Boyd T.J.M., Sanderson J.J. The Physics of Plasmas

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12.2 Dynamics of a many-body system 467

with q

= r

, p

= mv

and F

=−∂ V /∂r

from which the equivalence of (12.7)

and (12.5) is easily demonstrated.

The aim now is to integrate (12.5) over most of the space coordinates r

and

velocities v

to obtain equations for reduced distribution functions (containing less

information) which we may hope to determine. The reduced distribution functions

are deﬁned by

, v

,...,r

, v

, t) =

N !

(N − s)!



i=s+1

(12.9)

where the normalization constant has been chosen because there are N !/(N − s)!

ways of choosing s electrons from a total of N . This choice introduces a second,

rather more subtle, change in the nature of the distribution functions that we are

seeking. The ﬁrst change from f

to f

acknowledged the fact that we cannot ever

know the exact starting point in phase space so it makes more sense to consider a

distribution of initial points f

(t = 0) leading to a corresponding distribution

(t) at any later time. Of course, f

is supposed to be chosen to be consistent with

whatever information we do have about the plasma, but since such ‘macroscopic’

detail usually involves just one, or at most two, space and velocity coordinates that

leaves much uncertainty about f

. It is this indeterminate detail that we eliminate

by integrating over most of the phase space coordinates. Now, by our choice of

normalization constant in (12.9), we recognize that it makes no sense to talk

about speciﬁc electrons, labelled 1, 2,...,s being at (r

, v

), (r

, v

),...,(r

, v

respectively, but only about the probability of ﬁnding (any) electrons at these

coordinates since we have no way of distinguishing one electron from another.

The reduced distribution functions deﬁned by (12.9) are sometimes called generic

distribution functions as opposed to the speciﬁc distribution functions which would

be deﬁned by the choice of unit normalization constant.

Integrating (12.5) over all but s spatial and velocity coordinates, assuming that

vanishes on the boundaries of phase space and that the only velocity-dependent

forces are Lorentzian, we obtain

∂ f

∂t



i=1

∂ f

∂r

N !

(N − s)!m



i=1



∂ f

∂v

j=s+1

= 0 (12.10)

Separating F

into what we may call its internal component F

int

(the force due to

all the other electrons) and external component F

ext

(the force due to the ions and

any applied ﬁelds) it follows that

N !

(N − s)!m



ext

∂ f

∂v

j=s+1

ext

∂ f

∂v

(i = 1, 2,...,s) (12.11)

468 The classical theory of plasmas

since F

ext

may be taken outside the integral. In the non-relativistic approximation

considered here, the only internal forces are electrostatic

int

4πε



j=1

j=i

− r

=−



j=1

j=i

∂φ

∂r

(12.12)

where

4πε

− r

(12.13)

Hence

N !

(N − s)!m



i=1



int

∂ f

∂v

j=s+1



i=1



j=1

j=i

∂φ

∂r

∂ f

∂v

−



i=1



∂φ

is+1

∂r

∂ f

s+1

∂v

s+1

(12.14)

where we have separated the sum over j in (12.12) into the ﬁrst s terms and the

remaining (N − s) terms and then made use of the fact that all of the latter are

identical. Substituting (12.11) and (12.14) into (12.10) gives the general equation

for the reduced distribution functions. Using (12.6) this may be written



i=1



∂φ

is+1

∂r

∂ f

s+1

∂v

s+1

(s = 1, 2,...,N − 1) (12.15)

In (12.15) L

is the Liouville operator for s electrons and the right-hand side

represents electron interactions. We can see immediately the fundamental prob-

lem with this set of equations. The equation for f

contains f

s+1

so that the

system closes only with the Liouville equation (12.5) and no simpliﬁcation has

yet been achieved. This is not surprising since no approximations have been in-

troduced thus far and the problem is therefore the one with which we started.

The chain of equations represented by (12.15) is called the BBGKY hierarchy

after Bogolyubov (1962), Born and Green (1949), Kirkwood (1946), and Yvon

(1935).

