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The Boltzmann-Lorentz model for particles with spin
Physica 139A (1986) 80-100 North-Holland, Amsterdam
THE BOLTZMANN-LORENTZ
MODEL FOR PARTICLES WITH SPIN
II. THE PROJECTION-OPERATOR
TECHNIQUE
Marek DUDYNSKI Institute
for
Theoretical
Physics,
Polish
Academy of Sciences, Poland
Warszawa,
Al.
Lotnikdw
32146,
Received 12 February 1986
The projection-operator technique is applied to the linearized equation of the BoltzmannLorentz model for particles with spin in order to derive the equations of the hydrodynamics and the generalized hydrodynamics. The validity of the hydrodynamical description is studied for different time and space scales. The results are compared to the exact solution and to the Chapman-Enskog hydrodynamic equations.
1. Introduction In this paper we will continue the analysis of a linearized equation for the Boltzmann-Lorentz model for particles with spin. Our aim here is to apply the projection-operator method of Bixon, Dorfman and MO*) and compare the results obtained via this method with the exact solutions derived previously’). It is often stated in the literature’) that the projection-operator method results in hydrodynamical equations which are exact but that the solutions of these are as difficult to obtain as the solution of the Boltzmann equation itself. In order to simplify the problem one needs some sort of perturbation expansion usually using the space gradient as a small parameter. In the hydrodynamical limit such a procedure results in the linearized Navier-Stokes hydrodynamics equations, thus this expansion can be considered as a more rigorous derivation of the Navier-Stokes equations providing one can show that the perturbation series converges. For the general interaction potential one can at best expect that these series are asymptotic and such is the status of the Navier-Stokes equahydrodynamics. Many authors 3-6) try to obtain hydrodynamical-like tions for a larger number of dynamical variables than the five conserved densities in order to derive equations valid for intermediate times and space gradients. We will also consider this problem eliminating by successive re037%4371/86/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
BOLTZMANN-LORENTZ
MODEL FOR PARTICLES WITH SPIN II
81
summation the different time scales and obtaining equations valid for intermediate times. There is a definite time scale for our model where the reduced mode approximation scheme cannot be applied. We will show also that while improving the validity of the approximate equations in time we do not necessarily improve the validity of these equations for the space gradients and quite frequently the situation is reversed. As we have seen in the previous paper’) for our model the dependence of the eigenvalues of the linearized Boltzmann-Lorentz operator on the space gradients (or in the language of the Fourier transform, on the wave vector k) is analytical for small k for the first three eigenvalues. This fact is equivalent to the convergence of the perturbation series connected with these modes. We will show that only for these modes we can obtain hydrodynamical-like equations and incorporating other modes does not improve the approximation both in time and space. This paper is organized as follows: in section 2 we define the equation we will consider and write down the eigenfunctions and eigenvalues of the scattering operator, in section 3 we introduce the projection-operator technique with the modifications we need to handle nonsymmetric operators, in section 4 we calculate the equations for the hydrodynamical modes and analyze the solutions of these equations, in section 5 we use a resummation procedure in order to obtain equations valid for intermediate times and we show that they are equivalent to the equations obtained via generalized projection-operator method and in section 6 we derive the hydrodynamic equations by the Chapman-Enskog procedure and show that their linearized version agrees with the projection-operator equations.
2. The linear equation In the previous paper I’) we have introduced a set of linearized equations describing the Boltzmann-Lorentz model for particles with spin. It is convenient to write our basic linear equations as (a, + u - V)f= (a, +
LJ -
qF(P
V)g = u,F(P
- 1)f + 6,G,(P - 1)g , - 1)g + S,G,(P - l)f-
(2.la) uFg + uG,
f + U&G, (2.lb)
a,G = uFPg - aG,Pf
- ufPG
,
(2.lc)
where a,, u2, 6,) S,, and u are constants defined in I, F, G,, g,, and f, are the equilibrium states we have linearized around and given the average spin
M. DUDYtiSKI
82
polarization
density S and the light particle density n are
fs=n,
(2.2a)
(2.2b) (2.2c) F being the constant density of the scatterers P=-&
I
and
(2.3)
dL$,
the integration being carried on over the velocity angles. f = f(r, u, t), g(r, u, t) and G = G(r, t) are one-particle distribution functions for the light particles density, the spin of the light particles and the spin of the scatterers density respectively. Eqs. (2.la-c) can be conveniently analyzed in matrix notation after applying the Fourier transform in space and read d,?P=L?P,
(2.4)
where the state vector 9 is given as
!P=
0 f g I
(2.5)
G
and the linear operator
L =
L is
-ik* u + clF(P - 1) 6,G,(P - 1) + aG, -aG,P
%G,(P - 1) -ik * u + u2F(P - 1) - UF aFP
0
mf, . (2.6) - CfsP
Now we need to define a proper Hilbert space. To do so let us consider the operator L, defined by formula (2.6) with k = 0. We introduce the space Z-I, of functions !P depending on u and having the form
$1(u>
1
9 = J1Au) i A(u)
9
(2.7)
BOLTZMANN-LORENTZ
MODEL FOR PARTICLES WITH SPIN II
83
where &(u) are square integrable on R3 3 S3 = {u E R3; IuI = u,,} and G3(u) is a function of IV] only. The scalar product on HO is defined as
The operator L, was analyzed in the previous paper’). For the sake of convenience we write down its eigenvalues and corresponding eigenfunctions 1
A,=();
lq)=
() 6
s -G, F+f,
, (a,l=(Lo,O);
(2.9a)
(2.9b)
,
A,=-c+(F+f,);
A;, = -hF+
WC) ;
bdJ=
(a,[ = (3 F F+f,‘F+f,‘F+f,
-fe)
;
(2.9~)
$
(2.9d)
-W’-
4W ; (&,,I= &
1 (&A = ,(y;*,(eP
(2.9e)
44 - y;“,(e, 44 0) ,
where
(2z+ l)Cz- m)! “’ u,@4’~J (z+m)!
pm(cos
1
e)
eim+ 7
84
M. DUDYhKI
WY,
E=
t
1,
m
(2.10)
m20,
are normalized harmonics in velocity angles 8 and 4. The normalization chosen in such a way to have biorthogonality relation (ai 1ej) = Sij .
3. The projection-operator
was
(2.11)
technique
We begin by defining the projector
PH= le,)(4 + l%)(%l.
P, as (3.1)
This operator projects arbitrary functions defined on H, onto the subspace spanned by the hydrodynamical modes. We shall call this subspace a hydrodynamical one. It follows from eq. (3.1) that Pi = PH and we shall frequently use the notation QH = 1 - PH to denote the complementary operator to P,. The meaning of the operator PH is clear. If 14) is a solution of eq. (2.4) then
PJ~J)= 4~~Oled + Sk, f)le,) ,
(3.2)
with (3.3a)
bafI4) = Sk, t>>
(3.3b)
n(r, t) = P41(r, u, t) is the local particle density and S(r, t) = t) is the local spin density. According to the rules of the P.0 method we first apply the Laplace transform in time to the eq. (2.4) and obtain
where
Pc&(r, u, t) + &(r,
We now define
g,(k u, z) = P,4(k, u, 4,
(3Sa)
g,(k u, z) = QA4k
(3.5b)
u, 2) .
BOLTZMANN-LORENTZ
Using eqs. (3Sa,b)
g,(k u, 0) + p&z
The equation
85
we can easily obtain the following equation for g,(k, u, 2):
Zg,(k, r~, 4 - P&‘Hgr(k, =
MODEL FOR PARTICLES WITH SPIN II
u, z) - p&(z -
Q&Q,)-‘g,@,
- Q,~QJ’Q&Gg,(k
u, 2)
u, 0) .
(3.6)
can be written as
zn,(k, z) - n,(k, t = 0) + $ fiij(k, Z) + $_ K,(k, z)nj(k, Z) = Zi(k, Z) 7 ,=I
I=1
(3.7)
where we have used the notation
n,(k z) = n(k, z) ,
(3.8a)
n,(k, z) = S(k, z) .
(3.8b)
From the time reversal symmetry of the set le,) and (a,[ it follows that the frequency matrix Qj = (a,[Ljej) = 0. The kernel K,(k, z) and the initial value term Zi(k, z) are given by (3.9)
Zi(k, Z) = -(a,lL(z - QHLQH)-lQHI
+(k, u>’
=
0)) .
