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Separation of fast and slow variables for a linear system by the method of multiple time scales
Physica 146A (1987) 219-241 North-Holland, Amsterdam
SEPARATION OF FAST AND SLOW VARIABLES F O R A L I N E A R S Y S T E M BY T H E M E T H O D O F M U L T I P L E TIME SCALES* David WYCOFF **~ Department of Physics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA and N.L. BALAZS Department of Physics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA Received 21 January 1987
A matrix method, introduced by Geigenmuller, Titulaer and Felderhof, for separating fast and slow motions of a system of first order linear differential equations is shown to arise in a natural way from a multiple time scales expansion. The general method is then used to derive the Smoluchowski equation (and its initial data) for Brownian motion in a strong magnetic field from the corresponding Kramers-Chandrasekhar initial value problem.
1. Introduction In two previous papers 1'2) we used the multiple time scales technique 3-5) to obtain a p p r o x i m a t e solutions to the initial value p r o b l e m for the K r a m e r s C h a n d r a s e k h a r 6"7) e q u a t i o n in the case w h e n the Stokes fraction is large and the external force is slowly varying in space. It was f o u n d that for large times the configuration space density p r o p a g a t e s (from r e d u c e d initial data) via the S m o l u c h o w s k i 8) equation, while the higher coefficients in a H e r m i t e p o l y n o mial expansion of the distribution function are given as functionals of the * Research supported in part by the National Science Foundation. ** Current address: Department of Mathematics, State University College at Potsdam, Potsdam, NY 13676, USA. Submitted in partial fulfillment of the requirements for the Ph.D. degree at the State University of New York, Stony Brook (1985). 0378-4371 / 87 / $03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
220
D. WYCOFF AND N.L. BALAZS
density. The density thus provides a reduced causal description of the system. The essential feature of the Kramers-Chandrasekhar equation which leads to such behaviour is that the evolution of the density occurs on a much different time scale from that of all the other coefficients in the Hermite expansion of the distribution function. Physically this comes about because when the Stokes fraction is large the velocity distribution of the Brownian particle rapidly becomes Maxwellian and the position changes are then described by a diffusion process. Mathematically this is expressed by the fact that the ratio of the time derivitive of the density to the time derivitive of any higher Hermite coefficient is proportional to a small parameter. The density is thus a "slow" variable while the higher coefficients in the Hermite expansion are "fast" variables. In the present paper we will apply the multiple time scales technique to a system of linear first order differential equations which represents the evolution of an arbitrary number of "slow" and "fast" variables. The equation system we will examine was first introduced and studied by Geigemuller, Titulaer, and Felderhof 9-1~) (hereafter referred to as GTF) in the context of irreversible thermodynamics. GTF used the OnsagerCasimir 12-14) symmetries to show that, whenever two very different time scales are present, the phenomenological equations of motion for the variables characterizing the deviation of a system from equilibrium may be written in a form in which the coupling of the "slow" variables and the "fast" variables is small*. They then presented a functional ansatz which leads to a systematic perturbation scheme in which the evolution of the variables is decoupled into purely fast and purely slow parts. In section 2 we present a brief survey of the method of GTF. In section 3 we give the basic ideas and equations of the multiple time scales expansion. Sections 4 and 5 give the solutions for the slow and fast variables and section 6 gives a summary of the results, and a comparison to the results of GTF. As an example of the usefulness of the system of equations studied here we devote an appendix to applying them to the problem of Brownian motion of a charged particle in a strong magnetic field.
2. The method of GTF
In this section we will give a brief survey of the method of Geigenmuller, Titulaer and Felderhof and set the stage for the multiple time scales expansion of section 3. Consider the linear system of first order differential equations introduced by GTF 9-11), * "Fast" and "slow" refer to the behaviour the variables would have in the abscence of the coupling.
METHOD OF MULTIPLE TIME SCALES III i = - eAx
-
•By,
221 (la)
= -(F + •D)y - •Cx,
(lb)
x(0) =
0c)
y(0) = y .
(ld)
x and y are column vectors. A, B, C, D, and F are constant matrices, and e is a small parameter. We will assume that the eigenvalues of g all have positive real parts*. Although the time derivative of x is of order • 1 and the time derivative of y is of order •0, it is slightly misleading to call x and y the slow and fast variables of the system, as G T F do, since x picks up fast varying parts from the y in eq. (la), while the x in eq. (lb) results in slow parts in y. Intuitively we expect that the fast parts o f y will decay at least as fast as e -F°t, where F 0 is the smallest real part of g's eigenvalues, leaving only the slow parts of y for large t. Since these slow parts arise from the coupling of y to x, eq. (la) should then give an evolution equation for the slow parts of x alone. Let us quantify this intuitive expectation by assuming that for large t the time dependence of y is determined solely by a functional dependence of y on x,
x(t) ~ XB(t ) ,
(2a)
y(t)-*yB[xa(t)] = rxa(t),
(2b)
where F is independent of t. This is reminiscent of the functional ansatz made by Bogoliubov in his study of the B B G K Y hierarchy15-17): that for large times the time dependence of the n-particle distribution function comes about only through a functional dependence on the one-particle distribution function. Therefore it is natural to refer to eqs. (2) as the assumption that a "Bogoliubov-type" solution exists for the equation system (1). Eq. (la) then becomes an equation for x B alone,
-rB :
-
•(A
+
ar)x B ,
(3)
while substituting eqs. (2) and (3) into (lb) yields an equation for the matrix F, (F+ cO)F-eFA-
eFBF+ eC= 0.
(4)
If x has N s components and y has Nf components, then eq. (4) is N~ × Nf equations for the N s x Ne components of F, which can be solved as a power series in e. * An exception to this requirement would be the case when some of g's eigenvalues are purely imaginary. In that case time averaging over the fast oscillations plays a role analogous to the exponential decay which occurs if g's eigenvalues have positive real parts.
