Differential Form of the Collision Integral for a Relativistic Plasma

0.08 MB·English

by  Bastiaan J. Braams

PPPL–2467(Aug. 1987)
Phys. Rev. Lett. 59(16), 1817–1820(Oct. 1987)
DIFFERENTIAL FORM OF THE COLLISION INTEGRAL
FOR A RELATIVISTIC PLASMA
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0 Bastiaan J. Braams and Charles F. F. Karney
0
2 Plasma Physics Laboratory, Princeton University, Princeton, NJ 08544.

n
a The differential formulation of the Landau-Fokker-Planckcollision integral is developed for
J the case of relativistic electromagnetic interactions.

3 PACS numbers 52.25.Dg, 52.20.−j, 52.60.+h
1
Kinetic theory is founded upon the Boltzmannequation, which is a conservationequa-

] tion for the phase-space distribution function of each species in an ensemble of interacting
h
particles. Forthe caseofCoulombinteractions,Landau1 expressedthe collisionterminthe
p
- Fokker-Planck form. This mixed integro-differential representation was extended to rela-
m tivistic electromagnetic interactions by Beliaev and Budker.2 For the nonrelativistic case,
s it was shown by Rosenbluth et al.3 and by Trubnikov4 that the integrals appearing in the
a
collision term can be expressed in terms of the solution of a pair of differential equations.
l
p The present work extends that formulation to the relativistic collision integral. Using an
.
s expansion in spherical harmonics the relativistic differential formulation is then applied to
c
calculatethescatteringandslowingdownoffastparticlesinarelativisticequilibriumback-
i
s ground plasma. Our work is relevant to the study of high temperature plasma in fusion
y
energy research and in astrophysics.
h
In the work of Landau1 and that of Beliaev and Budker,2,5 the collision term that
p
[ occurs on the right-hand side of the Boltzmann equation for species a and describes the