(3.10)
We will solve the eqs. (3.7) perturbatively using ku as a small parameter. do so we rewrite the interaction part of L as L’ = iku(il,. G)Z’ ,
To
(3.11)
with
(3.12b)
(3.12~)
We are expanding now the Kij and Zi as
86
M. DUDYASKI
K, =
2 K; ,
(3.13a)
tl=?.
(3.13b) where the superscript indicates the order of the term with respect to k. Making use of the well-known operator identity
(z + QHLQH - Lo)-’ = (z - I,,)-’
- (z -
Lo)-'Q,L'(z
- I,,)-’ ,
+ Q,L'Q,
(3.14) we derive the following expressions for the nth order terms in these expansions: K;(k,
z) = (-l)“(a;ILQ,(z
La)-'Q&Q, .* * L’Q,(z
-
- L,)-’
x QHLIej) 3
(3.15a) - Lo)-‘Q,L’Q,
Z;(k, z) = (-l)“(a,(LQ,(z
x Q,+(k
* * * L’Q,(z
- L,)-’
u, t = 0) .
(3.15b)
The applicability of these expansions depends on the properties of the spectrum of operator L,. Recalling formulas (2.9a-d) we know that L, has four different eigenvalues namely
A,,, = 0 > A, = -4F A:, = -(ulF The value inequalities - Fq
+_a>
hi,,, = -(cr,F
+ S,G,)
and
- 6,G,) .
of 1G,I is bounded are fulfilled: < A;,$ s - F(min(o,
from
above
+ S,, a; - 6,)) .
by F and thus the following
(3.16)
This shows that A:: and A, are separated from A,,, by the gap but their relative positions depend on the magnitude of parameters a and J2 (see I) defining the spin-independent and spin-dependent parts of the interaction potential, respectively. Below we list the elementary operations we need for performing the calculations. In order to evaluate KS and ZGwe start with the right-hand side of eqs. (3.15a,b) and proceed toward the left. For Kij we always start with le,) or les) and we end with (a,\ or (a21. The first thing we have to calculate is then
BOLTZMANN-LORENTZ
Z’i,u[e,)
= 66”*((ei,)
MODEL FOR PARTICLES WITH SPIN II
+ let,,)) + 6-“2
&T
(lei,)
- ie:,,)),
87
(3.17a)
s Z’i&,)
=6-l’*
j&
S
Z’i,ulei,,) = -ylle&) +6-l’* I’i,ulef,)
l&J>T
(3.17b)
lel) + le,) +
(3.18a)
(
= -yllet,,) +6-l’*
(3.18b)
For n > 1 we have the following rules:
&4&J = .y,lei+l,o) + 74~L,o) y
(3.19a)
~‘h&) = 3/,lei+1,0> + dL,,)
(3.19b)
7
where
3/n =
l/2
2n + 1
n+1 (2n + 1)
2(n + 1) + 1
’
(3.20a)
112 rA
=
(2nR+ 1)
2(z):
1
.
(3.20b)
The modes with m # 0 do not couple perturbatively with lel) and le2) thus we do not need them any more. Note that in the above formulas we are allowed to omit the iku in L’ since the appropriate power of k appears in the front of the formulas (3.15a,b). With each operator L’ there is associated an operator QH operating on it from the right, thus we have
QAL’Iebd
= iku&‘J
QAL’l&J)
= ikua;k,) ,
a1
a2
=
=
6-“2
6-l’*
7
F-G
S
(3.21a) (3.21b) (3.22a)
F+f, ’ -(F + G,)
(3.22b)
F+.t We will also need L’le,) = ikrK”*(le~,)
- lef,,)) .
(3.23)
88
M. DUDYtiSKI
4. The hydrodynamic
equations
Using the formulas of section 3 and keeping the terms up to order k2 we obtain for K$(k, z)
W2
G
w++F+w-
6
z) =
K;(k,
L F+f,
w’
(4.1)
with
w+=(z + I/&l)-’ w- = (z
+ (z
+ 1/i;(J-* -(z
+
(h;,l)-’
+
Ih;ol)-’ .
(4.2a)
)
(4.2b)
Similar calculations give li(k, z) = B(k). In the hydrodynamical regime we are interested in the long time behaviour of the system which means that we are looking at z = 6(k). Thus we can disregard the z in the denominators of W* and we obtain the following set of parabolic equations for the hydrodynamical variables n(k, z) and S(k, z): z + Kf, (
(4.3)
.:~:,,(~~~~:~,=(~~~:~~)-
Kil
These equations can be easily solved and we see that the time dependence n(k, t) and S(k, t) is determined by the roots of the equation Kt2 z+K;,
= 0.
of
(4.4)
After simple algebra one obtains Zl,Z
=
-
y
{(cT,F)~ - (S,G,)2}-’
x!NQ’(F+ 2fs)(F -+{(qF2
- S,G:)‘(F
+ 4f,S;G;(F
+ L)-’
- &G:(F +
+ f,)-’
+ f,)-‘}““I
.
LX’1
- 4f,G;Fqb,(F
+f,)-2 (4.5)
It is convenient therefore to introduce two constants D, and D, which play the role of diffusion coefficients,
BOLTZMANN-LORENTZ
z1 =
-D,k2
22 =
-D,k*
MODEL FOR PARTICLES WITH SPIN II
89
(4.6a)
)
(4.6b)
,
In order to give some feeling of the relation between D, and D, we can expand Di in powers of the small parameter x = f,lF. To the lowest order in x we obtain
(4.7a)
crl
F6,G,2
+
-
qF2
We can evolution D, is of mode17) diffusion.
-D,+K:, D
2 n
_
e,
2
D
1
6,Gf
I+0(x2).
(4.7b)
see from these formulas that D, G D, and that for long times the of the distribution function is determined by the coefficient D,. The the same order as the diffusion constant for the ordinary Lorentz and we can see that the spin coupling results in slowing down the The solutions for n(k, t) and S(k, t) have the form
ex~(-D,k*f) 1=T(-
D,k%
K:,
-4+
+
+ n
D,
_
G
I
D,
exp(-~,k4)
exp(-D,k2r) 2
e -D,: +
K’
exp(K’
2
1
D,k*r)
ev(-D,k’r)
exp(-D,k’t)
+ & -‘D,
+ d
.‘I?
1
2
exp(-~,k*t)
(4.8) From eq. (4.8) we see that D, and D, cannot be considered as diffusion constants for particles and spin respectively. Diffusion in our model is a complex process which depends on D,, D, and on the relation between the initial values of la@, 0) and S(k, 0). To see this more clearly we consider two limit cases: A). At t = 0 all scatterers are polarized in one direction and light particles are also polarized in the same direction. In this case
M. DUDYrjSKI
90
D, = v*(127ru(a++)F)-‘ ,
(4.9a)
D, = u*(12~uF(c+__ + (++_))-r &
(4.9b)
. s
CT++, u__, u+_ were defined in the previous paper’) and are cross-sections for spin parallel, antiparallel and spin-flip respectively. Recalling also that the initial values n(k, t = 0) and S(k, t = 0) are related via S(k, 0) = n(k, 0) we finally obtain
n(k, t) = n(k, 0) exp(- D,k2t) ,
(4.1Oa)
S(k, t) = S(k, 0) exp(- D,k*t) .
(4.10b)
In this situation the only mechanism of spin diffusion is the diffusion of light particles (spin-flip is excluded). B). We consider G, = 0. This is the situation when at t = 0 the average polarizations of the scatterers and the light particles are equal but they are pointing in opposite directions. In this case we have D, = u2(3a,F)-’
,
(4.11a)
D, = u2(3a,F)-’
s
f * F+f, ’
(4.11b)
moreover
n(k, 0) and S(k, 0) are independent.
We obtain then
n(k, t) = n(k, 0) exp(- D,k*t) ,
(4.12a)
S(k, t) = S(k, 0) exp(- D2k2t) .