222
D. WYCOFF
AND
N.L. BALAZS
The Bogolubov-type solution has a major weakness; we do not know how to connect the initial data for the reduced evolution equation (3) to the initial data (lc), ( l d ) of the full problem. To correct this we must take into account the fact that x and y are both mixtures of slow and fast variables. G T F do this by an extension of the functional ansatz of eqs. (2). We would like to describe the system in terms of purely fast variables, yf, and purely slow variables, x~, in such a way that the slow and fast variables evolve independently,
X ~ SG s X s
'
(5)
.Of = Gfyf, where G~ and Gf are time-independent matrices. We express x and y as mixtures of x, and yf, X :
X s q- $ y f
,
(6) y
= y f -t- I x s ,
where the components of x S, which are the same in number as those of x, are the slow variables, and the components of yr., which are the same in n u m b e r as those of y, are the fast variables. The matrices s and f are assumed to be independent of time. s constructs the fast part of x from the fast part of y, while t constructs the slow part of y from the slow part of x. The matrices G s and 6~ propagate these slow and fast parts. We have used a different notation than G T F in order to make it explicitly clear how x and y are split into slow and fast parts. We expect that for t large enough the fast variables, yf, will have decayed, so that eqs. (6) will reduce to the Bogoliubov assumption of eq. (2) with Xs ~ XB ,
and f=F. For smaller values of t, the presence of the fast variables in eqs. (6) will allow us to connect the Bogoliubov solution to arbitrary initial data. Substituting eqs. (5) and (6) into (1) and separating slow and fast parts we get the four matrix equations
METHOD OF MULTIPLE TIME SCALES III
223
G~+ cA+ eBf=O, fGs + ( F + ED)f+ e C = 0 , (7)
sG~ + eAs + eB = 0, G~+ F + ~D+ ECs=0,
for the four unknown matrices Gs, Gf, s and f. Solution of the eqs. (7) order by order in ~ gives the results of GTF. Let us now see how a multiple time scales expansion leads automatically to solutions of the form assumed in eqs. (5) and (6).
3. A multiple time scales expansion The evolution of the physical system represented by eqs. (1) proceeds on many different time scales simultaneously. Let us recognize this explicitly by letting x and y in eqs. (1) depend on many time scales t 0, t 1, t 2. . . . instead of the single time t,
x(tl ~) ~ x(t o,
tI,
t2 ....
I ~) =- x ( t l ~) ,
(8) y(t l ~)-~
Y(to, tl, t2, . . . I e)
=---y(tl
e) .
Since the small parameter e regulates the relationship between the different time scales we will take (9)
ts = Est,
and thus Or.--> ~
~POtp .
(10)
P
We will treat all the different time scales as independent variables until a solution is obtained, then restrict that solution to the subspace of the t space defined by eqs. (9). We will also assume that the ~ dependence of x and y which is not incorporated into the definition of the time scales may be expanded in a power series,
224
D. WYCOFF
x(tl
AND
N.L. BALAZS
•) = ~ •~X(~)(t) , tr
y(t] •) = ~ e~y(~)(t) . The terms of order •~ in eqs. (1) then b e c o m e
~ 0 x (~-p) = tp
p=O
~0
p=O
tp
x(~)(0)
y(c~-p)=
- A x (~-1) - By (~-1) _Fy(O-) _ Dy(,~-l) _ Cx(~,-1)
(11) =
~(~),
y(~)(0) = ~7(~) . We have assumed that the initial data for x and y may be expanded in •. ~(~) and f(~) are then the terms of order • in those expansions. For the special, but typical, case that the initial data has no • dependence, only ~(0) and )7(°) are non-zero. The introduction of the m a n y different time scales as independent variables leaves the system of eqs. (11) underdetermined. The goal of the m e t h o d of multiple time scales is to impose extra conditions on x and y so as to m a k e the scheme deterministic1-5). In the spirit of section 2, let us write x and y as sums of "slow" and "fast" parts, where by "slow" and "fast" we now m e a n t0-independent and t 0dependent, respectively,
x(t)
= Xs(tl,
y(t)
= ys(tl, t 2 . . . .
t2, . . . ) + xf(to,
tl,
t 2 ....
),
(12) ) + yf(t0, tl, t 2 . . . .
).
Unlike the t r e a t m e n t of G T F , as discussed in section 2, we m a k e no assumption about the relationship of the fast parts of x to the fast parts of y or about the relationship of the slow parts of y to the slow parts of x, as we did in eqs. (6). In fact, eqs. (12) really m a k e no restriction o f x a n d y at all, since any function of to, t 1, t 2 , . . . may be written in the form (12). There is obviously a certain amount of ambiguity involved in eqs. (12). For example if x =
~(t0,
we could have x~ = qJ, X f = ~
" ,
tl,
t2, • • • ) +
q,(t,,
t2, • • • ),
METHOD OF MULTIPLE TIME SCALES III
225
or x s = a~,
xf = ~" + (1 - a)tO, and so on. In practice this ambiguity will cause no problems. We will simply absorb any t0-independent terms into x~ or y~. Substituting eqs. (12) into eqs. (11) we find _(,,-.) = _A(x~,,-1) + X~Cr-1))_ B(y~,-z) + y~O,-z)), ~ 0'. x f(~'-p) + ~ Otp.~ ~
p=0
p=l
(13a)
o.yf~.p'O'-P)+ ~ Otpy(~,-p) =_F(y(,,) + y~,,))_D(y~,,-a) + y~,,-1)) p=O
p=l
--C(x~ °'-1) @X~°'--I)),
(13b)
a7= xs(0) + xf(0),
(13c)
y=ys(0) +yf(0).
(13d)
Of course; eqs. (13) are even more underdetermined than eqs. (11), since we have added the arbitrariness of the division (12) into fast and slow parts. However, we will show that two assumptions are sufficient to make the scheme deterministic: a) We assume that the fast (t0-dependent) parts may be separated from the slow (t0-independent) parts in eqs. (13a) and (13b); and b) we require that the ~ expansions for x s, x~, Ys, and yf each be convergent for all values of t 0, tl, t2,... (The "closure" condition.)