effect of collisions with species b is written in the Fokker-Planckform,
1
v ∂ ∂f
7 Cab = ∂u ·(DDDDab· ∂ua −Fabfa), (1)
6
0 in which the coefficients DDDD and F are defined by
ab ab
1
0 q2q2
5 DDDDab(u)= 8πǫa2mb 2 logΛabZ UUUU(u,u′)fb(u′)d3u′, (2a)
0 0 a
/ q2q2 ∂
s F (u)=− a b logΛ ·UUUU(u,u′) f (u′)d3u′. (2b)
c ab 8πǫ2m m abZ (cid:18)∂u′ (cid:19) b
0 a b
i
s
y Here,faandfbarethedistributionfunctionsforthetwospecies,uistheratioofmomentum
h to species mass, q and q are the species charge, m and m are the species mass, ǫ is
a b a b 0
p the vacuum dielectric permittivity, and logΛ is the Coulomb logarithm. The kernel UUUU
ab
:
v is specified below. This form of the collision operator is only approximate because of the
i introduction of cutoffs in the collision integral. More accurate operators that take into
X
accountDebyeshieldingatlargeimpactparametersandlarge-anglescatteringandquantum
ar effects at small impact parameters have been derived.6,7 The purpose of this letter is to
presentadifferentialformulationfortheintegraltransformsthatoccurinEqs.(2). Toavoid
unnecessary clutter we discardthe factor that depends only on the species properties, drop
the species subscript, and consider the transforms
1
DDDD(u)= UUUU(u,u′)f(u′)d3u′, (3a)
8π Z
1 ∂
F(u)=− ·UUUU(u,u′) f(u′)d3u′. (3b)
8π Z (cid:18)∂u′ (cid:19)
1
For guidance, let us recall briefly the nonrelativistic theory.3,4 In that case the
momentum-to-mass ratios u and u′ reduce to the velocities v and v′, and the collision
kernel is the one given by Landau,1 UUUU =(|s|2IIII −ss)/|s|3, where s=v−v′. It may be seen
thatUUUU =∂2|s|/∂v∂vand(∂/∂v′)·UUUU =−2∂|s|−1/∂v. Toobtainthedifferentialformulation,
these representations are inserted into Eqs. (3), and the differentiation with respect to v is
moved outside the integration over v′. Defining the potentials h(v) = −(1/8π) |s|fd3v′
and g(v) = −(1/4π) |s|−1fd3v′, we have DDDD = −∂2h/∂v∂v and F = −∂g/∂v.R Further-
more, from ∆|s|=2|sR|−1 and ∆|s|−1 =−4πδ(s) it follows that h and g obey the equations
∆h = g and ∆g = f. (∆ denotes the Laplacian with respect to the variable v.) These
equations provide the differential formulation of the collision term in the nonrelativistic
case.
The Landau collision kernel was obtained in a semi-relativistic fashion, assuming
Coulomb collisions and relativistic particle kinematics. It is a good approximation to the
fully relativistickernelgivenbelowprovidedthat|v.v′|≪c2,whichistrue whenoneofthe
colliding particles is nonrelativistic. However, the reduction of the collision integral to the
differential form of Rosenbluth and Trubnikov relies on the stronger assumptions |v|2 ≪c2
and |v′|2 ≪ c2, and is therefore entirely nonrelativistic. A differential formulation that is
exactly equivalent to the Landau collision integral was given by Franz.8
We turn now to the differential formulation of the relativistic collision integral due to
Beliaev and Budker.2,5,6 They obtained the expression
r2/(γγ′)
UUUU(u,u′)= (r2−1)IIII −uu−u′u′+r(uu′+u′u) , (4a)
(r2−1)3/2
(cid:0) (cid:1)
in which γ = 1+|u|2, γ′ = 1+|u′|2, and r =γγ′−u.u′. (We set c=1 in this part of
the paper.) Opne finds p
∂ 2r2/(γγ′)
·UUUU(u,u′)= (ru−u′). (4b)
∂u′ (r2−1)3/2
Notice that r is the relativistic correction factor for the relative velocity between the two
particles (i.e., for the velocity of one particle in the rest frame of the other). Conversely,
this relative velocity is given by r−1(r2−1)1/2.
In developing a differential formulation for the collisionterm based on the Beliaev and
Budker kernel, it is helpful to work in terms of relativistically covariant quantities. The
expression γγ′UUUU is equal to the space part of a four-tensor W that depends on the four-
vectors u=(γ,u) and u′ =(γ′,u′),
r2
Wij(u,u′)= (r2−1)gij −uiuj −u′iu′j +r(uiu′j +u′iuj) , (5a)
(r2−1)3/2
(cid:0) (cid:1)