(4.12b)
The spin and particle diffusion are independent. For other situations a more intricate coupling is present. As we have stated in the previous paper it is possible to consider a slightly different system with a finite number of light particles in the infinite medium of scatterers. The stationary state minimalizing the H functional is now f, = 0, g, = 0 and G, unspecified. For such a system we expect that the finite number of particles cannot change significantly the spin state of the infinite system of the scatterers and a natural choice of G, is G(r, t = 0). This dependence on the spatial variable r adds significant complexity to the problem. We consider now only a system for which G(r, t = 0) differs only slightly from a constant value which we chose as G,. With such a definition of G, linearization of eqs. (1.2.7a-c) leads to
BOLTZMANN-LORENTZ
MODEL FOR PARTICLES WITH SPIN II
91
qF(P - 1)f + 6,G,(P - 1)g ,
(4.13a)
(a, + u - P)g = u2F(P - 1)g + &G,(P - 1)f - CrFg + uG, f ,
(4.13b)
d,G = uFPg - crG,Pf .
(4.13c)
(a, + u * V)f=
The two first equations of this set can be solved independently from the third one. A similar system of two equations has been analyzed in our note’), where we have found exact solutions, but we have used a different approximation scheme not allowing the scatterer spins to flip thus the present results are directly comparable for G, = 0 only. The long time limit of the solutions of eqs. (4.13a-c) can be obtained directly from formulas (4.5) in the limit f,lF = 0. We obtain the following expressions for the transport coefficients:
D, =
$ (ulF2- ~,G;)[(cT~F)~ -
(4.14a)
(6,G,)2]-1 ,
D,=O,
(4.14b)
and the solutions for n(k, t) and S(k, t) read n(k, t) = n(k, 0) exp(- D,k2t) , S(k, t) = S(k, 0) - II-
exp(-D,k2t)~(+G,F(~IF2
(4.15) - 6,Gf)-‘n(k,
0).
(4.16) These solutions show that the presence of the spin manifests itself by changes in particles diffusion as compared to the spinless Lorentz gas of hard spheres. Comparison of eqs. (4.13a-c) with eqs. (2.la-c) shows that the new and interesting features of the linearized model are connected with higher order terms in f,lF and are in fact connected with the conservation of the global spin of the system. Having the hydrodynamic equations (4.3) we are interested in the range of validity of these equations both in time and space variables. To answer these questions we calculate the k4 terms in expansion (3.15a-b). We obtain Kz + K>’ + K$2, where, explicitly,
KQ.’ =
’
92
M. DUDYhKI
where
u-i
- 6
IF-G F+f,
-
1 z+cr,F+6,G,
1 z+a,F-6,G,
1 1
(4.lSa)
’
(4.18b)
’
-$(ku)4[s+ +& s-1
K;’ =
-
& (ku)‘[S+& s
(4.19)
s+]
with
s+=(2 + JA;,I)-3+ (2 -+-JA;,()-3
(4.20a)
s- = (z + lA;01)-3-
(4.20b)
)
(z +
lh;,I)-3 .
The relevant terms of Z,,,(k, z), up to order k2, are
, (W2 1
6
1, (W2 6
(a3
w-
I 4J
(z+lA3()
(a3
’ (4.21a)
I (bo)
w+ (z+lA3()
.
(4.21b)
We can easily see now that terms in K:’ can become very large for z = A, or (ku)2 = A,A;& This means that the hydrodynamic equations are valid for t+ IA31-’ and k2 e (A3A~;ZI/u2. One can check by simple but tedious calculations that for such a t and space gradients both Kz and 1’ with II > 2 can be disregarded. Comparison with exact results shows that in the hydrodynamical limit defined by the above inequalities the behaviour of the hydrodynamic modes obtained from the exact dispersion relation and the projection-operator method agrees. We know from the analysis of the spectrum of L, that for (+ -=s(TV,6, the A, is close to zero and thus the region of validity of the hydrodynamical equations is very limited. If we want to consider the shorter times or larger space gradients we ought to sum the most divergent terms in the expansion (3.13). We do this in the next section.
BOLTZMANN-LORENTZ
5. Generalized
MODEL FOR PARTICLES WITH SPIN II
projection-operator
93
method
The hydrodynamic equations discussed in the previous section are valid, for small A,, for very long times and small space gradients only. Thus one is tempted to extend this description by incorporating the A, mode. Given the perturbation series with a certain class of terms diverging for z a - 1A,/, the natural method is to sum up these terms in all orders in k. We follow this method and next we will show that this is equivalent to the projection-operator technique with {le,), le,), le3)} t a k en as a basic set of variables. The most divergent terms are those containing a maximal number of denominators of the form (z + IA,())‘. Th ese terms are at least of order k4 and have the form
x [ i: (a~lQ,&Q,j
+
le;o)(u;olQn~~Q~le~)l" 0
n=l
(5.1) where N = 0, 1,2, . . . i, j = 1,2. The expression in square brackets is equal to
W2 ‘=-
6(F+f,)
F-
G,
F+
G,
1
z+~A:~[ + z+~A;~[ 1 z+[A,l
(5.2)
’
We can see then that K;:‘(k, z) = i
K;,?’ .
(5.3a)
N=O
The matrix elements form then a geometric obtain
series, we can sum it up and we
W2
6
f,
‘lw-
F+f,
@g&P,,+
s
(53b) where
94
M. DUDYfiSKI
F-
-PI = z+r~(F+f,)+
P2 =
6(F+f,)
z+a(F+f,)+
The appropriate
F-
F+G,
G,
G,
G,
2 + l&l
6(F+f,) F-
1
F+
(k#
W2
+ z+)A;,]
G,
z+&)
F+
z + lA:,,l - .z + lA:ol
W2
6(F+.fs)
F-
1
+ 2 + [A:J G,
G,
(5.4a)
’
1 F+G,
(5.4b)
1 ’
z + lA:Ol + z + lA:,,l
terms from Z1 and Z2 are (ku)‘/6
I:#, 2) = W-t+ 14,) z + (T(F + f,) +
Ft-G,
+
IGJlI
2+
’
(5Sa) and
I&,
2) =
(~31(b,)W2/6
w+ z+c+(F+f,)+
W2 6(F+f,)
F-
G,
F+G,
z + lA;Ol + z + [A:,,/
We obtain now a complicated set of nonhomogeneous dynamical variables of the form
equations
1 ’
(5.5b)
for hydro-
2 + ZC;, + K;1;’ ZC;, + ZC;; (5.6) We can see now that disregarding the z and k2 in the denominators is not a good approximation as far as the time scale t = 1h,J -’ is concerned. Eqs. (5.6) form a system of coupled integro-differential equations both in space and time variables. These equations can both be easier obtained and interpreted if we apply the generalized projection-operator method. To do so we include in the basic set a A, mode. It means that we define a projector P, as
pE = lelN31 + l4(4 and
+ ld(4
(5.7)
BOLTZMANN-LORENTZ
MODEL FOR PARTICLES WITH SPIN II
QE=l-PE.
95
(5.8)
With these operators we obtain now equations for three hydrodynamical variables, namely n(r, t), S(r, t), defined before, and n3(r, t), where
(5.9) Now we can write down equations for the set n(k, z), S(k, z) and n,(k, z) obtained as previously up to order k’. These equations are
(5.10) with
PI,2= PI,,
z+u(F+f,)+
(W2
F-
G,
6
z + IA:01
+ F+Gs z+lA:,l
11
(5.11)
.
It is an easy exercise to eliminate n,(k, z) from eqs. (5.10) and check that the resulting equations for n(k, z) and S(k, z) are exactly those given by eqs. (5.6). One can generalize these results and show that incorporating a larger number of nonhydrodynamical modes is equivalent to a partial summation of selected terms in the perturbation series for the hydrodynamical variables. The above system of equations is not in general parabolic but integrodifferential in time. If however A, < A:$, we can calculate the approximate solution of this system by dropping z in the denominators thus approximating these equations by a set of parabolic equations of the Navier-Stokes type. We solve these equations and then apply the inverse Laplace transform to find the time dependence of the solutions. We obtain three roots of the appropriate determinant zi, z2 and z3. For small k, z1 and z, agree with those obtained previously and the third z3 agrees with the value A,(k) we have determined in I from the exact dispersion relation up to terms of order A,lA:t. Explicitly, z3 reads
96
M. DUDYhKI
q=-u(F+f,)-
W2
ulF2
6(F+ f,)
(u,F)~
+ S,G; -
(6,G,)2
(5.12)
’
The parabolic hydrodynamic equations (5.10) are valid as a good approximation to the distribution function for t s 1A$) -I and for (ku) e (I A:$\ - 1A~(). From this we see that if &/A:$ < 1 we extend both regions of t and space gradients we are able to describe properly as compared to the equations for the hydrodynamical variables n(r, t) and S(r, t) alone. If however A, = Ai: we cannot use the parabolic approximation and our new equations can properly handle for short times only very small space gradients. From the exact solutions we know that there are two cuts in the complex z plane near z = A;b2+ iku. The existence of these spectral singularities is easily seen in the perturbation method and we cannot by adding the next nonhydrodynamical modes remove the divergencies for z = Ai;* due to the infinite number of different modes with the same denominators (z + I A$[)-‘. This shows that the time scale t 3 I A,1-’ is the limit we can describe by the perturbation method and by hydrodynamical-like equations thus there are not such equations for shorter times.