4. Solution for the slow variables
The slow parts of eqs. (13a) and (13b) are
p=l
atp x ('-p) = -Ax~ °-1) - By~°-x) s _
.
_
_
p=l
For or = 0 eq. (14a) gives us no new information. Eq. (14b) gives
(14a)
(14b)
226
D. WYCOFF AND N.L. BALAZS y~0) : 0.
(15)
For o-= 1 eq. (14a) gives (16a)
(E~tl -[- A)X~ 0) = 0 ,
while eq. (14b) gives y(sl) = - F - I C x ~ °~ .
(16b)
For or = 2, use of eq. (16b) in (14a) gives (at2 - B F - I C ) x ~ °) + (at1 + A)x~ 1) = 0 .
(17)
The dependence of x(s°) on the t~ scale is known from eq. (16a) therefore the terms involving X(s°) in eq. (17) may be treated as a known inhomogeneity in an equation determining the t I dependence of x~1). We can then show that eq. (17) has the particular solution ,I,Xs(1),)part. = -- t~1 vt2'~s a _(0) + E [(_)i/(i + 1)!]til+l(a,, + i=0
A)iBF-lCx(sO)
(18)
The requirement
,<,, < < 0 , which is a necessary condition for the convergence of the e expansion for x S, will be violated by eq. (18) for large values of t 1. Therefore we require that the first term on the right-hand side of eq. (18) be canceled by the term in the series which is linear in t~, (a,2 - BF-1e)X(s°~ = 0 .
(19)
BI= 1Cx~°~ in eq. (18) can then be replaced by at2x~ °~. Since t2 and t I are considered to be independent, all other terms in the series in eq. (18) then vanish automatically, since x~°~ satisfies eq. (16a). Substituting eq. (19) into eq. (17) we find (a,, + A)x~ '> = O. The part of eq. (ldb) which is second order in ¢ is
METHOD OF MULTIPLE TIME SCALES III .(1)
.(2)
, (1)
0,2y(~°)+0,1ys = - F y s - D y ~
_(1)
-C*s
227
.
Using eqs. (15), (16a), and (16b) we find y(2) s
=
_F-ICx~I) + (F-IDF-ZC _ F-2CA)x~O)
°
(20)
The expansion may be continued in this way to any order desired. It is clear from the structure of eqs. (14) and the results we have just obtained that (to o',, (tr) _ ( o ' - 1) order e ) Ys will be determined as a linear functional of x(s°), J~S _(1) ' " " " ' 'XS and x~ p) will be determined on the time scales to, t l , t 2 , . . . , G-p- It remains to be seen how the fast parts of x and y evolve, and what the initial data are for the slow and fast variables.
5. Solution for the fast variables
The fast parts of eqs. (13a) and (13b) are
p=O
0 X (°'-p) : tp f
--AX~ °'-1)
--
ay[ ~-1)
O y(O--p) = _Fy~O) _ D y ~ - l ) _ Cx{~-l)
p=O 9 f
(21a)
(21b)
Since eqs. (21) have the same form as eqs. (11) it may not be obvious what we have gained by making the separation into slow and fast parts, this will become clear when we apply the "closure" condition to the first order solutions of eqs. (21). For cr = 0 eq. (21a) becomes
OtoX~O) =
0
.
x~ ° ) is independent of t o, so we can take x~°) = 0 . Eq. (21b) for or = 0 gives (0,0 + F)y[ °) = 0 . For tr = 1 eq. (21a) becomes
(22)
228
D. WYCOFF AND N.L. BALAZS
0t0X~ 1) = - a y ~ °) .
(23)
Since y~0) is determined on the t o scale by eq. (22), eq. (23) may be integrated on the t o scale to give
x~')
= BF-ly[O)
.
(24)
We have set the homogeneous part of the solution to eq. (23) equal to zero, since it is independent of t o and can be absorbed into x~~). The first order part of eq. (21b) is (25)
(0t0 + F)y~ 1) + (0tl -]- D)y~ °) = 0
y~0) is known on the t o scale, so the terms involving y~0) in eq. (25) may be treated as an inhomogeneity in an equation determining the t o dependence of y~l). Integration of eq. (25) on the t o scale then gives the particular solution
Yf(1)~)part. =
L [(-)i/(i + 1)!]to+a(0to _tOOtly(1)f i=0
+ F)iDY~ °).
(26)
In order to insure
eye1) < y~O) ,
(27)
for all values of t 0, we require that the first term in eq. (26) be cancelled by the term in the series which is linear in t o. Thus (0q + D)y~°) = 0,
(28a)
(O,o + F)y~ ') = 0.
(28b)
Now we can understand the importance of separating eqs. (13) into fast and slow parts. Had we not done so, there would have been an extra term on the right-hand side of eq. (25), namely
-Ax~ °) . The particular solution (26) f o r y(1) would then have had the extra term
-F-1Ax~ °) .
(29)
METHOD OF MULTIPLE TIME SCALES III
229
Since y(O) decays as e - Ft°
on the t o scale (while x (°) has no t o dependence), the term (29) would be much larger than the terms in eq. (26) for large to; in fact it would be much larger than y(O). Thus the condition (27) cannot be fullfilled for large to, unless we set Ax~ °) = O.
Similar arguments give the second order solution to eqs. (21). For o" = 2 eq. (21a) gives (0,1 + A)x~ ') + O,oX~z)= -By~ ') .
(30)
Using eqs. (24) and (28) we find that eq. (30) may be integrated for x~2) on the t o scale, x~ 2) = B F - l y ~ ') + ( A B F - z - B F - ' D F - 1 ) y ~ ° ) .
Once again we have set the homogeneous part of the solution equal to zero, since it can be absorbed into the slow part of x. The second order part of eq. (21b) is
(0t0 "~ F)y~2) + (0,1 + D)y~ O + (0,5 + CBF-1)y~ °) = 0.
