Kinetic equations with soluble moment equations

Volume 81A, number 6

PHYSICS LETITERS

2 February 1981

KINETIC EQUATIONS WITH SOLUBLE MOMENT EQUATIONS M.H. ERNST and E.M. HENDRIKS Instituut voor Theoretische Fysica der Rijksuniversiteit, Utrecht, The Netherlands Received 25 November 1980

If the moments of the kernel in a kinetic equation are polynomials, then the moment equations have a symmetry property, which permits the general solution in the form of a Laguerre series, and its coefficients (Laguerre moments) satisfy the same equations as the ordinary moments.

1. In the last few years the Boltzmann equation has been solved for several models: for a so-called very hard particle model the general solution was given in closed form [1]; for Maxwell models a special solution (BKW-mode) was found by Bobylev, Krook and Wu [2], as well as the general solution [3,4]. The latter was obtained in the form of a series in orthogonal polynomials of which the coefficients, the so-called polynomial moments, can be solved sequentially from a set of nonlinear moment equations. In this way the general solution of the Boltzmann equation for an isotropic distribution functionf(v, t) with u = lul was found in the form of a Laguerre series, in close analogy with the Hermite series solution for the distribution functionf(v, t), obtained a long time ago by Kac [5] for a one-dimensional model Boltzmann equation where v E (_oo, oo) The purpose of this note is to introduce a new class of kinetic equations for which the polynomial moments can be derived and solved in a simple manner, so that the distribution function can be obtained by inversion from these moments.

gy-like scalar E (0, F(x) =

~

oc),

will in general have the form

ff du dy [K(xy; u)F(y)F(u

(1)

—

K(yx; u)F(x)F(u x)]. The scattering kernel K(xy;u) represents the transition rate in the binary interaction (y, u y) (x, u x) [which in general is different from the rate for (y, u y) (u x, x)]. From the conservation of energy it follows that the kernel vanishes outside the region where x
—

—

—

-+

-~

—

—

00)

x

e~’/F(m).

The kernel should have the following symmetry properties [6] It should: (i) show the interactional symmetry K(xy; u) = K(u x, u y; u);(ii) obey the detailed balance condition ~.

—

—

K(xy; u)F~(y)F~m)(u y) —

2. We consider a system of particles interacting only through binary interactions in which the total energy is conserved. The energy might be the kinetic energy of a particle withd translational degrees of freedom (d = 2m = 1, 2, 3, ), or the internal energy ‘of a molecule with m degrees of freedom in which energy can be stored (m = 1, 2, The time evolution of the distribution function F(x, t), in which x is an ener...

...).

=

K(yx; U)F~m)(X)F~m)(U

—

x);

and (iii) integrate to a well-defined collision rate m0(y, u)

f~K(xy;u)dy. We will further restrict ourselves to Maxwell models, such that the collision rate is independent of the energy, i.e. m0(y, u) = constant, and define a particular kernel in the class by specifying its moments m~(y,u) =

0 031—9163/81 /0000—0000/$ 02.50 © North-Holland Publishing Company

315

Volume 81A, number 6

PHYSICS LETTERS

saf dxxflK(xy; u) in the following way: for all n we require m0(y, u) to be a polynomial of degree n in the energy variablesy and u, and since m~(y,u) has the dimension [energyJ~?,it must have the homogeneity property m~(Xy,Xu) = X’~m~(y; u). This yields

where 1F1 is confluent hypergeometric function. The Laguerre moments are defined through (5) as c~(t) = (1F1(—n, m;x)). Using the explicit polynomial form of the function 1F1(—n, m;x) one sees: c,~=

f dxx??K(xy; u)= i~ n

U

m~(y,u) =

k=0

0

(2)

(3)

PnkA”kMn_k

where M~(t)= (~~)/(~°>=, with (x’~)given by fdxxrz X F(x, t) and (x’~ = (m)~ F(m +n)/F(n) is the nth moment of the equilibrium distribution F0(m)(x), and Pnk

=

A~k(m)k (m)flk/(m)n.

