On the equivalence between the probabilistic, kinetic, and scattering kernel formulations of the Boltzmann equation

Physica A 164 (19911) 41X1-4111 North-Holland

ON THE EQUIVALENCE BETWEEN THE PROBABILISTIC, KINETIC, AND SCATTERING KERNEL FORMULATIONS OF THE BOLTZMANN EQUATION V.C. BOFFI I)ipartimento di Matemath'a dell'Univer~itit di Boh~gua. via Saragozza ,~¢.4111Z~ Ih~hJgua. haly

V. PROTOPOPESCU I'nghu,ering Physh's and Mathemath's l)ivi,~hm. Oak Rhlge ?v'athmal Lahoratory. Oak Ridge. TN 37,~'31. USA

G. SPIGA l)iparl#llenlo di Matetttath'a dell'Universitb di flari, vhl (;. l'brtu/lato, 70125 Ihlri. hah Received 18 ()ctober 198~,I We prove the equivalcqce between the probabilislic, kinetic, zmd seancring kerqcl fi~rmttlalions of the Boltzmann equation, and we discuss in more detail the features of 1lie scattering

kernel fl)rmulation in both the deterministic and the stochastic ,,ersions.

I. Introduction The evolution equation for the one particle distribution function f ( x , v , t) of a monoatomic rarefied gas not very far from equilibrium was proposed in 1872 by Boltzmann II]. The equation expresses the mass balance (conservation of mass) as a result of free flow and binary collisions in the form

aI'(x, v, t) iJt

+ v.

aI'(x, v, t) iJx

= J(J', J ' ) .

(i.1)

U n d e r certain conditions for the intermolecular potential the collision term can be written down as the difference between two terms, J,,.i - J,,, accounting for the outcome and income of the binary collisioq, respectively 121. If the collision of the molecules is deterministic, then the postcollisional velocities o, w and the precollisional velocities v', w' are related through the conservation laws 0378-4371/90/$03.50 ~ Elsevier Science Publishers ILV. (North-Holhmd)

V.C. Boffi et al. I Equivalence between different formulations of Boltzmann equation

v + w = v' + w' v 2 + w2 = v '2 + w '2

(momentum), (energy).

401

(1.2)

If momentum and/or energy are not preserved, then the scattering process is called stochastic. Although not considered originally by Boltzmann in his theory of gases, stochastic collisions (and the stochastic models generated by them) became lately of importance, in connection with (i) several extensions of the Boltzmann equation to heat baths and inelastic collisions and (ii) finding exact solutions for the Boltzmann equation [3]. Boltzmann expressed the collision term in what is now called the kinetic formulation. Two more formulations have been developed since, known as the probabilistic [4] and the scattering kernel [5, 6] formulation, respectively. Partial connections between these three formulations have been established in refs. [6-8], but their complete equivalence has not been considered yet and thus there remain some fuzzy areas which may lead to possible confusion. The aim of this paper is (i) to develop a complete and coherent presentation of the equivalence relationships existing between the three formulations above, and (ii) to discuss in more detail the implications of the conservation laws on the scattering kernel formulation. The equivalence proof will allow one to use, in different contexts, the formulation most convenient for the problem at hand. (We mention, for instance, that the scattering kernel formulation turned out to be advantageous in proving global existence [9] and global stability [10] for certain stochastic models of the Boltzmann equation.)

2. The three formulations of the collision term

Historically, the kinetic formulation was proposed first [1]; the probabilistic formulation was developed in connection with it [4], and the scattering kernel formulation was deduced from the probabilistic one [7]. We shall adopt here a more "axiomatic" viewpoint, namely we shall enunciate the three formulations, postulate the relationships between W, ! and H (see the notation henceforth) and we shall prove the equivalence. (i) In the kinetic formulation, the collision term reads [2]

J(f, f)--

dw f g/(g, x) [f(v') f(w')- f(O f(w)l dO', R3

(2.1)

4~

where the postcollisional velocities v, w and the precollisional velocities v', w' are related through