(31)
The t o dependence of y~O) and y[X) is known from eqs. (22) and (28) so the terms involving y~0) and y~X) in eq. (31) may be treated as known inhomogeneities in an equation determining the t o dependence of y~O). If we are to have E- ( 2 ) / . . (1)
yf ~- yf
,
for all values of t 0, a repetition of the argument following eq. (26) shows that eq. (31) should be split into two parts, (0,0 + F)y~2) = 0 , (Oil +
D ) Y ~ 1)
(32)
+ (0, z + CBF-1)Y~ °) = 0.
The t 1 dependence of y~O) is known from eq. (28a), so eq. (32) may be integrated for y~l) on the t I scale. To insure e" (i) I - . (o) l,f -~.yf ,
230
D. W Y C O F F A N D N.L. B A L A Z S
for all values of t l, eq. (32) must be further split into two parts,
(Oq + D ) y l
l)=0,
(Ot2+CBF-1)yl°)=0.
As was the case for the slow variables, the expansion for the fast variables may be continued in this way to any order desired. It is apparent from the structure of eqs. (21) and the results obtained through second order in • that at order • ~ we will find xl ~) determined as a linear functional of y~0), y~l) , . . . , Y f .(~-1) , a n d y~P) determined on the time scales to, t l , . . . , G p .
6. Summary and conclusions Collecting our results and identifying the time scales as in eqs. (9) and (10) we find
x(t) = xs(t) + [eBF ~ + E2(ABF -2 - B F - ~ D F -~) + ©~(E3)]yf(t),
y(t) : y f ( t )
(33) + [-•F
1C+ E 2 ( F - 1 D F - 1 C - F eCA)
+ C(E3)]Xs(t) ,
J?s = [ - E A + •2BF-1C + •(E3)]Xs , (34) .vf = - [ F + •D + • 2 C B F - ' + ~ ( e 3 ) ] y f . This is precisely the form (5), (6) assumed by GTF; the fast part of x is given as a linear functional of the fast part of y, the slow part of y is given as a linear functional of the slow part of x and the fast and slow parts evolve independently. Since the initial data for eqs. (34) may be found by inverting eqs. (33) at t = 0, it will also agree with the results of GTF. The multiple time scales technique gives the same results as the matrix methods of GTF. For practical calculations the matrix methods are much easier to apply, but the multiple time scales approach gives us a fundamental physical insight into how the functional ansatz (5), (6) arises as a consequence of the existence of the different time scales.
Acknowledgements Both authors thank Professor Max Dresden for discussions, and professors G. Brown and A. Jackson for the use of their computing facilities. NLB thanks the National Science Foundation for their support; D W was supported in part by a Mumford Fellowship.
METHOD OF MULTIPLE TIME SCALES III
231
Appendix Brownian motion in a strong magnetic field In this appendix we will apply the general results of this paper to a specific physical problem: Brownian motion of a charged particle in a strong external magnetic field. We will use the functional approach of GTF, as discussed in section 2, since it gives the same results as the multiple time scales method, but provides a more convenient tool for calculation. We take as our starting point the Kramers-Chandrasekhar 6'7) equation for a single Brownian particle in the presence of an external magnetic field B(x) and non-magnetic force F(x),
Off + v. Vxf + [m-iF + (e/mc)v × BI" V~f = OK" (v + (kT/m)V~)f . (A.1)
f ( x , v , t) is the probability of the Brownian particle having position x and velocity v at time t. f is positive and normalized to unity and is specified initially as f(x, v, o) =
v).
It is convenient to expand the velocity dependence of f in a series of three-dimensional Hermite polynomials2), m
]3/2
f(x, v, t) = \~--~-~/
Z ~ e-'~2/2kr(aN(X, t ) ) i , . . , iN N = 0 {i}
X
(A.2)
VA/i 1 . . . iN ~
where .2 = (--1)N e
(nN[u]),l...iN
0
d
OUil OUi2"'"
0 OUi'~N e -"2
•
Hn is totally symmetric, so a N may also be taken to be so. a N is given in terms of the distribution function, f, by (BN(X, t))il..
'
iN ~-"
(2Na!b!c!) -1 f d3v f(x, v, t) (HN[ ( m ]1/2
232
D. WYCOFF AND N.L. BALAZS
where a, b, and c are the number of times the indices one, two, and three respectively appear in the set {i 1, i 2 , . . . , iN}. The coefficients in the expansion (A.2) have simple physical interpretations: a o is the configuration space density, a 1 is proportional to the current density, and a 2 is proportional to the deviation of the kinetic energy density from the value it has when the velocities have a Maxwellian distribution. In general, a N is a linear combination of the velocity moments of f. We substitute the expansion (A2) into eq. (A.1) and use the rescaling x--Ly, ~'= f l t , e = (2kT/m)l/2(flL)
1,
where L is the scale length of the external non-magnetic force. The coefficients a:~(y, t) then satisfy*
o~.a:v+NaN+e[(N+ l)V'alV+l-{J®a:v_~}-N{O.a:v}]=O.
(A.4)
H e r e V is the gradient with respect to y, J is the Smoluchowski current operator, j = _ 1 (v + (kr)-l(vu)),
where U is the potential of the external force, and O is an antisymmetric 3 × 3 matrix defined in terms of the magnetic field by Oi: = ( e L / m c ) ( m / 2 k T ) l / 2 % k B
k .
We use an index-free notation for the direct product of two tensors,
(M ~ N)il... i s J l . .
"
jp ==--M i l . . " iN~1/p
,
and the symmetrization of a tensor, (M}i,i2...
i, ~
(l/s!) ~ M~.l~p,..i.,
where the sum is over all permutations P of the numbers 1, 2, 3 . . . . . , s. The initial data for a N is given by evaluating eq. (A.3) at 1" = 0, and will be denoted * For details, see ref. 2.
METHOD OF MULTIPLE TIME SCALES III
aN(y, 0) = ~i,,(y).