(4)

By choosing proper time units, we can set ji~~equal to 1. Of course, the equations for the energy moments can be solved sequentially, starting at n = 2, 3, since M0(t) = M1 (t) = 1. However, the purpose of this letter is to show that a different set of moments, namely the Laguerre moments c~(t)(to be defined below), satisfies the same set of moment equations (3), i.e. the coefficientsAflk or ~nk Bobylev symmetry [2]. Bobylev observed thathave the the Boltzman equation for Maxwell molecules is invariant under a one parameter semigroup of transformations. We use the term Bobylev symmetry for an invariance property formulated in terms of the coefficients ~nk’ which is a consequence of it. Once the equations for the Laguerre moments Cn(t) can be solved for given initial conditions {c~(0)}, the general solution can be found by inversion, i.e. ...

F(x, t)

=

~

F~m)(x)

C~(t)E~m_l)(X),

n=0

(5)

where the generalized Laguerre polynomials are defined as .e(m_l)(x)

=

1F1(—n, m;x)(m)~/n!, 316

~

(;:)

(6)

(_)kMk

Aflkyk(u _y)~~.

As this homogeneity property holds for all n, the kernel satisfies the homogeneity relation K(\x, Xy Xii) = X~lK(xy; u). From (1) and (2) the equations for the normalized moments M~(t)follow: Mn + ,i00M~=

2 February 1981

We define the combinatorial factor to be zero outside the region where 0 ~ k ~ n. The inverse relation is obtained by simply interchanging the symbols c and M Note that c0 = I and c1 = 0. 3. The Bobylev symmetry, implying that the nonlinear set of coupled moment equations is invariant under the transformation (6), requires that the coefficients ~~nksatisfy the relations ~

=

)

(P)(k)(n —k Pnk n (_)n+X+v,

(7)

for all (p, X, p) as follows from inserting (6) into (3). We define ~nk = 0 for k > n and let the summation variables k and n run through all non-negative integers. For a given p there are ~(p + l)(p + 2) different coefficients ~nk occurring in these equations. In order to count the number of equations, we see that every summand on the rhs of (7) is only non-vanishing for 0 ~ X ~ k ~ n ~ p and 0 ~ v n k, so that 0 ~ X ~ p and v + X ~
...

PnX~/1kk(k)(X)(Y~”’

(8)

which is equivalent to (7). Next, we will show that for the kernels defined by (2) the symmetry (7) is a consequence of the detailed balance condition. In view of this condition, and the homogeneity property of the kemelKit is convenient to introduce a function Q(x,y) by means of the relation: 3 [y(u — y)]l—mQ(xu—l ;yu l). (9) From balance it follows that Q(x,y) = K(uy; detailed u) = u2m

Volume 81A, number 6

PHYSICS LETTERS

Q(y, x). On multiplying (2) withy’~and integrating over the range 0 ~y ~ u, one verifies that the condi-

~nn

=

2 February 1981

f d2~(P)(~4~)~

(13)

tion of detailed balance gives for the coefficients sunk,

(m+ff)k

n

~I~nk

(m+n)k

‘~

1nk

(m)k

kOl

(m)k

and we deduce directly from (8) that

(10)

valid for all (n, n). For a given fixed p one has 0 ~ n
k

Since (2) and (4) imply that

~nn

=

(14)

f~dxx’K(x,1; 1)

we are led to the following identification: g(ji) = ~K(~(l + ji), 1; 1) (15) This construction of the function g(p) for the models defined by (2) brings them within the class of Maxwell .

(n\ 1

(I) Mnk=\1JPll~

k

(n~ I n—k Pnk= k/f dpg(p)(~)(i~_~_~)

(11)

One can solve this set by multiplying with (~)(_)1+X and summing over X. The solution is identical to (8), which proves that the ansatz (2) together with the detailed balance property implies the Bobylev symmetry. Proof Taking ñ = 0 in (10) shows that (11) is true for / = 0 and all n. Next assume that (11) is true for all l<ñ, and use the identity

models solved in ref. [3]. Hence the special solution found by Bobylev, Krook and Wu (BKW-mode), F(x,

x

t) =xm_le_~ram [1 + m — mci + xa(a

—

1)]/F(m),

(16)

is an exact solution in which a(t) is given by a ‘(t) = (1 be~ where b is a positive constant ana X = 1 X f 1 dii g(p)(1 ,i2). Furthermore, the eigenfunctions and eigenvalues of the linearized form of(l)11n0 are 5n0) respectivelyi2~m_fl(x)andpn=p00(1 + iinn~This spectrum is bounded from above by ii By a minor extension of Cornille and Gervois’ 00. arguments [4] we may drop the requirement that is fmite, and relax 2) being it tofinite. the value For actual of the Maxwell integral f~ molexcules g(j.z)(l p ~ diverges, since it is proportional to the total classical cross section for the intermolecular force law —r5 where r is the relative distance and s 2d 1. With this modification the models with property (2) include all Maxwell models, which have been solved up till now by the La~uerreseries. —