To reduce the complexity of the theoretical description we want, in principle,

some physical approximation enabling us to write f

s+1

in terms of f

, f

,..., f

for some small value of s, so obtaining a solvable set of equations – that is, a set of

s equations, s − 1 of which may be used to eliminate f

,..., f

leaving a single

(kinetic) equation for f

. Even for s = 2 this is a formidable task. Putting s = 1, 2

12.2 Dynamics of a many-body system 469

in (12.15) the pair of equations is

∂ f

∂t

+ v

∂ f

∂r

ext

∂ f

∂v



∂φ

∂r

∂ f

∂v

(12.16)

∂ f

∂t

+ v

∂ f

∂r

+ v

∂ f

∂r

ext

∂ f

∂v

ext

∂ f

∂v

−

∂φ

∂r



∂ f

∂v

−

∂ f

∂v







∂φ

∂r

∂ f

∂v

∂φ

∂r

∂ f

∂v



(12.17)

where we have used ∂φ

/∂r

=−∂φ

/∂r

in (12.17).

12.2.1 Cluster expansion

As a ﬁrst step towards the expression of f

in terms of f

and f

we introduce the

cluster expansion by means of which we deﬁne new functions f , g and h such that

, v

, t) ≡ f (1) (12.18)

, v

, r

, v

, t) = f (1) f (2) + g(1, 2) (12.19)

, v

, r

, v

, r

, v

, t) = f (1) f (2) f (3) + f (1)g(2, 3) + f (2)g(3, 1)

+ f (3)g(1, 2) + h(1, 2, 3) (12.20)

For convenience we have also simpliﬁed the notation, suppressing the time depen-

dence in f, g, h and writing (1) for (r

, v

), (2) for (r

, v

), etc. The idea behind the

cluster expansion is easily seen. If electrons 1 and 2 were completely independent

of each other then the probability of ﬁnding 1 at (r

, v

) at the same time that

2isat(r

, v

) would simply be the product f (1) f (2). Thus, g(1, 2), being the

difference between f

and f (1) f (2), is a measure of the extent to which electrons

1 and 2 are not independent but correlated and it is called the pair correlation

function. In a similar manner the ﬁve terms in the expansion in (12.20) represent

the contributions to f

corresponding to all three electrons being independent of

each other, each electron in turn being independent while the remaining two are

correlated, and ﬁnally all three being correlated.

The cluster expansion was ﬁrst introduced to deal with molecular interactions

for which the range of interaction r

 λ

mfp

, the mean free path of the molecules.

Thus, throughout most of its motion a molecule is unaware of the presence of other

molecules and g(1, 2) ≈ 0 except when |r

− r

|  r

. Likewise, h(1, 2, 3) ≈ 0

unless all three molecules are within a sphere of radius ≈ r

. In these circumstances

we expect that binary interactions will be far more signiﬁcant than three particle

interactions and it can be shown that it is valid to ignore h in (12.20) thus achieving

the desired truncation of the BBGKY hierarchy.

470 The classical theory of plasmas

In a plasma the Coulomb interaction is anything but short range; indeed, the long

range nature of electron interactions is a dominant feature of plasma dynamics.

Nevertheless, the cluster expansion works for plasmas, too, but in a quite different

way. First of all, it separates out the dominant long range part of electron interac-

tions in the form of a ﬁeld force. The short range interactions, the ‘collisions’, are

then deﬁned in terms of the pair correlation function g(1, 2) and we shall see that

this is again vanishingly small over most of phase space provided that the number

of electrons in a Debye sphere is large. To see how this comes about let us substitute

(12.18) and (12.19) in (12.16) giving

∂ f (1)

∂t

+ v

∂ f (1)

∂r

ext

∂ f (1)

∂v

−

∂ f (1)

∂v



∂φ

∂r

f (2)dr



∂φ

∂r

∂g(1, 2)

∂v

(12.21)

The ﬁrst three terms in (12.21) are comparatively straightforward. The fourth term

contains the average electric ﬁeld experienced by one electron due to other elec-

trons and may be written

−

∂ f (1)

∂v



∂φ

∂r

f (2)dr

=−

∂ f (1)