where gij is the metric tensor, with signature −+++. (r =−u u′i is clearly a four-scalar.)
i
The tensor W is symmetric (Wij = Wji), symmetric in u and u′, satisfies u Wij =0, and
i
satisfies Wi =2r2(r2−1)−1/2. Likewise γγ′(∂/∂u′)·UUUU is the space part of the four-vector
i
V, where
2r2
Vi(u,u′)= (rui−u′i). (5b)
(r2−1)3/2
If the relativistic differential formulation is to parallel most closely the nonrelativistic
formulation,thenoneshouldfindarepresentationoftheformWij =HijψandVi =−2Giϕ,
where ψ and ϕ are four-scalars depending on u and u′, and Hij and Gi are covariant
differential operators acting on the variable u. In the nonrelativistic limit, ψ should reduce
to |v−v′| and ϕ should reduce to |v−v′|−1. It should be possible to transform ψ and
ϕ to delta functions by a sequence of second-order differential operators. The potentials
2
would be defined as h = −(1/8π) (ψf/γ′)d3u′ and g = −(1/4π) (ϕf/γ′)d3u′; these
expressions define four-scalars (cf. RRef. 5). The differential equationsRsatisfied by h and g
follow immediately from those satisfied by ψ and ϕ. Finally, DDDD would be obtained as the
space part of −γ−1Hijh and F as the space part of −γ−1Gig. In fact, it will turn out that
the relativistic formulation has to be somewhat more complicated, but not fundamentally
different from the outline just sketched.
′
A function of the four-vectors u and u that is a four-scalar must be a function of
r = −u.u′ alone. The form of the differential operators Hij and Gi is restricted because
these should be interior operators on the surface u2 = −1 in four-space. In addition, it is
required that Hij =Hji and u Hij =0. Under those restrictions it is found that the most
i
generalformofHij andGi,uptoamultiplicativeconstant,isHijχ=Lijχ+α(gij+uiuj)χ
and Giχ=Kiχ+βuiχ. Here, α and β are arbitrary constants, and
∂2χ ∂χ
Lijχ=(gik+uiuk)(gjl+ujul) +(gij +uiuj)um , (6a)
∂uk∂ul ∂um
∂χ
Kiχ=(gik+uiuk) . (6b)
∂uk
The spatial part of Lijχ is LLLLχ and that of Kiχ is Kχ where
∂2χ ∂χ ∂χ
LLLLχ=γ−2 −v − v, (7a)
∂v∂v ∂v ∂v
∂χ
Kχ=γ−1 , (7b)
∂v
in which v=u/γ, and ∂/∂v=γ(IIII +uu)·∂/∂u. If χ is a function of r alone then
d2χ dχ
Lijχ= (rui−u′i)(ruj −u′j)+r (gij +uiuj)
dr2 dr
and Kiχ=(dχ/dr)(rui−u′i). One is thereby led to the representations
Wij = Lij +gij +uiuj r2−1
(cid:2) − Lij −gij −u(cid:3)piuj rcosh−1r− r2−1 , (8a)
Vi =−2K(cid:2)i r(r2−1)−1/2−(cid:3)(cid:0)cosh−1r . p (cid:1) (8b)
(cid:0) (cid:1)
These representations for W and V are only suitable for constructing a differential
formulation of the collision term if the functions that occur on the right-hand sides can be
reduced to delta functions by some sequence of differential operators. For that purpose the
contraction L=Li is needed; in terms of the three-space variables it is
i
∂2χ ∂χ
Lχ=(IIII +uu): +3u· . (9)
∂u∂u ∂u
If χ is a function of r alone, then Lχ=(r2−1)(d2χ/dr2)+3r(dχ/dr) away from r =1; at
r =1 (or u=u′) there may be a singularity. Specifically, it is found that
L r(r2−1)−1/2 =−4πγδ(u−u′),
[L+1(cid:0)] (r2−1)−1/2(cid:1)=−4πγδ(u−u′),
(cid:0) L cosh−1r(cid:1)=2r(r2−1)−1/2,
[L−3](cid:0) r2−1(cid:1)=2(r2−1)−1/2,
[L−3] rcosh−1r−(cid:0)pr2−1(cid:1)=4 r2−1.
(cid:0) p (cid:1) p
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TheexplicitformofthedifferentialrepresentationofEqs.(3)basedonthe Beliaevand
Budker collision kernel2 follows: The potentials are
h =−(1/4π) (r2−1)−1/2f(u′)/γ′d3u′, (10a)
0
Z
h =−(1/8π) r2−1f(u′)/γ′d3u′, (10b)
1
Z
p
h =−(1/32π) rcosh−1r− r2−1 f(u′)/γ′d3u′, (10c)
2
Z
(cid:0) p (cid:1)
g =−(1/4π) r(r2−1)−1/2f(u′)/γ′d3u′, (10d)
0
Z
g =−(1/8π) cosh−1rf(u′)/γ′d3u′. (10e)
1
Z
These potentials satisfy the differential equations
[L+1]h =f,
0
Lg =f,
0
[L−3]h =h , (11)
1 0
Lg =g .
1 0
[L−3]h =h ,
2 1
Finally one obtainsDDDD and F as
DDDD(u)=−γ−1[LLLL+IIII +uu]h +4γ−1[LLLL−IIII −uu]h , (12a)
1 2
F(u)=−γ−1K(g −2g ). (12b)
0 1