4112 V.C. Boffi et al. I Equivalence between different formulations of Boltzmmm equation

v'=G+

~ggt',

w ' = G - ~ggt',

(2.2)

where G = ~(v + w ) = ~ ( v ' + w') is the velocity of the center of mass of the two colliding molecules, and g = Iv - w[ = Iv ' - w' I is the common modulus of the relative velocities g = v - w and g' = v' - w'. (Since collisions are local in both time and space, from now on we shall specify only the velocity among the arguments of f.) Notice that the deflection angle of the relative motion, X = a r c o s ( O . ,O'), with g] = ( v - w ) / g and , O ' = ( v ' - w ' ) / g ' . instead of the more common angle 0 between v - w and the apse line, has been used as independent variable to account for the angular dependence of the differential scattering cross section /. The angles are simply related by )¢ = ~ - 20. The cross section / ( g , X) enjoys a special symmetry, namely

(2.3)

t ( g , X) = t ( g , ~ r - x ) .

only when the collision outcome is not sensitive to the exchange v' ~ w'. Even if this symmetry is not true, one can still use a symmctrized differential cross section, (2.4)

l'(g, X) = ½[l(g, X) + l ( g , ~r - X)I, when computing the collision term, since i ( g , X) and I ( g , ~ r - X ) same result when integrated out in (2.1).

yield the

(ii) in the probabilistic version of the Boltzmann equation, the collision term reads [4]

J(r, f)= f f f wto,, w,; v ,

w)

[f(v')

f(w') - f(v)

R3 R 3 R3

J'(w)] dw d v '

dw' ,

(2.5)

where W ( v ' , w'; v, w), the microscopic probability distribution for the scattering process (v', w')---'-(v, w), is given by [2, 31

W(v', w';

v , w) = 8 1 ( I v ' - w'[, x ) 8 ( v + w - v' - w ' )

x 8(v 2 + w 2 - v'" - w ' : ) .

(2.6)

In terms of the center of mass and relative velocities before and after the collision we rewrite W ( v ' , w'; o, w) in the form [3,7]

V.C. Boffi et al. I Equivalence between different formulations of Boltzmann equation 403 W(v', w'; v, w) = l ( g , X) 8(G - G') 8( l g 2 - / g , 2 )

1 = ~ l(g, X) 8(G - G') 8(g - g').

(2.7)

W satisfies the following symmetry properties [3]: (a) interactional symmetry of identical particles, W(v', w'; v, w) = W(w', v'; w, v) ;

(2.8)

(b) detailed balance symmetry (microscopic reversibility), W(v, w; v', w ' ) = W(v', w'; v, w) ;

(2.9)

(This symmetry is not valid, in general, for polyatomic gases.) (c) special symmetry, W(v', w'; v, w) = W(w', v ,*" v, W).

(2.10)

This symmetry is also not of a very general nature. However, even when (2.10) is not satisfied, we can use a symmetrized probability distribution in the collision term (2.5) by noticing, as before, that their contributions in the collision integral are identical. (iii) In the scattering kernel formulation,

J(f, f) is expressed as [7]

J(f. f)= f f v.(v', w') rl(v', w'; v) f(v') f(w')dv' dw' R3 R3

- f(v) f v.(v, w) f(w) dw,

(2.11)

R3

where

f W(v',w';v,w)dv'dw'

(2.12)

R3 R3

and r

P

.def /

v,(v', w') lI(v', w ; v) =

d

R3

W(v', w'; v, w) d w .

(2.13)

4114 V.C. Bo].'fi et al. / Equivalence between different ]ormulations o f Boltzmatm equation By taking into account expression (2.6), one can rewrite [6.71

-gl ( g , x ) 8 ( g - g ' ) ~ ( G - G ' l g ' - d g ' d g F d G '

p,(v, w) = Ri R~

- g S I(g,

(2.14)

X) d$'] = ,,.11o - wl)

R

and e,(v',

w') ll(v',

w'; v) = 4/(Iv' -

w'l. X*) ~[(v -

v ' ) . (v -- w' )1 {2.15)

with i~ ~ ~

. -., v~ ' -.~ w'- ) =- arccos( (v - w)-{v', - w') a r c c o s / 2 v - ( v ' -. . .w')

Iv'- w I-

;: P

-)

~2.16)

3. The equivalence of the three formulations With the above notation and definitions, the equivalence between the kinetic, probabilistic and scattering kernel formulation can be established quite easily, it is sufficient to show that (i)<--,(ii) and (iii),,--,(ii).

(ii)---, (i) This implication is well known [4, 71 and we summarize it here briefly fl~r the reader's convenience. Introducing W(v', w'; v, w), given by (2.7)• into the probabilistic fl~rm of the collision term (2.5), we obtain the kinetic form (2. I ).