∂v

where the ﬁeld E is given by

(−e)E(r

, t) =−



∂φ

∂r

f (2)dr

(12.22)

and (−e) is the electronic charge. Note, however, that E is computed assuming that

electrons are uncorrelated; the average is taken over all positions and velocities of

electron 2 with no reference to any interaction between 1 and 2. This is in fact the

electron contribution to the self-consistent ﬁeld. Thus (12.21) may be written

∂ f (1)

∂t

+ v

∂ f (1)

∂r

∂ f (1)

∂v



∂ f

∂t



(12.23)

where F now includes all external and internal ‘ﬁeld’ forces and



∂ f

∂t





∂φ

∂r

∂g(1, 2)

∂v

(12.24)

is called the collision term or collision integral.

The separation of electron interactions into the self-consistent ﬁeld (12.22) and

the collision integral (12.24), brought about by splitting f

into its uncorrelated and

correlated parts, is fundamental. Given that we are treating the ions as a uniform

background of positive charge, any charge imbalance in the plasma must appear as

12.2 Dynamics of a many-body system 471

an inhomogeneity in f and hence lead to a non-zero self-consistent ﬁeld. There is

no sense in which this term is small; indeed, we know that the tendency of a plasma

to resist charge imbalance is one of its dominant features. On the other hand, as

discussed in Section 7.1, it is very often an entirely satisfactory approximation

to neglect the collision term (12.24) completely. This, of course, corresponds to

truncation at s = 1 by expressing f

in terms of f

, namely f

(1, 2) = f (1) f (2).

Substituting (12.19) and (12.20) in (12.17), but setting h(1, 2, 3) = 0 to truncate

the expansion, we obtain, after some cancellation on using (12.21),

∂g(1, 2)

∂t



∂

∂r

+ v

∂

∂r



g(1, 2)



ext

∂

∂v

ext

∂

∂v



g(1, 2)

−

∂φ

∂r



∂ f (1)

∂v

f (2) −

∂ f (2)

∂v

f (1) +

∂g(1, 2)

∂v

−

∂g(1, 2)

∂v







∂φ

∂r

∂

∂v

[ f (1)g(2, 3) + f (3)g(1, 2)]

∂φ

∂r

∂

∂v

[ f (2)g(1, 3) + f (3)g(1, 2)]



(12.25)

Next, consistent with the neglect of h(1, 2, 3), we may drop the g terms com-

pared with the ff terms in the last expression on the left-hand side of (12.25).

Neglecting these terms is equivalent to the assumption that in the cluster expansion

|h||g| f  fff (12.26)

and is known as the weak coupling approximation. This reduces our pair of equa-

tions to (12.23) and

∂g(1, 2)

∂t



∂

∂r

+ v

∂

∂r



g(1, 2)



ext

∂

∂v

ext

∂

∂v



g(1, 2)

−

∂φ

∂r



∂ f (1)

∂v

f (2) −

∂ f (2)

∂v

f (1)







∂φ

∂r

∂

∂v

[ f (1)g(2, 3) + f (3)g(1, 2)]

∂φ

∂r

∂

∂v

[ f (2)g(1, 3) + f (3)g(1, 2)]



(12.27)

Although we have achieved closure we are still a long way from obtaining the

desired kinetic equation for f alone. The pair of simultaneous equations (12.23)

472 The classical theory of plasmas

and (12.27) for f and g are far too complicated for the elimination of g. Assum-

ing Bogolyubov’s hypothesis, which states that the time scale for changes in g is

much shorter than that for f , Balescu (1960) and Lenard (1960) obtained a kinetic

equation for the special case of a homogeneous plasma in the absence of external

forces. The Balescu–Lenard equation is

∂ f

∂t

=−

∂

∂v



kφ

(k)dk

|(k, k · v

k ·





∂ f (v

)

∂v

f (v

) −

∂ f (v

)

∂v

f (v

)



δ[k · (v

− v

)]dv

(12.28)

where

φ(k) =

(2π)

3/2

(12.29)

and

(k,ω)= 1 +

k ·



∂ f /∂v

(ω − k ·v)

dv (12.30)

are the Fourier transforms of the Coulomb potential and the plasma dielectric func-

tion, respectively. We shall not give the derivation of (12.28) for it is not the most

appropriate nor the most useful kinetic equation for most plasmas. That description

is provided by the Landau kinetic equation, which we shall derive in Section 12.4,

where we also discuss the Bogolyubov hypothesis.