Equations (11–12) together with the definitions, Eqs. (7) and (9), provide the differential
formulation in the relativistic case.
In order to proceed further analytically, it is useful to decompose the distribution
function and the potentials in spherical harmonics, e.g.,
∞ n
f(u,θ,φ)= f (u)Pm(cosθ)exp(imφ). (13)
nm n
nX=0mX=−n
Here u = |u| (different from the convention used earlier), θ is the polar angle, and φ is
the azimuthal angle. The equation [L−α]g = f is equivalent to the system of separated
equations [L −α]g =f , where
n nm nm
d2y dy n(n+1)
[L −α]y =(1+u2) +(2u−1+3u) − +α y. (14)
n du2 du (cid:18) u2 (cid:19)
After the changeof variablex=sinh−1u and the change of unknown z =(sinhx)−ny, then
the equation [L −α]y =w transforms to [D −a2]z =(sinhx)−nw, where a2 =α+1 and
n n
d2z dz
[D −a2]z = +2(n+1)(cothx) + (n+1)2−a2 z. (15)
n dx2 dx
(cid:0) (cid:1)
Thesolutiontothehomogeneousequation[D −a2]z =0isrequiredinordertoconstructa
n
Green’s function for the problem. To obtain this solution we note the following recurrence:
If zn−1,a solves [Dn−1−a2]z = 0, then zn,a = (sinhx)−1(d/dx)zn−1,a solves [Dn−a2]z =
0. Furthermore, for n = −1 the homogeneous equation is trivial to solve. However, the
recurrence breaks down in the case that a is an integer. If a = n, then zn−1,a = 1 solves
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[Dn−1 −a2]z = 0, and differentiation produces the null solution to [Dn −a2]z = 0. The
recurrence must then be restarted from the general solution to [D −n2]z =0, which is
n
x
z =(sinhx)−2n−1 C +C (sinhx′)2ndx′ .
n,n 1 2
(cid:18) Z (cid:19)
0
The integral that occurs here can be expressed in closed form.
The Green’s function allows us to reduce the separated ordinary differential equations
to quadrature. An important special application for these results is in the treatment of
collisions off an equilibrium background distribution. Assuming that f is a stationary
b
Maxwellianwithdensityn andtemperatureT andthattheenergyofthecollidingparticles
b b
greatly exceeds T , we obtain
b
K u2 K u2
D =Γ 1 tb 1− 0 tb ,
uu abK v3 (cid:18) K γ2c2(cid:19)
2 1
1 K u2 u2 K u2 u2
D =Γ 1− 1 tb + tb + 0 tb tb ,
θθ ab2v(cid:20) K (cid:18)u2 γ2c2(cid:19) K u2 γ2c2(cid:21)
2 2
and F = −(m v/T )D . (The other components of DDDD and F vanish.) Here we have put
u a b uu
theexpressionsforDDDD andFintodimensionalformasinEqs.(2),K isthenth-orderBessel
n
functionofthe secondkind,the argumentforthe Besselfunctions is m c2/T , u2 =T /m ,
b b tb b b
and Γ =n q2q2logΛ /(4πǫ2m2). The errors are exponentially small in u/u .
ab b a b ab 0 a tb
To conclude, we have presented a differential formulation for the Beliaev and Budker2
relativistic collision integral. This permits the rapid numerical evaluation of the collision
term. Adecompositionintosphericalharmonicsisusefulincarryingoutanalyticalwork. It
alsoprovidesaconvenientmethodforcalculatingtheboundaryconditionsforthepotentials.
We are grateful to N. J. Fisch for several enlightening discussions. This work was
supported by DoE contract DE–AC02–76–CHO–3073.
1L. D. Landau, Phys. Z. Sowjet. 10, 154 (1936).
2S. T. Beliaev and G. I. Budker, Sov. Phys. Doklady 1, 218 (1956).
3M. N. Rosenbluth, W. M. MacDonald, and D. L. Judd, Phys. Rev. 107, 1 (1957).
4B. A. Trubnikov, Sov. Phys. JETP 7, 926 (1958).
5E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics, Course of Theoretical Physics, Vol.
10 (Pergamon,Oxford, 1981), Section 50.
6Yu. L. Klimontovich, The Statistical Theory of Non-Equilibrium Processes in a Plasma
(Pergamon, Oxford, 1967).
7B. Bezzerides and D. F. DuBois, Ann. Phys. (NY) 70, 10 (1972).
8M. R.Franz, “TheNumericalSolutionof the Relativistic Bounce-AveragedFokker-Planck
EquationforaMagneticallyConfinedPlasma,”UniversityofCaliforniaReportUCRL–
96510 (April, 1987).
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