(i)--,(ii) The proof of this implication is based on the simple observation that all the steps made in the previous calculations are reversible. Indeed, ,~tarting from (2.1), we can write

f dw S gl(g, X)[f(v')f(w')...f(v)/'(w)] I~. 4

d"'

4~ '.

P,~

4rr

li

× j ~(G - G ' ) [ f ( v ' )

g

jIw') -f(v) f(w)ldG'

V.C. Boffi et at. I Equivalence between different formulations of Boltzmann equation 405

R3

R3

× [f(v')

R3

f(w')-f(v)

f(w)l dG'

= f dw f dv' f W(v',w';v, w)[f(v')f(w')-f(v)f(w)ldw'. R3

R3

(3.1)

R3

( i i ) ~ (iii) This implication was proved already in ref. [6]. Actually, it is sufficient to use into (2.5) definitions (2.12), (2.13) in order to get (2.11).

(iii)--> (ii) Starting from the expressions of/-/(v', w'; v) and v,(v', w') given by (2.12) and (2.13) we transform the collision term in (2.11) as follows:

,( ~,(v', w') n(v', w'; ~) f(o') f(w')do' dw'-f(~) f ~,(v, w) f(w) dw R3 R3

R3

= f f f(o') f(w')do'dw' f W(o',w';~,w)dw R3 R3

R3

-,
R3 R3

= f f f w
(3.2) In conclusion, the three formulations of the Boltzmann equation considered here are completely equivalent, provided the correct expressions (2.6), (2.14), (2.15) of W, v~ and H, respectively, are adopted. Of course, other expressions could be used for special purposes, but they only give models or approximations of the Boltzmann equation.

4. Properties of H(v', w ", v) If W(v', w,r° v, w) possesses the symmetry (2.10), then from the definitions (2.13) and (2.15) follows that

406

V. C'. Boffi et al. I Equivalence between different formulations of Boltzmwm equation

ll(v', w '", o) = / / ( w ' .

v '", o)

(4.1)

w'; v ' + w' - v ) ,

(4.2)

and ll(v',

w': v ) = I I ( v ' ,

expressing indistinguishability of the colliding particles. An important characteristic feature of H is that

f

g/(v) ll(v', w'; v ) d v = ~ [ ~ ( v ' )

(4.3)

+ ~(w') I ,

R.t

¢ representing any of the collisional invariants 1, v, v" [21. Eq. (4.3) follows if one notices that

ff

[~b(v) +

qffw)]W(v', w'; v, w) dvdw

R~ R~

=2f ~b(v)dv f W(v',w';v,w)dw R~

R3

= 2p,(lv' - w'l)

f ll(v', w'; v) ~(v) dv

(4.4)

R~

and simultaneously, because of the definition (2.6),

f f l~(o) + ~,(w)lW(v'. w': o. w)dvdw R3 R3

= [ ~ ( v ' ) + q,(w')]

f W(v', w'; v, w) dv dw R3

(4.5)

= ~,(Iv' - w'l) i¢,(v') + ¢ , ( w ' ) l .

Comparison between (4.4) and (4.5) leads just to (4.3). In particular, we get

f ll(v', w'"l v ) d, v R3

f R3

=

•

f vn(v',w',v)dv''. = 2(v

+ w'),

R~,

v"ll(v', w",

v)dv = ~(v"- + w'-'),

(4.6)

V.C. Boffi et al. I Equivalence between different formulations of Boltzmann equation

407

of clear physical meaning (average momentum or energy of considered particle after collision equal to half of the total momentum or energy before a collision). We further note that the property (4.3) is sufficient for

f

(4.7)

~b(v) J(f, f ) dv = O,

R3

namely for conservation of particles, momentum and energy in the whole process (in the spatially homogeneous case). In fact

f q,(v) J~(f, f) dv

R3

-- f f ~s ,<~,> ,~o, ~w, f n ~~o R3 R3

R3

- -~ f f ~l~'-w'l) [~o') + ~w,)]f~.,)f~w,)~o,

,w,

R3 R3

R3

R3

R3

(4.8) Finally, it is worth commenting that essentially all the results of the present section remain true even if one were dealing with an unsymmetrized differential cross section 1 (or, equivalently, transition probability W). There are only two differences to be pointed out. Due to the lack of the property (2.10), eqs. (4.1) and (4.2) would not be in order, but it would be easy to obtain l l ( w ' , v'; v' + w' - v) = l l ( v ' , w'; v) .