233
(A.5)
The initial value problem defined by eqs. (A.4) and (A.5) was studied in ref. 2 using the method of multiple time scales. There it was shown that the slowly decaying part of a o provides a reduced causal description of the system for t>>/3 -1, provided E is small and the magnetic field is weak in the sense that ojc//3
-
• ,
where coc = ( e / m c ) l B I
is the (position-dependent) cyclotron frequency of the external magnetic field and/3 is the Stokes friction coefficient. We can study the effect of a stronger field by rescaling B with a factor of • -1, B -= • - 1 ~
.
Eqs. (A.4) then become O,a N + N a N - N { I ~ . aN) + • [ ( N + 1)V. aN+ 1 - ( J ~ aN_l} ] = 0,
(A.6)
a s ( y , O) = a N ( Y ) ,
where 12ij =-- •ijk CrPk=-- ( e L / m c ) ( m / 2 k T ) l / 2 • i j k ~
k .
Now we may consider fields for which the magnetic term in eq. (A.6) is of order • 0:
I@1-1, or
(2)C ~
/3 °
The magnetic force on the particle is of the same order of magnitude as tht Stokes force. Eqs. (A.6) are of the same form as eqs. (1) if we identify
234
D. WYCOFF AND N.L. BALAZS
x = a0 ,
y =
(A.7a)
3 ,
(A.7b)
A = 0,
(A.7c)
a N T = (~N1 V" T ,
(A.7d)
C u T = - ~ul {J @ T } ,
(A.7e)
DMNT= ( m + a)~$M+I,N ~7"
T-
~M_I,N{J@
T},
FMNT= M6MN(T- {11. T } ) ,
(A.Yf) (A.7g)
where T is an arbitrary tensor. Notice that for this case x = a 0 is a scalar while the c o m p o n e n t s of y are tensors of differing ranks, as being a symmetric tensor of rank S. F -1 does exist for F given by eq. (A.7g), (F-I)MN-=
M-16MUA-I
,
where A-1 is the inverse of a matrix A defined by A - r-= T - {11- T } , for a symmetric tensor T. If T has c o m p o n e n t s Ta,,2 . . . . , in the frame where 11 is diagonal, then ({1"~ • T } ) a,a2 . . . . ~ = s - l ( (j)a, -}- (-Oa 2 " J r - ' ' ' -~- O.)as) r a la2 . . . . , ,
where % are the eigenvalues of 1~. A-1 is then given in this frame by
l I has eigenvalues +ilcP I and zero, so A -1 is guaranteed to exist. For the special case that T is of rank one the symmetrization brackets are unneeded and we have
A. T= ( I - ~ ) - T ,
235
M E T H O D O F M U L T I P L E T I M E S C A L E S III
where I is the unit matrix, f~ has the special property ~ 3 ---~ -- (~ 2~'~ ,
(A.9)
where ~2_
~.
ff~ ,
SO
A -1
=
(I
-
n ) -1 = E n' ---t + (1 + , 2 ) - 1 [ n + n 2 ] .
When T is of rank two or higher the more general form of eq. (A.8) or its equivalent in a non-diagonal frame, must be used. Using eqs. (A.7) in eq. (7) we find that the matrix f satisfies Nf N
-
N(I~ "fN} = ¢[fNV" fl + {J®fN-1} -- (N + 1)V. fN+l] + ~SNIJ. (A.10)
Solving eq. (A.10) order by order in ~ we find f(0) = 0 N
f (Nl ) ~. (~NI(I -- l - ~ ) - I . j
{(I - 1~). f~)} = ( 1 / 2 ) S m { J ® [(I - 1~) -1- J ] } . For any tensor T, s satisfies NSN( T - I I . T} = ~N1 V" T + e [ - - N S N _ 1 V" "r + S N + I { J @
T}
+ $1{J~$NT}]
,
(A.11) SO
s N( o=) 0 s2 ) T = (~N1 ~'° s (N' ) { r -
(l -- ,~)-i
.
T,
1~- T} = --SNEV" (I--1~) -1 "V" T
In principle s and f may be found to any order in E from eqs. (A.10) and (A.11), but the complicated form of F -~ will make the solution quite difficult except for special cases.
236
D. WYCOFF AND N.L. BALAZS
If we write a 0 = b + ~ SM'OM , N=1,2,3,...
8N = 'ON -JC-I N b ,
the initial value problem (A.6) is equivalent to the separate initial value problems for the slow and fast variables, 19rb --It- E2V , [] @ (1
+ ~2)-1(O + 02)] .Jb
+ ~(4)
__~0 ,
b(y, O) = ~o - elY. [i + (1 + q~:)-~(O + 02)] • a, + ~ ( 2 ) ,
(A.11a) (A.11b)
and Cg'ON + N'ON - { N i l " ,8N} - e[(N + 1)V. 'ON+I -- { J (~ 'ON-1 }]
q- aNl[--~2{J~)(V • (]-I- (1 + @ 2 ) - ' ( n + 012) • 'Ol)} + ~(E4)] ~- 0, 'ON(Y, O) = aN -- e6Nl[I + (1 + ~ 2 ) - 1 ( O + f12)] "Jao + 0~(e2) •
Though they are rapidly oscillating (with the cyclotron frequency of the external field) the fast variables 'ON are also quickly damped out, so that for r >> 1 (t>> 13-1) the slow part of the density gives a reduced causal description of the system, with eqs. (A. 11) as the reduced initial value problem and aN(y, t)--+e6u~[l+ (1 + @ 2 ) - ' ( O + 0 2 ) ] ' J b ( y ,
r ) + G(e2).