A simpler version of (12.27) is obtained by setting the right-hand side equal to

zero, the advantage of which is that the equation can then be solved for g without

assuming plasma homogeneity. However, throwing away terms just because they

are inconvenient is mathematically unconvincing, to say the least. Nevertheless,

there is another special case for which (12.27) can be solved without further reduc-

tion. This is the equilibrium plasma and by solving it we shall gain insight into the

signiﬁcance of the terms that we wish to discard.

12.3 Equilibrium pair correlation function

In this section we consider a plasma in equilibrium and evaluate f and g.Itis

a well-known result of statistical mechanics that in thermodynamic equilibrium a

solution of the Liouville equation (12.5) is the Gibbs distribution function

= C exp(−H/θ) (12.31)

12.3 Equilibrium pair correlation function 473

where C and θ are constants and H is the Hamiltonian (12.8) expressed in terms

of v

and φ

H =



i=1









j=1

j=i







It is easily veriﬁed that (12.31) satisﬁes (12.5) and, assuming that f

like f

normalized to unity, it follows that the constant C is given by

C = 1



exp(−H/θ )

i=1

Knowing f

, we may in principle calculate all the reduced equilibrium distribu-

tion functions though in practice only the calculation of f

is simple. From (12.9)

and (12.31)

(1) =



exp(−H/θ )

i=2



exp(−H/θ )

i=1

(12.32)

The velocity integrations are separable and thus trivial. The substitutions



= r

− r

(i = 2, 3,...,N ) (12.33)

remove r

from both integrands in (12.32) with the result

(1) =

N exp(−mv

/2θ)



exp(−mv

/2θ)

= n



2πθ



3/2

exp(−mv

/2θ) (12.34)

where n is the electron number density. The constant θ may be identiﬁed with the

deﬁnition of temperature T (see Section 12.5)

T =



giving

θ = k

and the equilibrium distribution function (12.34) is thus the Maxwell distribution

(1) = f

(1) ≡ n



2πk



3/2

exp(−mv

/2k

T ) (12.35)

Direct integration of (12.31) to obtain higher-order reduced distribution func-

tions proves to be a formidable task requiring approximation techniques (see Mont-

gomery and Tidman (1964)). However, the equilibrium pair correlation function

474 The classical theory of plasmas

may be obtained indirectly by solving the truncated equation for f

. From (12.27),

with no external forces, this is

∂g(1, 2)

∂t



∂

∂r

+ v

∂

∂r



g(1, 2)

−

∂φ

∂r



∂ f (1)

∂v

f (2) −

∂ f (2)

∂v

f (1)







∂φ

∂r

∂

∂v

[

f (1)g(2, 3) + f (3)g(1, 2)

]

∂φ

∂r

∂

∂v

[

f (2)g(1, 3) + f (3)g(1, 2)

]



(12.36)

which is to be solved for g when f = f

. Note ﬁrst that, since the velocity

integrations in

(1, 2) =

N (N − 1)



exp(−H/θ )

i=3



exp(−H/θ )

i=1

are separable, f

must be of the form

(1, 2) = f

) f

)[1 + p(r

)]

i.e.

g(1, 2) = f

) f

) p(r

) (12.37)

where r

=|r

− r

|. Hence, (12.36) reduces to

− v

) ·



∂p(r

)

∂r

∂φ

∂r



−





∂φ

∂r

p(r

) + v

∂φ

∂r

p(r

)



This equation is valid for arbitrary v

and v

, so choosing v

= 0wehave

∂p(r

)

∂r

∂φ

∂r

=−



p(r

)

∂φ

∂r

(12.38)

The divergence of (12.38) with respect to r

, using (12.13) and

∇

(1/r) =−4πδ(r) (12.39)

then gives

∇

p(r

) −

δ(r

) =



p(r

)δ(r

)