(4.9)

Correspondingly, the conservation property (4.3) should be modified in

f

¢,(v) [//(v', w'; v) +//(w', v'; v)] dv = ¢,(v') + ¢,(w')

(4.10)

R3

with ~ -- 1, v, v 2 and would imply again conservation of mass, momentum and energy.

408

V.C. BoJ]i et al. I Equivalence between different fonnulmiol~s of Bohzmann equathm

5. Discussion Among the three formulations discussed above, the scattering kernel formulatkm of the Boltzmann equation lent itself to a somewhat controversial use and interpretation. The formulation was originally conceived [5] as an extension to the nonlinear Boltzmann equation of the linear neutron transport formalism, where momentum and energy are not supposed to be preserved during the scattering of the neutrons by the background. The scattering kernel formalism remains perfectly valid in the nonlinear case (with either stochastic or deterministic collisions), and we have proved here that it is indeed equivalent with the other two formulations in the deterministic case. As mentioned before, stochastic models have been used [3,9.10] in the framework of the nonlinear Boltzmann equation for different technical reasons. In particular, the continuity assumption made in ref. [10] (theorem 7) to prove stability for the solutions of the Boltzmann equation cannot be satisfied if ll(v', w'; v) is given the deterministic interpretation (see (2.15)). For stochastic models in which 11 satisfies only the relationships

ll(v', w': v)

f

=

ll(w', v': v) ,

(5.1)

(5.2)

li(v', w': v) d v = I .

R3

with v', w'. v independent velocities, one cannot expect conservation of momentum and energy. On the other hand, if ,,(v, w) and ll(v', w': v) are derived from eqs. (2.12) and (2.13) [6, 7], then all the conservation laws will be automatically satisfied; in this case only eight of the nine components of v. v', w' are independent. If one wants a stochastic model to satisfy also momentum and energy conservation, further assumptions must be imposed on 11 in addition to (5. l) and (5.2). A sufficient condition for that would be the second and the third of eqs. (4.6). We remark that the further addition of property (4.2) to (5.2) implies property (4.3) for all functions ~(v) that are linear in v. in particular for ~t,(v) = v, and guarantees thus not only mass, but also momentum conservation. In fact

f

R3

~(v)ll(v',

w'; v) dv = f

R3

~ ( v ' + w ' - v ) I I ( v ' . w '", V t + W' - v )

dv

V.C. Boffi et al. I Equivalence between different formulations of Boltzmann equation 409

--

f ~ [¢(v') + ¢(w') - ¢,(v)]n(v', w'; v) dv R3

= ~(v') + ¢,(w') - ~ ~(v)/-/(v', w ,'. v) dv .

(5.3)

R:~

An equivalent, more complicated, proof can be found in ref. [8]. Energy conservation does not follow from (4.2) and (5.2). It is, indeed, a consequence of the actual structure of H, in particular of the delta factor in the r.h.s, of (2.15) with support on the sphere centered at (v' + w')/2 and with radius Iv'-w'l12. In the special case of a constant collision frequency, the scattering kernel is explicitly given by

n(v', w'; v)= 1 e(x*) ~ [ ( v - v ' ) . ( v Iv'-wq

w')l

(5.4) '

namely the same result obtained in a different way in ref. [8] for Maxwell particles by an "ad hoc" hypothesis on invariance under velocity rescaling. Eq. (5.4) is easily verified by bearing in mind that, if v,(lv'- w'[) is a constant, then

l(g, X) = ~ "g e ( x ) ,

21 a e ( x ) sin X dx = 1.

(5.5)

0 In conclusion, by their very nature, stochastic models should not be expected to preserve momentum and energy. If one wants to preserve these conservation laws, one has to start, consistently, from the deterministic definition and interpretation, in which case no further assumptions are necessary.

Acknowledgements We acknowledge the partial support of C.N.R.'s National Group for Mathematical Physics, Project PS-MMAI for V.C.B. and G.S. and C.N.R.'s Visiting Professor Program for V.P.

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410

V.C. BoJ'fi et a L I

Equivtdence between different formtdations of Boltzmutm equatiott

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