particles moving in an electric and magnetic field, the total force is evidently

F = q(E + vxB) (6.37)

incorporating self-consistent fields.

If particle production and/or destruction also take place and particles can leave and/or enter the reaction volume under consideration, then we account for these sources and sink processes on the right-hand side of Eq.(6.36) by an additional term (5f/5t)s to be specified according to the appropriate conditions.

In the statistical sense [f(r, v,t) d3r d3v] is interpreted as the probability of finding at time t a particle of the species of interest in the differential volume element d3r d3v about the point (r, v) in the 6-dimensional phase space, Fig.6.1. We may now conclude that the expected number of such particles in this differential volume element is

N(r,,t)d3rd3y = N* f(r, v,t)d3rd3v (6.38)

where N is the total number of particles in the entire volume at some reference time. Hence

6 expected number of particles in dr3 у about r and in dv about v at time t,

Obviously the normalization applied to Eq.(6.39) is

N(t)

N with the number density obtained by

N( r, t) = J N( r, v, t) d3 v = N * J /( r, v, t) d3 v.

The transport equation for a charged particle density distribution N(r, v,t) in an electric and magnetic field and subjected to collisional effects as well as to particle production/destruction is therefore finally written as

(6.42)

A solution analysis of this equation for N = N(r, v,t) requires the self-consistent specification of E and В (determined in conjunction with Maxwell’s Equations) everywhere, identification of the form of the collision and source/sink terms and the imposition of initial and boundary conditions. A reaction domain will generally contain more than one species of particles so that the determination of the overall collision kinetics requires several equations of the form of Eq.(6.42)~ one for each species. All of these may be coupled by several collision terms since collisions will invariably occur among the same and other species. Needless to say, the dimensionality of this problem is formidable. A simplification is possible by a reduction of the number of independent variables and elimination of some terms depending upon the problem of interest.

It is useful to note that the left-hand side of Eq.(6.42) constitutes, by our classification of processes, continuous changes in phase space while the right — hand parts are discontinuous changes. Thus, particle collisions, particle reactions and particle destructions are viewed as discontinuous-that is discrete processes — whereas particle acceleration can vary smoothly in the coordinate-velocity space. If only binary collisions are considered, the analysis is labeled “classical” or “neoclassical” if applied to toroidal geometry. As plasma wave-particle interactions become dominant such that the transport of mass, momentum or energy results therefrom, then the label “anomalous transport” is used.