6. The Chapman-Enskog
expansion
In this section we apply the conventional Chapman-Enskog’) method to the analysis of our kinetic model. We begin with rewriting the nonlinear equations’) in the form (a, + u*V)f=
f (qF(P-
(a, + u - V)g = i (r2F(P a,G = i (CrFPg - CGPf)
l)f+
S,G(P-
- 1)g + S,G(P - l)f.
(6.la)
l)g), aFg + aGf)
,
(6lb) (6.1~)
We have introduced the formal small parameter E to scale the u~,~F, 6,,,G, and aF, UC,, which are of the order of the inverse mean free paths for various scattering processes, with the aim to investigate the limit with these mean free paths going simultaneously to zero. This limit turns out for linearized equation to be equivalent to the application of the projector-operator method supplemented with an expansion in small gradients. As usual in the ChapmanEnskog procedure we assume that the solutions of eqs. (6.la-c) are of the form
BOLTZMANN-LORENTZ
MODEL FOR PARTICLES WITH SPIN II
97
(6.2a) g =
&dr,u; {n(r, 4
w, q>>>
(6.2b) (6.2~)
G = G(r; {n(r, 9, S(r, t)>) . The equations for the time evolution
of particle and spin densities are
ate, t>= --v*[Pof(r, u, f)l = D,({n, s>>,
(6.3a)
O,S(r, t) = -v* [PzJg(r, u, t)] = Ds({n, S}) .
(6.3b)
Applying relations (6.2a-c)
to eqs. (6.la-c)
af D, + as af D, + u .Vf= ; [a,F(P an 2
D, + z
D, + us Vg = ; [(r,F(P
and using (6.3a,b) we obtain
- 1)f + S,G(P - l)g] ,
(6.4a)
- 1)s + 6,G(P - 1)f - crFg + uGf]
,
(6.4b) gDn+g
Expanding
D, = 5 [vFPg
- aGPf] .
(6.4~)
now the functions f, g and G in powers of E as
f = i& srnfCrn)>
(6Sa)
m g = c emgCm)) m=O
(6.5b)
G = 2
(6.5~)
smG@) ,
m=O we will use the condition that the hydrodynamical zero order in E. This means that for m # 0 Pf(“)(r, P( g’“‘(r,
u, t) = 0 )
(6.6a)
u, t) + G(“+(r, t)) = 0 .
Inserting (6.5a-c) obtain
into (6.4a-c)
variables are exact up to
(6.6b)
and collecting terms in the same order in E we
98
M. DUDYrjSKI
In order E-~: cr,F(P - 1)f’“’ + 6,G”‘(P
- l)g”’
@(P
- l)f’“’ - aFg(‘) + aGCo)fCo) ,
- l)g”’
&‘g(‘)
+ 6,G”‘(P
- ,G(‘$f(‘)
= 0,
(6.7a)
=0,
(6.7b) (6.7~)
with solutions f(O) = const., g(O) = const. and G(O) = const. These constants are related by the equation co) = G(O)f(O) .
Fg
From eqs. (3.6a,b) we obtain f(O) = Iz(T, t) )
(6.8a)
g(O) = S(r, t)n(r, t)(F + n(r, t))-’ G(O) = S(r, t)F(F + n(r, t))-’ In this order the equations dp(r, t) = 0 a&,
,
(6.8b)
.
(6.8~)
of hydrodynamics
assumed the trivial form (6.9a)
)
t) = 0 .
(6.9b)
In order e(O) we obtain Vf(O) =
u.
u
.
a,F(P- l>f (l)
vg(0) = a,F(P
_ ~lG’o’(P
_ l)g’l’
,
- 1)g”’ - c+Fg(l) + 6,G(‘)(P
+ ,f(o,G(‘)
- 1)f”’
>
0 = (+Fpg(‘) _ ,G(‘$f(‘)
(6.1Oa) + ,G(‘)f(‘) (6. lob)
_ oG(*)f(O) .
(6.10~)
This set of equations is equivalent to a simpler one, v
.
Vf(O) = qF(P
- 1)f”’
”
.
vg(0) = qF(P
- 1)g”’ + 6,G”‘(P
+ 6,G”)(P
- 1)g”’ ,
(6.11a)
- 1)f”’
(6.11b)
.
BOLTZMANN-LORENTZ
pf(l) = 0, Pg@) + G(l) = 0 are
The solutions of eqs. _ (6.11a,b) with constraints of the following form: f’yr,
u, t) = - 5
gyr,
u, t) = - 5
G(l)(r,
1
99
MODEL FOR PARTICLES WITH SPIN II
+ g(O)) u . V( f(O) + g(O)) + u - V( f-(O) us V(fO’ + g(O)) - u . V( f(O) + g(O)) alF - 6,G”’
olF + SIG”’
1 1 ’
u,F - SIG”’
ulF + c$G’~
’
t) = 0 .
(6.12a)
(6.12b) (6.12~)
The distribution functions f(r, u, t), g(r, u, t) and G(r, t) are thus uniquely determined to the order co by the solutions of the following hydrodynamic equations:
d,n(r, t) =
$
V-
v(n+& > qF
a,qr,
t) = g
v*
f S, &
V n+& ( a,F+S,
nS F+n
>
+ u,F - S, fi
> &
(
V n--
nS V n-( F+n - ulF-6,
&
)
1’ 1*
(6.13a)
(6.13b)
We note here the unusual form of the nonlinear hydrodynamic equations. Now we linearize these equations around the stationary solutions, viz.
n(r, t) = n, + 6n(r, t) , and with use of the stationary
=
S(r, t) = S, + SS(r, t)
values n, = f,, G, = FS,I(F + n,) we obtain
0,
(6.14a)
= 0.
(6.14b)
100
M. DUDYrjSKl
Eqs. (6.14a,b) are identical to eqs. (4.3) obtained by the projection-operator method which shows that in the long time limit, where we can prove that they are a good approximation to the distribution function, both schemes are equivalent. The problem how they are related to the solution of the nonlinear equation will be considered in a future paper”).
7. Conclusions We have analyzed the application of the projection-operator technique and we have shown that this method can be applied both to describe the hydrodynamic long time limit and the intermediate region of times scale provided we are away from the spectral singularity of the scattering operator Lo. By adding nonhydrodynamical modes to the set we project on we can obtain more accurate equations but we pay a price losing the simple differential structure of the resulting equations. In the next paper”) we will show that these equations of generalized hydrodynamics can be viewed as a reduction of a greater system of differential equations of the hyperbolic type connected with Grad’s method of moments.
Acknowledgements
The author wishes to thank Prof. LA. Turski for very helpful discussions and encouragements during this work. This work was partially supported by Polish Ministry of Sciences and Technology; Grant MR.1.7.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)
M. Dudynski, Physica 139A (1986) 57, to be referred to as I. M. Bixon, J.R. Dorfman and K.C. MO, Phys. Fluids 14 (1971) 1049. I. Muller, Z. Phys. 198 (1967) 339. H. Grad, Comm. Pure and Appl. Math. 2 (1949) 331, and Principles of the kinetic theory of gases, Handbuch der Physik XII, S. Flugge, ed. (1958), pp. 205-294. W. Israel and J.M. Steward, in: General Relativity, vol 2, A. Held, ed. (Plenum, New York, 1980); Ann. Phys. 118 (1979) 334; Proc. R. Sot. London A365 (1979) 43. W. Israel, Annls. Phys. 100 (1976) 310. E.H. Hauge, Phys. Fluids 13 (1970) 1201. M. Dudydski and L.A. Turski, ZFT Report (1983). cf. J.H. Ferciger and H.G. Kaper, Mathematical Theory of Transport Processes in Gases (North-Holland, Amsterdam, 1972). M. Dudy&ki, in preparation. M. Dudydski, submitted to Physica A.