—

—

—

(m

+ ‘~k-

(m)k

—

(k) I! 1=0 (1) 1 (in)1’ fl

(12)

on both of(10). For all the terms with l
tion (II), and observe that they cancel on both sides. The remaining terms on the right-hand side are only non-vanishing for 1 ~ k ~ ñ and 1 ~ ñ. Hence, only the term with / = k = ñ survives, and one obtains (11) for i = ñ. Then by induction (11) is true for all n and!. 4. Inversion ofthe problem. Can one express ~nk in terms of the kernel K? We have seen that the ansatz (2) together with the detailed balance condition allows as the only freedom in K that the sequence ‘1nn is supplied. The definition of 11nk implies [take u y = 1 in (2)] that all ~nn are positive and form a totally monotone sequence [7] [provided K(xy; u) is not a single delta function]. This sequence uniquely defines [7] a positive function g(p) through

—

—

5. It is of interest to apply our results to the persistent scattering model of Futcher and Hoare. describing molecules with m = p + q internal degrees of freedom, q of which participate in the binary interaction. The kernel for this model is given by [6] ~‘ ~ Notice the printing errors on the sIts of eq. (7), where x andy should be interchanged and where the limit should be replaced by

u.

317

Volume 81A, number 6

PHYSICS LETTERS a

min(x,y)

f

K(xy; u) =

dv

0

late first the moments (2). As we have seen in (9) the

f

dw Wpq(v,y)

max(x,y)

X Wqp(W — y,

U

—

y) Wqq(X

—

V, W

—

v),

(17)

where Wpq(X, u)

=

2 February 1981

kernel K(xy; u), depending on three variables, is determined by a function Q(x,y) of two variables due to the homogeneity property. The moment property (2) together with the detailed balance symmetry shows that K is actually determined by a single function g~p) of one variable only. In a separate publication [8] we have constructed the kernel K(xy; u) in terms of a

xPl(u _x)~’ul—P~/B(p,q),

B(p, q) = F(p)F(q)F(p + q).

Futcher and Hoare have shown that the moments of (17) satisfy the eqs. (2). However, they did not succeed in showing that the I.aguerre (or Sonine) moments obey the same simple equations as the ordinary

giveng(p) for a given dimensionality m. The relation can be used as an a priori criterion for testing whether K(s, 1; 1) reproduces the original K(xy; u). However, this test seems to be of little practical value, since the calculations for simple choices ofK(s, 1; 1) are already quite complicated.

moments, as follows from our result. Furthermore

g~.z)in (15) can be calculated directly as

References

K(s,1;1)

2F1(p,q;p+q;l —s)

x s~ (1

(18) —

s)P+~l/B(q,q),

where 2F1 is Gauss’ hypergeometric function. 6. Consider next the question: what are the conditions on the kernel K(xy; u) that the BKW mode is an exact solution to the kinetic equation (1)? By inserting the BKW-mode in (1) it appears to be sufficient that the kernel Khas the interactional symmetry and detailed balance property and that its first three moments satisfy eq. (2). It would also be of interest to formulate an a priori

criterion, under which the kinetic equation is soluble from the moment equations, without having to calcu-

318

[1] M.H. Ernst and E.M. Hendriks, Phys. Lett. 70A (1979) 183. [2] A.V. Bobylev, Soy. Phys. DokI. 20(1976) 820; M. Krook and T.T.~ Wu, 69A Phys.(1979) Fluids 390. 20 (1977)1589. [3] Mdl. F~nst, ~.

[4] H. Cornille and A. Gervois, J. Stat. Phys. 23 (1980) 167. [5] M. Kac, Proc. Berkeley Symp. on Mathematics, statistics and probability III (Univ. of California Press, Berkeley, 1955) P. 171. [6] E.J. Futcher and M.R. Hoare, Phys. Lett. 75A (1980) 443. [7] H.S. Wall, Analytic theory of continued fractions (Van

Nostrand, New York, 1967) p. 267.

[8] M.H. Ernst and E.M. Hendriks, Interrelations between soluble Boltzmann equations Phys. Lett. 81A (1981), to be published.