12.3 Equilibrium pair correlation function 475

(∇

− λ

−2

) p(r

) =

δ(r

) (12.40)

It is easily veriﬁed that the solution of (12.40) is

p(r

) =−

4πε

exp(−r

/λ

)

and hence from (12.37) the equilibrium pair correlation function is

g(1, 2) =−f

) f

)

exp(−r

/λ

) (12.41)

This result allows us to examine the assumption that |g|ff. Clearly this is

true for r

>λ

and breaks down only when electrons are sufﬁciently close that

the potential energy of their interaction φ

is of the order of, or greater than, their

mean kinetic energy, i.e. for



4πε

T λ

4πnλ

(12.42)

Provided that the number of electrons in a Debye sphere (4πnλ

/3)  1, this

shows that the approximation is good even within the Debye sphere except for a

tiny region at the centre given by (r

/d)<1/4π(nλ

)

2/3

, where d ≡ n

−1/3

the mean distance between electrons. The chance of two (or more!) electrons being

this close to each other is clearly very small and the dimensionless parameter that

ensures this is the number of particles in the Debye sphere. The inverse of this

number is the small parameter in the weak coupling approximation as can be seen

from (12.41); generally, within the Debye sphere

|g|

∼

4πnλ

Even more pleasing than the consistency of the result of our calculation with

the assumption underlying it is the precise form of (12.41) for it demonstrates

Debye shielding. This is worthy of further examination for we can see exactly

where the shielding has arisen. It is clear from (12.38) that had the integral term on

the right-hand side of this equation been set equal to zero we should have obtained

the solution p(r

) =−φ

T . The effect of this term, therefore, has been to

replace the Coulomb potential by the shielded potential. With hindsight this is not

surprising. The integral term in (12.38) has arisen directly from the integral terms

in (12.36) and these are the only terms in that equation which retain any effect

of the rest of the electrons (labelled 3) on the correlation between electrons 1 and

2; these terms ‘sum up’ such effects and describe the shielding which the other

476 The classical theory of plasmas

electrons provide. In the next section we use this insight to obtain an intuitive and

relatively simple method for ﬁnding g in the general case.

12.4 The Landau equation

The general solution of (12.27) for g as a functional of f ,aswehavealready

observed, is no simple matter. But if we use the insight obtained in the equilibrium

case and assume that the major, indeed the only important effect of the rest of

the electrons on the correlation of electrons 1 and 2 is to replace the Coulomb

potential by the shielded potential then we are left with an equation that is solvable.

Thus, we put the right-hand side of (12.27) equal to zero and replace the Coulomb

potential (12.13) by the shielded potential

4πε

− r

exp(−|r

− r

|/λ

) (12.43)

giving

∂g

∂t

+ v

∂g

∂r

+ v

∂g

∂r

∂φ

∂r



∂ f (1)

∂v

f (2) −

∂ f (2)

∂v

f (1)



(12.44)

In solving this equation we shall use Bogolyubov’s hypothesis which is based

on the observation that time and length scales for changes in g and f are widely

separated. Since particle correlations are limited to the Debye sphere, the length

scale for g is λ

and the corresponding time scale is ω

−1

, the time for an electron

with mean thermal energy to cross the Debye sphere. In contrast, f relaxes to a

Maxwellian under the inﬂuence of collisions on a time scale of τ

, the collision

time, and the corresponding length scale is the electron mean free path λ

, the

distance travelled by a thermal electron between collisions. Since ω

 1 and

 λ

, the assumption of Bogolyubov’s hypothesis is justiﬁed.

Equation (12.44) may be solved by Green function techniques. Deﬁning

G(r

, r



, r



, t, t



) as the solution of

∂G

∂t

+ v

∂G

∂r

+ v

∂G

∂r

= δ(t − t



)δ(r

− r



)δ(r

− r



)

it is easily veriﬁed that G is given by

G = "(t − t



)δ[r

− r



− v

(t − t



)]δ[r

− r



− v

(t − t



)]

where "(t) is the step function

"(t) =



1 t > 0

0 t < 0