THE BOLTZMANN-LORENTZ
MODEL FOR PARTICLES WITH SPIN
II. THE PROJECTION-OPERATOR
TECHNIQUE
Marek DUDYNSKI Institute
for
Theoretical
Physics,
Polish
Academy of Sciences, Poland
Warszawa,
Al.
Lotnikdw
32146,
Received 12 February 1986
The projection-operator technique is applied to the linearized equation of the BoltzmannLorentz model for particles with spin in order to derive the equations of the hydrodynamics and the generalized hydrodynamics. The validity of the hydrodynamical description is studied for different time and space scales. The results are compared to the exact solution and to the Chapman-Enskog hydrodynamic equations.
1. Introduction In this paper we will continue the analysis of a linearized equation for the Boltzmann-Lorentz model for particles with spin. Our aim here is to apply the projection-operator method of Bixon, Dorfman and MO*) and compare the results obtained via this method with the exact solutions derived previously’). It is often stated in the literature’) that the projection-operator method results in hydrodynamical equations which are exact but that the solutions of these are as difficult to obtain as the solution of the Boltzmann equation itself. In order to simplify the problem one needs some sort of perturbation expansion usually using the space gradient as a small parameter. In the hydrodynamical limit such a procedure results in the linearized Navier-Stokes hydrodynamics equations, thus this expansion can be considered as a more rigorous derivation of the Navier-Stokes equations providing one can show that the perturbation series converges. For the general interaction potential one can at best expect that these series are asymptotic and such is the status of the Navier-Stokes equahydrodynamics. Many authors 3-6) try to obtain hydrodynamical-like tions for a larger number of dynamical variables than the five conserved densities in order to derive equations valid for intermediate times and space gradients. We will also consider this problem eliminating by successive re037%4371/86/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
BOLTZMANN-LORENTZ
MODEL FOR PARTICLES WITH SPIN II
81
summation the different time scales and obtaining equations valid for intermediate times. There is a definite time scale for our model where the reduced mode approximation scheme cannot be applied. We will show also that while improving the validity of the approximate equations in time we do not necessarily improve the validity of these equations for the space gradients and quite frequently the situation is reversed. As we have seen in the previous paper’) for our model the dependence of the eigenvalues of the linearized Boltzmann-Lorentz operator on the space gradients (or in the language of the Fourier transform, on the wave vector k) is analytical for small k for the first three eigenvalues. This fact is equivalent to the convergence of the perturbation series connected with these modes. We will show that only for these modes we can obtain hydrodynamical-like equations and incorporating other modes does not improve the approximation both in time and space. This paper is organized as follows: in section 2 we define the equation we will consider and write down the eigenfunctions and eigenvalues of the scattering operator, in section 3 we introduce the projection-operator technique with the modifications we need to handle nonsymmetric operators, in section 4 we calculate the equations for the hydrodynamical modes and analyze the solutions of these equations, in section 5 we use a resummation procedure in order to obtain equations valid for intermediate times and we show that they are equivalent to the equations obtained via generalized projection-operator method and in section 6 we derive the hydrodynamic equations by the Chapman-Enskog procedure and show that their linearized version agrees with the projection-operator equations.
2. The linear equation In the previous paper I’) we have introduced a set of linearized equations describing the Boltzmann-Lorentz model for particles with spin. It is convenient to write our basic linear equations as (a, + u - V)f= (a, +
LJ -
qF(P
V)g = u,F(P
- 1)f + 6,G,(P - 1)g , - 1)g + S,G,(P - l)f-
(2.la) uFg + uG,
f + U&G, (2.lb)
a,G = uFPg - aG,Pf
- ufPG
,
(2.lc)
where a,, u2, 6,) S,, and u are constants defined in I, F, G,, g,, and f, are the equilibrium states we have linearized around and given the average spin
M. DUDYtiSKI
82
polarization
density S and the light particle density n are
fs=n,
(2.2a)
(2.2b) (2.2c) F being the constant density of the scatterers P=-&
I
and
(2.3)
dL$,
the integration being carried on over the velocity angles. f = f(r, u, t), g(r, u, t) and G = G(r, t) are one-particle distribution functions for the light particles density, the spin of the light particles and the spin of the scatterers density respectively. Eqs. (2.la-c) can be conveniently analyzed in matrix notation after applying the Fourier transform in space and read d,?P=L?P,
(2.4)
where the state vector 9 is given as
!P=
0 f g I
(2.5)
G
and the linear operator
L =
L is
-ik* u + clF(P - 1) 6,G,(P - 1) + aG, -aG,P
%G,(P - 1) -ik * u + u2F(P - 1) - UF aFP
0
mf, . (2.6) - CfsP
Now we need to define a proper Hilbert space. To do so let us consider the operator L, defined by formula (2.6) with k = 0. We introduce the space Z-I, of functions !P depending on u and having the form
$1(u>
1
9 = J1Au) i A(u)
9
(2.7)
BOLTZMANN-LORENTZ
MODEL FOR PARTICLES WITH SPIN II
83
where &(u) are square integrable on R3 3 S3 = {u E R3; IuI = u,,} and G3(u) is a function of IV] only. The scalar product on HO is defined as
The operator L, was analyzed in the previous paper’). For the sake of convenience we write down its eigenvalues and corresponding eigenfunctions 1
A,=();
lq)=
() 6
s -G, F+f,
, (a,l=(Lo,O);
(2.9a)
(2.9b)
,
A,=-c+(F+f,);
A;, = -hF+
WC) ;
bdJ=
(a,[ = (3 F F+f,‘F+f,‘F+f,
-fe)
;
(2.9~)
$
(2.9d)
-W’-
4W ; (&,,I= &
1 (&A = ,(y;*,(eP
(2.9e)
44 - y;“,(e, 44 0) ,
where
(2z+ l)Cz- m)! “’ u,@4’~J (z+m)!
pm(cos
1
e)
eim+ 7
84
M. DUDYhKI
WY,
E=
t
1,
m
(2.10)
m20,
are normalized harmonics in velocity angles 8 and 4. The normalization chosen in such a way to have biorthogonality relation (ai 1ej) = Sij .
3. The projection-operator
was
(2.11)
technique
We begin by defining the projector
PH= le,)(4 + l%)(%l.
P, as (3.1)
This operator projects arbitrary functions defined on H, onto the subspace spanned by the hydrodynamical modes. We shall call this subspace a hydrodynamical one. It follows from eq. (3.1) that Pi = PH and we shall frequently use the notation QH = 1 - PH to denote the complementary operator to P,. The meaning of the operator PH is clear. If 14) is a solution of eq. (2.4) then
PJ~J)= 4~~Oled + Sk, f)le,) ,
(3.2)
with (3.3a)
bafI4) = Sk, t>>
(3.3b)
n(r, t) = P41(r, u, t) is the local particle density and S(r, t) = t) is the local spin density. According to the rules of the P.0 method we first apply the Laplace transform in time to the eq. (2.4) and obtain
where
Pc&(r, u, t) + &(r,
We now define
g,(k u, z) = P,4(k, u, 4,
(3Sa)
g,(k u, z) = QA4k
(3.5b)
u, 2) .
BOLTZMANN-LORENTZ
Using eqs. (3Sa,b)
g,(k u, 0) + p&z
The equation
85
we can easily obtain the following equation for g,(k, u, 2):
Zg,(k, r~, 4 - P&‘Hgr(k, =
MODEL FOR PARTICLES WITH SPIN II
u, z) - p&(z -
Q&Q,)-‘g,@,
- Q,~QJ’Q&Gg,(k
u, 2)
u, 0) .
(3.6)
can be written as
zn,(k, z) - n,(k, t = 0) + $ fiij(k, Z) + $_ K,(k, z)nj(k, Z) = Zi(k, Z) 7 ,=I
I=1
(3.7)
where we have used the notation
n,(k z) = n(k, z) ,
(3.8a)
n,(k, z) = S(k, z) .
(3.8b)
From the time reversal symmetry of the set le,) and (a,[ it follows that the frequency matrix Qj = (a,[Ljej) = 0. The kernel K,(k, z) and the initial value term Zi(k, z) are given by (3.9)
Zi(k, Z) = -(a,lL(z - QHLQH)-lQHI
+(k, u>’
=
0)) .