(A.12)
Eq. (A.11a) is thus the Smoluchowski equation for Brownian motion in a strong magnetic field. We see once again that the reduced initial data (A.11b) for this equation differs by terms of order e from the true initial data for the density. For O = 0 eqs. (A.11) reduce to O,b + ~ 2 V ' j b
+
~(E 4) = 0 ,
b(y, O) = a o - , V . al + G(e2),
in agreement with earlier results1'18'19). If 0--+0,
then eqs. (A.11) become
METHOD OF MULTIPLE TIME SCALES III
237
O¢b+ EEv'(I + ,O)'Jb+ ~(,4) = 0, b(y,O) = a o - ' V ' a l
+ ~(2),
which agrees with the results of the multiple time scales expansion of ref. 2. It is interesting to examine the solution of eq. (64a) for some special cases. If the magnetic field is constant and directed along the z-axis,
= (0, 0, ~ ) ,
(A.13)
and there is no external force, eq. (A.11a) becomes
O,b = [DLO~ + DT(O 2 + 0~)lb, where the transverse and longitudinal diffusion coefficients are D T = 1 , 2 ( 1 q_ t~)2)-i ,
DL = 1 2 .
D L has the value it would have in the absence of the field while
DL/D, r = 1 + cO2 > 1, so the transverse variables thermalize more slowly than they do in the absence of a magnetic field. If • is constant and the external force has the Hooke's law form F = -ay,
(A.14)
eq. (A.11a) can be solved exactly. Rather than consider the full exact solution, it is instructive to examine the time evolution of the electric dipole moment of the Brownian particle's charge distribution,
d(r) =- f Yao(y , r) d3y. For ~">> 1 a o can be replaced by b, so
d('r)---> f yb(y, r) d3y = ds(~") . Using eq. (A.11a) and one integration by parts we find d s = ,211 + (1 + cO2)-I(~ .~_ ~ 2 ) ] . f Jb(y, r) d3y
238
D. WYCOFF AND N.L. BALAZS
which for the Hooke's law force becomes
,2( 2 ~o ) [I + (1 + 4 2)
1(~-~+ ~-~2)1 ° ds '
(A.15)
Eq. (A.11b) shows that d~ is given initially by
ds(0) =
f yao(y)d3y+ e[I + ( 1+ 1 ~ 2 ) i ( ~ - ~
_1_ ~-~2)] ,
f al(Y)d3y +
(~,(~.2) ,
(A.16) which is of course in general different from the true initial value of the dipole moment. Eq. (A.15) is easily integrated using the property (A.9) of ~ . If • is given by eq. (A.13) we find
e axt cos tOTt ds(t) = ~ e -ax' sin ~oxt \ 0
0° ) e -av' cos O-~rt 0
• ds(0),
(A.17)
e -aLt
where
AT=
0) ~-
2
(/)2) (1 +
1__
0")2/~ j~ 2 _~. (./)2 ,
~L
2
60 = -"~- ,
2
(.Oc0) 60T = ~/~T -- j~ 2 ..{_ 60 ~ •
We have returned to the dimensional time, t =/3-1r, used the fact that if F is given by eq. (A.14) then a is related to the usual harmonic frequency by
ot = mo)2L 2 and identified
We see from eq. (A.17) that the component of d along the magnetic field decays with the rate it would have in the absence of the field from an initial value d3(0 ) = f Y3ao(y) d3y + E f (a,(y)) 3 d3y, which, although it is different from its true initial value, is the same as it would be in the absence of the magnetic field. The decay of the transverse components of d is affected by the field in two ways: (a) AT is smaller than it would be
M E T H O D O F M U L T I P L E T I M E SCALES III
239
in the absence of the magnetic field, so the decay is slower, and (b) the transverse components of d rotate around the z-axis with the frequency tar as they decay. In addition the initial data (A.16) for the transverse components of d differs by terms of order ~ from the value it would have in the absence of the magnetic field. If there are n non-interacting Brownian particles per unit volume, the total currrent density produced is al(y, 7") d3y.
j = n e ( v ) = ne
If the external force is due to a constant electric field E, F=eE,
then eq. (A.12) shows, for a constant magnetic field, that j has the stationary value j=o"E,
where the conductivity tensor or is given by ne 2
or = mfl [I + (1 + (I)2)-1([I + 02)].
(A.18)
If the magnetic field is directed along the z-axis, eq. (A.18) becomes 1
0
09c
or=~--~
1+
fl 0
1
0to 2 .
(A.19)
0
The conductivity (A.19) displays both magneto-resistance and Hall effect terms. Eq. (A.11a) may also be written in two other convenient forms: O,b + e 2 V ' [ J + (l + q b 2 ) - ~ ( - ~ x J + ~ x ( ~ x g ) ] b = O ,
(A.20)
O.~b + e2V- [(1 + ~ 2 ) - ~ ( j _ ~ x J + @ ( ~ . J ) ) l b = 0 .
(A.21)
or
240
D. WYCOFF AND N.L. BALAZS
Using any of the forms (A.11a), (A.20) or (A.21) it can be shown that eq. (Alla) has an H-theorem. Define
f
=- b(y, r) ln[eU(Y)/krb(y, r)] d3y.
"F"(r)
" F " is like a free energy for the coordinates y:
kT"F"= "E" - kT"S", "E"
= f U(y)b(y, r) day,
"S" =
- f b(y, r)ln[b(y, r)] day.
For b determined by eq. (A.11a) we find t"
e-U(y)/kT
" P " ( r ) = - 2 ~ 2 J - -b
(1 + q,2)-l[IJbl2 + 1~ "Jbl2].
(A.22)
We see that
"k"(~) ~<0, "P"(r) = 0
iffJb(y, r) = O,
so all solutions of eq. (14.11a) go to the equilibrium solution
e-V(y)/krd3y
bequilibrium(Y ) =
for any initial data
e-V(y)/~T
b(y, 0). If we write
Jb(y, r) = JL + J T , where JL is directed along the magnetic field and JT is orthogonal to it, eq. (A.22) becomes "/~"'(7) = - - 2 ~ 2
b(y, r) [J2L+ f e-U(y)/kT
(1
+ ~)2) 1j2].
