The parallel to the Boltzmann theory has been used to good e¬ect in the

study of plasma physics, as in the work of Rosenbluth, et al. (1957). As

Spiegel & Thi¬eault

380

Clemmow & Dougherty (1969) explain, those authors obtained their results

by “expanding the Boltzmann collision operator under the approximation

that all the de¬‚ections are small angle and cutting o¬ the impact parameter

at about the Debye length. . . . At ¬rst sight the success of that method is

surprising, as any treatment dealing with binary collisions would seem to be

discredited. The physical reason for the agreement is that, for the major-

ity of particles, there is little di¬erence between a succession of numerous

small-angle collisions (regarded as instantaneous and occurring at random)

and the stochastic de¬‚ections due to the presence of many nearby particles

continually exerting weak forces. These two pictures of the dynamics are of

course represented respectively by the Boltzmann and the Fokker“Planck”

approaches.

Similar thoughts have been expressed in the context of stellar dynamics,

most recently by Griv et al. (2001) and by kinetic theorists generally. As

E. G. D. Cohen reports (1997), “when Academician Bogolubov and I dis-

cussed the nature of kinetic equations, he mentioned a discussion he had

had with Professor A. Vlasov, where they had agreed that: Yes, in ¬rst

approximation the kinetic equations for gases with strong short-range forces

(i.e. the Boltzmann equation) and for gases with long-range forces (i.e. the

Vlasov equation) di¬er, but in higher approximations they will become more

and more similar. How right they were.” A formal theory to buttress these

remarks would be very comforting, but though we do not have one we shall

adopt the point of view that the Fokker“Plank terms capture the essence of

interactions of the stars in the system. Having thus supported our approach

by the appeal to authority, we turn to the main purpose of this work, the

derivation of continuum mechanics from the microscopic theory in a way

that is not severely restricted to the case of short mean free paths.

26.2 The collision term

In the spirit of standard kinetic theories, we shall suppose that the e¬ect of

the collision term is to drive the system toward a local equilibrium, though

the correct equilibrium of a stellar system is not known on purely theoretical

grounds. The tendency to approach an equilibrium seems not even to require

a collision term of the usual kind since the violent relaxation described by

Lynden-Bell apparently can do the job. Nevertheless, we shall proceed in

terms of the kinetic theory under the assumption that the spreading of

the phase ¬‚uid through the phase space may be e¬ected by a collision term.

Furthermore, though the gravitational force has long range in physical space,

we shall presume that this spreading takes the form of a di¬usion of f

Continuum equations for stellar dynamics 381

through velocity space, that is, by the agency of a Fokker“Planck form of

the collision term. (This may not be completely unfounded since there seems

to exist a form of gravitational shielding (Spiegel, 1998) that may support

the idea of local behaviour.) In this spirit, we write

‚ ‚

C[f ] = ’Ai f + B ij f

1

. (26.7)

2

i j

‚v ‚v

The coe¬cients Ai and B ij are generally functions of (x, v, t) and they

may also be functionals of f . In this discussion, the choice of these coe¬-

cients in the Fokker“Planck description is adapted to the equilibrium that

is expected or assumed. This equilibrium is a local one that satis¬es the

condition C[f0 ] = 0.

Our goal is to ¬nd equations that govern the dynamics of the macroscopic

properties of the ¬‚uid embodied in the density, temperature and velocity.

These are de¬ned as:

mf d3 v,

ρ :=

Mass density (26.8)

1

mvf d3 v,

u :=

Mean velocity (26.9)

ρ

m

c2 f d3 v,

T :=

Temperature (26.10)

3R ρ

where the peculiar velocity is

c(x, v, t) := v ’ u(x, t) , (26.11)

and R = k/m, k being Boltzmann™s constant.

As to the nature of C[f ], we shall design it so that it produces what

may be the simplest plausible equilibrium, namely the Maxwell“Boltzmann

distribution

c2

ρ

exp ’

f0 (x, v, t) = . (26.12)

m(2πRT )3/2 2R T

Since T, ρ and u generally depend on x and t, this is a local equilibrium and

we choose (Clemmow and Dougherty (1969))

Ai = ’„ ’1 (v i ’ ui ), B ij = 2„ ’1 R T δij , (26.13)

so that C[f0 ] = 0. We assume that the mean-free-time „ is a constant so

that the Fokker“Planck operator is linear in f .

Spiegel & Thi¬eault

382

The collision term adopted here ensures the conservation of mass, mo-

mentum and energy in the system. This is re¬‚ected in the property

ψ ± C[f ] d3 v = 0, ± = 0, . . . , 4 , (26.14)

where

ψ ± = m (1, v, 1 v 2 ) . (26.15)

2

Thus we neglect the possible e¬ects of dissipative processes and of evapora-

tion of stars from the system.

The macroscopic quantities (26.8)“(26.10) enter the equilibrium distribu-

tion in (26.12) about which we are expanding. To ensure that the same

macroscopic quantities that follow from f are those that determine f0 , we

impose a consistency requirement known as the matching conditions,

ψ ± f d3 v = ψ ± f0 d3 v, ± = 0, . . . , 4 . (26.16)

26.3 Fluid equations

When we multiply the kinetic equation (26.6) by the collisional invariants

(26.15) and integrate over v, the right-hand side does not contribute to the

outcome, and we are left with

‚t ρ + ∇ · (ρ u) = 0, (26.17)

‚t u + u · ∇u = ’ρ’1 ∇ · P ’ ∇φ, (26.18)

+ u · ∇T ) = ’P : ∇u ’ ∇ · q,

3

2 ρR (‚t T (26.19)

where ∇ means ∇x. Here the pressure tensor P and heat ¬‚ux q are de¬ned

as

P := mccf d3 v, 2

d3 v .

1

q := 2 mc cf (26.20)

We see that the form of the macroscopic equations is just that of the usual

¬‚uid equations. This result is independent of the rarity of the medium.

The usefulness of these equations depends entirely on how well we can pre-

scribe the higher-order moments P and q. A standard way to proceed is to

solve (26.6) approximately for f . We shall follow this route also, but will

deviate from the normally used prescription at a certain point.

We let f = f0 +„ f1 +. . . and look ¬rst at order „ 0 . We ¬nd that C[f0 ] = 0,

and the solution f0 is the Maxwell“Boltzmann equilibrium (26.12). From

(26.20), we see that P0 = p I and q0 = 0, where the scalar pressure is given

Continuum equations for stellar dynamics 383

by p := ρR T . If we stop at this order, Equations (26.17)“(26.19) are then

the Euler equations for an ideal ¬‚uid.

At order „ 1 , it is convenient to factor out the Maxwell“Boltzmann solution

from f1 and write the equation to be solved as