(3.10)
We will solve the eqs. (3.7) perturbatively using ku as a small parameter. do so we rewrite the interaction part of L as L’ = iku(il,. G)Z’ ,
To
(3.11)
with
(3.12b)
(3.12~)
We are expanding now the Kij and Zi as
86
M. DUDYASKI
K, =
2 K; ,
(3.13a)
tl=?.
(3.13b) where the superscript indicates the order of the term with respect to k. Making use of the well-known operator identity
(z + QHLQH - Lo)-’ = (z - I,,)-’
- (z -
Lo)-'Q,L'(z
- I,,)-’ ,
+ Q,L'Q,
(3.14) we derive the following expressions for the nth order terms in these expansions: K;(k,
z) = (-l)“(a;ILQ,(z
La)-'Q&Q, .* * L’Q,(z
-
- L,)-’
x QHLIej) 3
(3.15a) - Lo)-‘Q,L’Q,
Z;(k, z) = (-l)“(a,(LQ,(z
x Q,+(k
* * * L’Q,(z
- L,)-’
u, t = 0) .
(3.15b)
The applicability of these expansions depends on the properties of the spectrum of operator L,. Recalling formulas (2.9a-d) we know that L, has four different eigenvalues namely
A,,, = 0 > A, = -4F A:, = -(ulF The value inequalities - Fq
+_a>
hi,,, = -(cr,F
+ S,G,)
and
- 6,G,) .
of 1G,I is bounded are fulfilled: < A;,$ s - F(min(o,
from
above
+ S,, a; - 6,)) .
by F and thus the following
(3.16)
This shows that A:: and A, are separated from A,,, by the gap but their relative positions depend on the magnitude of parameters a and J2 (see I) defining the spin-independent and spin-dependent parts of the interaction potential, respectively. Below we list the elementary operations we need for performing the calculations. In order to evaluate KS and ZGwe start with the right-hand side of eqs. (3.15a,b) and proceed toward the left. For Kij we always start with le,) or les) and we end with (a,\ or (a21. The first thing we have to calculate is then
BOLTZMANN-LORENTZ
Z’i,u[e,)
= 66”*((ei,)
MODEL FOR PARTICLES WITH SPIN II
+ let,,)) + 6-“2
&T
(lei,)
- ie:,,)),
87
(3.17a)
s Z’i&,)
=6-l’*
j&
S
Z’i,ulei,,) = -ylle&) +6-l’* I’i,ulef,)
l&J>T
(3.17b)
lel) + le,) +
(3.18a)
(
= -yllet,,) +6-l’*
(3.18b)
For n > 1 we have the following rules:
&4&J = .y,lei+l,o) + 74~L,o) y
(3.19a)
~‘h&) = 3/,lei+1,0> + dL,,)
(3.19b)
7
where
3/n =
l/2
2n + 1
n+1 (2n + 1)
2(n + 1) + 1
’
(3.20a)
112 rA
=
(2nR+ 1)
2(z):
1
.
(3.20b)
The modes with m # 0 do not couple perturbatively with lel) and le2) thus we do not need them any more. Note that in the above formulas we are allowed to omit the iku in L’ since the appropriate power of k appears in the front of the formulas (3.15a,b). With each operator L’ there is associated an operator QH operating on it from the right, thus we have
QAL’Iebd
= iku&‘J
QAL’l&J)
= ikua;k,) ,
a1
a2
=
=
6-“2
6-l’*
7
F-G
S
(3.21a) (3.21b) (3.22a)
F+f, ’ -(F + G,)
(3.22b)
F+.t We will also need L’le,) = ikrK”*(le~,)
- lef,,)) .
(3.23)
88
M. DUDYtiSKI
4. The hydrodynamic
equations
Using the formulas of section 3 and keeping the terms up to order k2 we obtain for K$(k, z)
W2
G
w++F+w-
6
z) =
K;(k,
L F+f,
w’
(4.1)
with
w+=(z + I/&l)-’ w- = (z
+ (z
+ 1/i;(J-* -(z
+
(h;,l)-’
+
Ih;ol)-’ .
(4.2a)
)
(4.2b)
Similar calculations give li(k, z) = B(k). In the hydrodynamical regime we are interested in the long time behaviour of the system which means that we are looking at z = 6(k). Thus we can disregard the z in the denominators of W* and we obtain the following set of parabolic equations for the hydrodynamical variables n(k, z) and S(k, z): z + Kf, (
(4.3)
.:~:,,(~~~~:~,=(~~~:~~)-
Kil
These equations can be easily solved and we see that the time dependence n(k, t) and S(k, t) is determined by the roots of the equation Kt2 z+K;,
= 0.
of
(4.4)
After simple algebra one obtains Zl,Z
=
-
y
{(cT,F)~ - (S,G,)2}-’
x!NQ’(F+ 2fs)(F -+{(qF2
- S,G:)‘(F
+ 4f,S;G;(F
+ L)-’
- &G:(F +
+ f,)-’
+ f,)-‘}““I
.
LX’1
- 4f,G;Fqb,(F
+f,)-2 (4.5)
It is convenient therefore to introduce two constants D, and D, which play the role of diffusion coefficients,
BOLTZMANN-LORENTZ
z1 =
-D,k2
22 =
-D,k*
MODEL FOR PARTICLES WITH SPIN II
89
(4.6a)
)
(4.6b)
,
In order to give some feeling of the relation between D, and D, we can expand Di in powers of the small parameter x = f,lF. To the lowest order in x we obtain
(4.7a)
crl
F6,G,2
+
-
qF2
We can evolution D, is of mode17) diffusion.
-D,+K:, D
2 n
_
e,
2
D
1
6,Gf
I+0(x2).
(4.7b)
see from these formulas that D, G D, and that for long times the of the distribution function is determined by the coefficient D,. The the same order as the diffusion constant for the ordinary Lorentz and we can see that the spin coupling results in slowing down the The solutions for n(k, t) and S(k, t) have the form
ex~(-D,k*f) 1=T(-
D,k%
K:,
-4+
+
+ n
D,
_
G
I
D,
exp(-~,k4)
exp(-D,k2r) 2
e -D,: +
K’
exp(K’
2
1
D,k*r)
ev(-D,k’r)
exp(-D,k’t)
+ & -‘D,
+ d
.‘I?
1
2
exp(-~,k*t)
(4.8) From eq. (4.8) we see that D, and D, cannot be considered as diffusion constants for particles and spin respectively. Diffusion in our model is a complex process which depends on D,, D, and on the relation between the initial values of la@, 0) and S(k, 0). To see this more clearly we consider two limit cases: A). At t = 0 all scatterers are polarized in one direction and light particles are also polarized in the same direction. In this case
M. DUDYrjSKI
90
D, = v*(127ru(a++)F)-‘ ,
(4.9a)
D, = u*(12~uF(c+__ + (++_))-r &
(4.9b)
. s
CT++, u__, u+_ were defined in the previous paper’) and are cross-sections for spin parallel, antiparallel and spin-flip respectively. Recalling also that the initial values n(k, t = 0) and S(k, t = 0) are related via S(k, 0) = n(k, 0) we finally obtain
n(k, t) = n(k, 0) exp(- D,k2t) ,
(4.1Oa)
S(k, t) = S(k, 0) exp(- D,k*t) .
(4.10b)
In this situation the only mechanism of spin diffusion is the diffusion of light particles (spin-flip is excluded). B). We consider G, = 0. This is the situation when at t = 0 the average polarizations of the scatterers and the light particles are equal but they are pointing in opposite directions. In this case we have D, = u2(3a,F)-’
,
(4.11a)
D, = u2(3a,F)-’
s
f * F+f, ’
(4.11b)
moreover
n(k, 0) and S(k, 0) are independent.
We obtain then
n(k, t) = n(k, 0) exp(- D,k*t) ,
(4.12a)
S(k, t) = S(k, 0) exp(- D2k2t) .