Thus the approach to "parallel equilibrium", Jt=0,
METHOD OF MULTIPLE TIME SCALES III
241
is faster than the approach to "transverse equilibrium",
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
D. Wycoff and N.L. Balazs, Physica 146A (1987) 175, this volume. D. Wycoff and N.L. Balazs, Physica 146A (1987) 201, this volume. E.A. Frieman, J. Math. Phys. 4 (1963) 410. G. Sandri, Ann. Phys. 24 (1963) 332. A.H. Nayfeh, Perturbation Methods (Wiley, New York, 1973) chap. 6. H.A. Kramers, Physica 7 (1940) 284. S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. M.V. Smoluchowski, Ann. Physik 48 (1915) 1103. U. Geigenmuller, U.M. Titulaer and B.U. Felderhof, Physica lI9A (1983) 41. U. Geigenmuller, U.M. Titulaer and B.U. Felderhof, Physica llgA (1983) 53. U. Geigenmuller, B.U. Felderhof and U.M. Titulaer, Physica 120A (1983) 635. L. Onsager, Phys. Rev. 37 (1931) 405. L. Onsager, Phys. Rev. 38 (1931) 2265. H.G.B. Casimir, Rev. Mod. Phys, 17 (1945) 343. N.N. Bogoliubov, Studies in Statistical Mechanics, vol. 1, de Boer, ed. (North-Holland, Amsterdam, 1952) p. 1. G.E. Uhlenbeck, Boulder Lectures, (1951). S.T. Choh and G.E. Uhlenbeck, The Kinetic Theory of Dense Gases, Univ. of Michigan (1958). G. Wilemski, J. Stat. Phys. 14(1976) 2. U.M. Titulaer, Physica 91A (1978) 321.
SEPARATION OF FAST AND SLOW VARIABLES F O R A L I N E A R S Y S T E M BY T H E M E T H O D O F M U L T I P L E TIME SCALES* David WYCOFF **~ Department of Physics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA and N.L. BALAZS Department of Physics, State University of New York at Stony Brook, Stony Brook, NY 11794, USA Received 21 January 1987
A matrix method, introduced by Geigenmuller, Titulaer and Felderhof, for separating fast and slow motions of a system of first order linear differential equations is shown to arise in a natural way from a multiple time scales expansion. The general method is then used to derive the Smoluchowski equation (and its initial data) for Brownian motion in a strong magnetic field from the corresponding Kramers-Chandrasekhar initial value problem.
1. Introduction In two previous papers 1'2) we used the multiple time scales technique 3-5) to obtain a p p r o x i m a t e solutions to the initial value p r o b l e m for the K r a m e r s C h a n d r a s e k h a r 6"7) e q u a t i o n in the case w h e n the Stokes fraction is large and the external force is slowly varying in space. It was f o u n d that for large times the configuration space density p r o p a g a t e s (from r e d u c e d initial data) via the S m o l u c h o w s k i 8) equation, while the higher coefficients in a H e r m i t e p o l y n o mial expansion of the distribution function are given as functionals of the * Research supported in part by the National Science Foundation. ** Current address: Department of Mathematics, State University College at Potsdam, Potsdam, NY 13676, USA. Submitted in partial fulfillment of the requirements for the Ph.D. degree at the State University of New York, Stony Brook (1985). 0378-4371 / 87 / $03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
220
D. WYCOFF AND N.L. BALAZS
density. The density thus provides a reduced causal description of the system. The essential feature of the Kramers-Chandrasekhar equation which leads to such behaviour is that the evolution of the density occurs on a much different time scale from that of all the other coefficients in the Hermite expansion of the distribution function. Physically this comes about because when the Stokes fraction is large the velocity distribution of the Brownian particle rapidly becomes Maxwellian and the position changes are then described by a diffusion process. Mathematically this is expressed by the fact that the ratio of the time derivitive of the density to the time derivitive of any higher Hermite coefficient is proportional to a small parameter. The density is thus a "slow" variable while the higher coefficients in the Hermite expansion are "fast" variables. In the present paper we will apply the multiple time scales technique to a system of linear first order differential equations which represents the evolution of an arbitrary number of "slow" and "fast" variables. The equation system we will examine was first introduced and studied by Geigemuller, Titulaer, and Felderhof 9-1~) (hereafter referred to as GTF) in the context of irreversible thermodynamics. GTF used the OnsagerCasimir 12-14) symmetries to show that, whenever two very different time scales are present, the phenomenological equations of motion for the variables characterizing the deviation of a system from equilibrium may be written in a form in which the coupling of the "slow" variables and the "fast" variables is small*. They then presented a functional ansatz which leads to a systematic perturbation scheme in which the evolution of the variables is decoupled into purely fast and purely slow parts. In section 2 we present a brief survey of the method of GTF. In section 3 we give the basic ideas and equations of the multiple time scales expansion. Sections 4 and 5 give the solutions for the slow and fast variables and section 6 gives a summary of the results, and a comparison to the results of GTF. As an example of the usefulness of the system of equations studied here we devote an appendix to applying them to the problem of Brownian motion of a charged particle in a strong magnetic field.
2. The method of GTF
In this section we will give a brief survey of the method of Geigenmuller, Titulaer and Felderhof and set the stage for the multiple time scales expansion of section 3. Consider the linear system of first order differential equations introduced by GTF 9-11), * "Fast" and "slow" refer to the behaviour the variables would have in the abscence of the coupling.
METHOD OF MULTIPLE TIME SCALES III i = - eAx
-
•By,
221 (la)
= -(F + •D)y - •Cx,
(lb)
x(0) =
0c)
y(0) = y .