(4.12b)
The spin and particle diffusion are independent. For other situations a more intricate coupling is present. As we have stated in the previous paper it is possible to consider a slightly different system with a finite number of light particles in the infinite medium of scatterers. The stationary state minimalizing the H functional is now f, = 0, g, = 0 and G, unspecified. For such a system we expect that the finite number of particles cannot change significantly the spin state of the infinite system of the scatterers and a natural choice of G, is G(r, t = 0). This dependence on the spatial variable r adds significant complexity to the problem. We consider now only a system for which G(r, t = 0) differs only slightly from a constant value which we chose as G,. With such a definition of G, linearization of eqs. (1.2.7a-c) leads to
BOLTZMANN-LORENTZ
MODEL FOR PARTICLES WITH SPIN II
91
qF(P - 1)f + 6,G,(P - 1)g ,
(4.13a)
(a, + u - P)g = u2F(P - 1)g + &G,(P - 1)f - CrFg + uG, f ,
(4.13b)
d,G = uFPg - crG,Pf .
(4.13c)
(a, + u * V)f=
The two first equations of this set can be solved independently from the third one. A similar system of two equations has been analyzed in our note’), where we have found exact solutions, but we have used a different approximation scheme not allowing the scatterer spins to flip thus the present results are directly comparable for G, = 0 only. The long time limit of the solutions of eqs. (4.13a-c) can be obtained directly from formulas (4.5) in the limit f,lF = 0. We obtain the following expressions for the transport coefficients:
D, =
$ (ulF2- ~,G;)[(cT~F)~ -
(4.14a)
(6,G,)2]-1 ,
D,=O,
(4.14b)
and the solutions for n(k, t) and S(k, t) read n(k, t) = n(k, 0) exp(- D,k2t) , S(k, t) = S(k, 0) - II-
exp(-D,k2t)~(+G,F(~IF2
(4.15) - 6,Gf)-‘n(k,
0).
(4.16) These solutions show that the presence of the spin manifests itself by changes in particles diffusion as compared to the spinless Lorentz gas of hard spheres. Comparison of eqs. (4.13a-c) with eqs. (2.la-c) shows that the new and interesting features of the linearized model are connected with higher order terms in f,lF and are in fact connected with the conservation of the global spin of the system. Having the hydrodynamic equations (4.3) we are interested in the range of validity of these equations both in time and space variables. To answer these questions we calculate the k4 terms in expansion (3.15a-b). We obtain Kz + K>’ + K$2, where, explicitly,
KQ.’ =
’
92
M. DUDYhKI
where
u-i
- 6
IF-G F+f,
-
1 z+cr,F+6,G,
1 z+a,F-6,G,
1 1
(4.lSa)
’
(4.18b)
’
-$(ku)4[s+ +& s-1
K;’ =
-
& (ku)‘[S+& s
(4.19)
s+]
with
s+=(2 + JA;,I)-3+ (2 -+-JA;,()-3
(4.20a)
s- = (z + lA;01)-3-
(4.20b)
)
(z +
lh;,I)-3 .
The relevant terms of Z,,,(k, z), up to order k2, are
, (W2 1
6
1, (W2 6
(a3
w-
I 4J
(z+lA3()
(a3
’ (4.21a)
I (bo)
w+ (z+lA3()
.
(4.21b)
We can easily see now that terms in K:’ can become very large for z = A, or (ku)2 = A,A;& This means that the hydrodynamic equations are valid for t+ IA31-’ and k2 e (A3A~;ZI/u2. One can check by simple but tedious calculations that for such a t and space gradients both Kz and 1’ with II > 2 can be disregarded. Comparison with exact results shows that in the hydrodynamical limit defined by the above inequalities the behaviour of the hydrodynamic modes obtained from the exact dispersion relation and the projection-operator method agrees. We know from the analysis of the spectrum of L, that for (+ -=s(TV,6, the A, is close to zero and thus the region of validity of the hydrodynamical equations is very limited. If we want to consider the shorter times or larger space gradients we ought to sum the most divergent terms in the expansion (3.13). We do this in the next section.
BOLTZMANN-LORENTZ
5. Generalized
MODEL FOR PARTICLES WITH SPIN II
projection-operator
93
method
The hydrodynamic equations discussed in the previous section are valid, for small A,, for very long times and small space gradients only. Thus one is tempted to extend this description by incorporating the A, mode. Given the perturbation series with a certain class of terms diverging for z a - 1A,/, the natural method is to sum up these terms in all orders in k. We follow this method and next we will show that this is equivalent to the projection-operator technique with {le,), le,), le3)} t a k en as a basic set of variables. The most divergent terms are those containing a maximal number of denominators of the form (z + IA,())‘. Th ese terms are at least of order k4 and have the form
x [ i: (a~lQ,&Q,j
+
le;o)(u;olQn~~Q~le~)l" 0
n=l
(5.1) where N = 0, 1,2, . . . i, j = 1,2. The expression in square brackets is equal to
W2 ‘=-
6(F+f,)
F-
G,
F+
G,
1
z+~A:~[ + z+~A;~[ 1 z+[A,l
(5.2)
’
We can see then that K;:‘(k, z) = i
K;,?’ .
(5.3a)
N=O
The matrix elements form then a geometric obtain
series, we can sum it up and we
W2
6
f,
‘lw-
F+f,
@g&P,,+
s
(53b) where
94
M. DUDYfiSKI
F-
-PI = z+r~(F+f,)+
P2 =
6(F+f,)
z+a(F+f,)+
The appropriate
F-
F+G,
G,
G,
G,
2 + l&l
6(F+f,) F-
1
F+
(k#
W2
+ z+)A;,]
G,
z+&)
F+
z + lA:,,l - .z + lA:ol
W2
6(F+.fs)
F-
1
+ 2 + [A:J G,
G,
(5.4a)
’
1 F+G,
(5.4b)
1 ’
z + lA:Ol + z + lA:,,l
terms from Z1 and Z2 are (ku)‘/6
I:#, 2) = W-t+ 14,) z + (T(F + f,) +
Ft-G,
+
IGJlI
2+
’
(5Sa) and
I&,
2) =
(~31(b,)W2/6
w+ z+c+(F+f,)+
W2 6(F+f,)
F-
G,
F+G,
z + lA;Ol + z + [A:,,/
We obtain now a complicated set of nonhomogeneous dynamical variables of the form
equations
1 ’
(5.5b)
for hydro-
2 + ZC;, + K;1;’ ZC;, + ZC;; (5.6) We can see now that disregarding the z and k2 in the denominators is not a good approximation as far as the time scale t = 1h,J -’ is concerned. Eqs. (5.6) form a system of coupled integro-differential equations both in space and time variables. These equations can both be easier obtained and interpreted if we apply the generalized projection-operator method. To do so we include in the basic set a A, mode. It means that we define a projector P, as
pE = lelN31 + l4(4 and
+ ld(4
(5.7)
BOLTZMANN-LORENTZ
MODEL FOR PARTICLES WITH SPIN II
QE=l-PE.
95
(5.8)
With these operators we obtain now equations for three hydrodynamical variables, namely n(r, t), S(r, t), defined before, and n3(r, t), where
(5.9) Now we can write down equations for the set n(k, z), S(k, z) and n,(k, z) obtained as previously up to order k’. These equations are
(5.10) with
PI,2= PI,,
z+u(F+f,)+
(W2
F-
G,
6
z + IA:01
+ F+Gs z+lA:,l
11
(5.11)
.
It is an easy exercise to eliminate n,(k, z) from eqs. (5.10) and check that the resulting equations for n(k, z) and S(k, z) are exactly those given by eqs. (5.6). One can generalize these results and show that incorporating a larger number of nonhydrodynamical modes is equivalent to a partial summation of selected terms in the perturbation series for the hydrodynamical variables. The above system of equations is not in general parabolic but integrodifferential in time. If however A, < A:$, we can calculate the approximate solution of this system by dropping z in the denominators thus approximating these equations by a set of parabolic equations of the Navier-Stokes type. We solve these equations and then apply the inverse Laplace transform to find the time dependence of the solutions. We obtain three roots of the appropriate determinant zi, z2 and z3. For small k, z1 and z, agree with those obtained previously and the third z3 agrees with the value A,(k) we have determined in I from the exact dispersion relation up to terms of order A,lA:t. Explicitly, z3 reads
96
M. DUDYhKI
q=-u(F+f,)-
W2
ulF2
6(F+ f,)
(u,F)~
+ S,G; -
(6,G,)2
(5.12)
’
The parabolic hydrodynamic equations (5.10) are valid as a good approximation to the distribution function for t s 1A$) -I and for (ku) e (I A:$\ - 1A~(). From this we see that if &/A:$ < 1 we extend both regions of t and space gradients we are able to describe properly as compared to the equations for the hydrodynamical variables n(r, t) and S(r, t) alone. If however A, = Ai: we cannot use the parabolic approximation and our new equations can properly handle for short times only very small space gradients. From the exact solutions we know that there are two cuts in the complex z plane near z = A;b2+ iku. The existence of these spectral singularities is easily seen in the perturbation method and we cannot by adding the next nonhydrodynamical modes remove the divergencies for z = Ai;* due to the infinite number of different modes with the same denominators (z + I A$[)-‘. This shows that the time scale t 3 I A,1-’ is the limit we can describe by the perturbation method and by hydrodynamical-like equations thus there are not such equations for shorter times.
6. The Chapman-Enskog
expansion
In this section we apply the conventional Chapman-Enskog’) method to the analysis of our kinetic model. We begin with rewriting the nonlinear equations’) in the form (a, + u*V)f=
f (qF(P-
(a, + u - V)g = i (r2F(P a,G = i (CrFPg - CGPf)
l)f+
S,G(P-
- 1)g + S,G(P - l)f.
(6.la)
l)g), aFg + aGf)
,
(6lb) (6.1~)
We have introduced the formal small parameter E to scale the u~,~F, 6,,,G, and aF, UC,, which are of the order of the inverse mean free paths for various scattering processes, with the aim to investigate the limit with these mean free paths going simultaneously to zero. This limit turns out for linearized equation to be equivalent to the application of the projector-operator method supplemented with an expansion in small gradients. As usual in the ChapmanEnskog procedure we assume that the solutions of eqs. (6.la-c) are of the form
BOLTZMANN-LORENTZ
MODEL FOR PARTICLES WITH SPIN II
97
(6.2a) g =
&dr,u; {n(r, 4
w, q>>>
(6.2b) (6.2~)
G = G(r; {n(r, 9, S(r, t)>) . The equations for the time evolution
of particle and spin densities are
ate, t>= --v*[Pof(r, u, f)l = D,({n, s>>,
(6.3a)
O,S(r, t) = -v* [PzJg(r, u, t)] = Ds({n, S}) .
(6.3b)
Applying relations (6.2a-c)
to eqs. (6.la-c)
af D, + as af D, + u .Vf= ; [a,F(P an 2
D, + z
D, + us Vg = ; [(r,F(P
and using (6.3a,b) we obtain
- 1)f + S,G(P - l)g] ,
(6.4a)
- 1)s + 6,G(P - 1)f - crFg + uGf]
,
(6.4b) gDn+g
Expanding
D, = 5 [vFPg
- aGPf] .
(6.4~)
now the functions f, g and G in powers of E as
f = i& srnfCrn)>
(6Sa)
m g = c emgCm)) m=O
(6.5b)
G = 2
(6.5~)
smG@) ,
m=O we will use the condition that the hydrodynamical zero order in E. This means that for m # 0 Pf(“)(r, P( g’“‘(r,
u, t) = 0 )
(6.6a)
u, t) + G(“+(r, t)) = 0 .
Inserting (6.5a-c) obtain
into (6.4a-c)
variables are exact up to
(6.6b)
and collecting terms in the same order in E we
98
M. DUDYrjSKI
In order E-~: cr,F(P - 1)f’“’ + 6,G”‘(P
- l)g”’
@(P
- l)f’“’ - aFg(‘) + aGCo)fCo) ,
- l)g”’
&‘g(‘)
+ 6,G”‘(P
- ,G(‘$f(‘)
= 0,
(6.7a)
=0,
(6.7b) (6.7~)
with solutions f(O) = const., g(O) = const. and G(O) = const. These constants are related by the equation co) = G(O)f(O) .
Fg
From eqs. (3.6a,b) we obtain f(O) = Iz(T, t) )
(6.8a)
g(O) = S(r, t)n(r, t)(F + n(r, t))-’ G(O) = S(r, t)F(F + n(r, t))-’ In this order the equations dp(r, t) = 0 a&,
,
(6.8b)
.
(6.8~)
of hydrodynamics
assumed the trivial form (6.9a)
)
t) = 0 .
(6.9b)
In order e(O) we obtain Vf(O) =
u.
u
.
a,F(P- l>f (l)
vg(0) = a,F(P
_ ~lG’o’(P
_ l)g’l’
,
- 1)g”’ - c+Fg(l) + 6,G(‘)(P
+ ,f(o,G(‘)
- 1)f”’
>
0 = (+Fpg(‘) _ ,G(‘$f(‘)
(6.1Oa) + ,G(‘)f(‘) (6. lob)
_ oG(*)f(O) .
(6.10~)
This set of equations is equivalent to a simpler one, v
.
Vf(O) = qF(P
- 1)f”’
”
.
vg(0) = qF(P
- 1)g”’ + 6,G”‘(P
+ 6,G”)(P
- 1)g”’ ,
(6.11a)
- 1)f”’
(6.11b)
.
BOLTZMANN-LORENTZ
pf(l) = 0, Pg@) + G(l) = 0 are
The solutions of eqs. _ (6.11a,b) with constraints of the following form: f’yr,
u, t) = - 5
gyr,
u, t) = - 5
G(l)(r,
1
99
MODEL FOR PARTICLES WITH SPIN II
+ g(O)) u . V( f(O) + g(O)) + u - V( f-(O) us V(fO’ + g(O)) - u . V( f(O) + g(O)) alF - 6,G”’
olF + SIG”’
1 1 ’
u,F - SIG”’
ulF + c$G’~
’
t) = 0 .
(6.12a)
(6.12b) (6.12~)
The distribution functions f(r, u, t), g(r, u, t) and G(r, t) are thus uniquely determined to the order co by the solutions of the following hydrodynamic equations:
d,n(r, t) =
$
V-
v(n+& > qF
a,qr,
t) = g
v*
f S, &
V n+& ( a,F+S,
nS F+n
>
+ u,F - S, fi
> &
(
V n--
nS V n-( F+n - ulF-6,
&
)
1’ 1*
(6.13a)
(6.13b)
We note here the unusual form of the nonlinear hydrodynamic equations. Now we linearize these equations around the stationary solutions, viz.
n(r, t) = n, + 6n(r, t) , and with use of the stationary
=
S(r, t) = S, + SS(r, t)
values n, = f,, G, = FS,I(F + n,) we obtain
0,
(6.14a)
= 0.
(6.14b)
100
M. DUDYrjSKl
Eqs. (6.14a,b) are identical to eqs. (4.3) obtained by the projection-operator method which shows that in the long time limit, where we can prove that they are a good approximation to the distribution function, both schemes are equivalent. The problem how they are related to the solution of the nonlinear equation will be considered in a future paper”).
7. Conclusions We have analyzed the application of the projection-operator technique and we have shown that this method can be applied both to describe the hydrodynamic long time limit and the intermediate region of times scale provided we are away from the spectral singularity of the scattering operator Lo. By adding nonhydrodynamical modes to the set we project on we can obtain more accurate equations but we pay a price losing the simple differential structure of the resulting equations. In the next paper”) we will show that these equations of generalized hydrodynamics can be viewed as a reduction of a greater system of differential equations of the hyperbolic type connected with Grad’s method of moments.
Acknowledgements
The author wishes to thank Prof. LA. Turski for very helpful discussions and encouragements during this work. This work was partially supported by Polish Ministry of Sciences and Technology; Grant MR.1.7.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)
M. Dudynski, Physica 139A (1986) 57, to be referred to as I. M. Bixon, J.R. Dorfman and K.C. MO, Phys. Fluids 14 (1971) 1049. I. Muller, Z. Phys. 198 (1967) 339. H. Grad, Comm. Pure and Appl. Math. 2 (1949) 331, and Principles of the kinetic theory of gases, Handbuch der Physik XII, S. Flugge, ed. (1958), pp. 205-294. W. Israel and J.M. Steward, in: General Relativity, vol 2, A. Held, ed. (Plenum, New York, 1980); Ann. Phys. 118 (1979) 334; Proc. R. Sot. London A365 (1979) 43. W. Israel, Annls. Phys. 100 (1976) 310. E.H. Hauge, Phys. Fluids 13 (1970) 1201. M. Dudydski and L.A. Turski, ZFT Report (1983). cf. J.H. Ferciger and H.G. Kaper, Mathematical Theory of Transport Processes in Gases (North-Holland, Amsterdam, 1972). M. Dudy&ki, in preparation. M. Dudydski, submitted to Physica A.