(ld)
x and y are column vectors. A, B, C, D, and F are constant matrices, and e is a small parameter. We will assume that the eigenvalues of g all have positive real parts*. Although the time derivative of x is of order • 1 and the time derivative of y is of order •0, it is slightly misleading to call x and y the slow and fast variables of the system, as G T F do, since x picks up fast varying parts from the y in eq. (la), while the x in eq. (lb) results in slow parts in y. Intuitively we expect that the fast parts o f y will decay at least as fast as e -F°t, where F 0 is the smallest real part of g's eigenvalues, leaving only the slow parts of y for large t. Since these slow parts arise from the coupling of y to x, eq. (la) should then give an evolution equation for the slow parts of x alone. Let us quantify this intuitive expectation by assuming that for large t the time dependence of y is determined solely by a functional dependence of y on x,
x(t) ~ XB(t ) ,
(2a)
y(t)-*yB[xa(t)] = rxa(t),
(2b)
where F is independent of t. This is reminiscent of the functional ansatz made by Bogoliubov in his study of the B B G K Y hierarchy15-17): that for large times the time dependence of the n-particle distribution function comes about only through a functional dependence on the one-particle distribution function. Therefore it is natural to refer to eqs. (2) as the assumption that a "Bogoliubov-type" solution exists for the equation system (1). Eq. (la) then becomes an equation for x B alone,
-rB :
-
•(A
+
ar)x B ,
(3)
while substituting eqs. (2) and (3) into (lb) yields an equation for the matrix F, (F+ cO)F-eFA-
eFBF+ eC= 0.
(4)
If x has N s components and y has Nf components, then eq. (4) is N~ × Nf equations for the N s x Ne components of F, which can be solved as a power series in e. * An exception to this requirement would be the case when some of g's eigenvalues are purely imaginary. In that case time averaging over the fast oscillations plays a role analogous to the exponential decay which occurs if g's eigenvalues have positive real parts.
222
D. WYCOFF
AND
N.L. BALAZS
The Bogolubov-type solution has a major weakness; we do not know how to connect the initial data for the reduced evolution equation (3) to the initial data (lc), ( l d ) of the full problem. To correct this we must take into account the fact that x and y are both mixtures of slow and fast variables. G T F do this by an extension of the functional ansatz of eqs. (2). We would like to describe the system in terms of purely fast variables, yf, and purely slow variables, x~, in such a way that the slow and fast variables evolve independently,
X ~ SG s X s
'
(5)
.Of = Gfyf, where G~ and Gf are time-independent matrices. We express x and y as mixtures of x, and yf, X :
X s q- $ y f
,
(6) y
= y f -t- I x s ,
where the components of x S, which are the same in number as those of x, are the slow variables, and the components of yr., which are the same in n u m b e r as those of y, are the fast variables. The matrices s and f are assumed to be independent of time. s constructs the fast part of x from the fast part of y, while t constructs the slow part of y from the slow part of x. The matrices G s and 6~ propagate these slow and fast parts. We have used a different notation than G T F in order to make it explicitly clear how x and y are split into slow and fast parts. We expect that for t large enough the fast variables, yf, will have decayed, so that eqs. (6) will reduce to the Bogoliubov assumption of eq. (2) with Xs ~ XB ,
and f=F. For smaller values of t, the presence of the fast variables in eqs. (6) will allow us to connect the Bogoliubov solution to arbitrary initial data. Substituting eqs. (5) and (6) into (1) and separating slow and fast parts we get the four matrix equations
METHOD OF MULTIPLE TIME SCALES III
223
G~+ cA+ eBf=O, fGs + ( F + ED)f+ e C = 0 , (7)
sG~ + eAs + eB = 0, G~+ F + ~D+ ECs=0,
for the four unknown matrices Gs, Gf, s and f. Solution of the eqs. (7) order by order in ~ gives the results of GTF. Let us now see how a multiple time scales expansion leads automatically to solutions of the form assumed in eqs. (5) and (6).
3. A multiple time scales expansion The evolution of the physical system represented by eqs. (1) proceeds on many different time scales simultaneously. Let us recognize this explicitly by letting x and y in eqs. (1) depend on many time scales t 0, t 1, t 2. . . . instead of the single time t,
x(tl ~) ~ x(t o,
tI,
t2 ....
I ~) =- x ( t l ~) ,
(8) y(t l ~)-~
Y(to, tl, t2, . . . I e)
=---y(tl
e) .
Since the small parameter e regulates the relationship between the different time scales we will take (9)
ts = Est,
and thus Or.--> ~
~POtp .
(10)
P
We will treat all the different time scales as independent variables until a solution is obtained, then restrict that solution to the subspace of the t space defined by eqs. (9). We will also assume that the ~ dependence of x and y which is not incorporated into the definition of the time scales may be expanded in a power series,
224
D. WYCOFF
x(tl
AND
N.L. BALAZS
•) = ~ •~X(~)(t) , tr
y(t] •) = ~ e~y(~)(t) . The terms of order •~ in eqs. (1) then b e c o m e
~ 0 x (~-p) = tp
p=O
~0
p=O
tp
x(~)(0)
y(c~-p)=
- A x (~-1) - By (~-1) _Fy(O-) _ Dy(,~-l) _ Cx(~,-1)
(11) =
~(~),
y(~)(0) = ~7(~) . We have assumed that the initial data for x and y may be expanded in •. ~(~) and f(~) are then the terms of order • in those expansions. For the special, but typical, case that the initial data has no • dependence, only ~(0) and )7(°) are non-zero. The introduction of the m a n y different time scales as independent variables leaves the system of eqs. (11) underdetermined. The goal of the m e t h o d of multiple time scales is to impose extra conditions on x and y so as to m a k e the scheme deterministic1-5). In the spirit of section 2, let us write x and y as sums of "slow" and "fast" parts, where by "slow" and "fast" we now m e a n t0-independent and t 0dependent, respectively,
x(t)
= Xs(tl,
y(t)
= ys(tl, t 2 . . . .
t2, . . . ) + xf(to,
tl,
t 2 ....
),
(12) ) + yf(t0, tl, t 2 . . . .
).
Unlike the t r e a t m e n t of G T F , as discussed in section 2, we m a k e no assumption about the relationship of the fast parts of x to the fast parts of y or about the relationship of the slow parts of y to the slow parts of x, as we did in eqs. (6). In fact, eqs. (12) really m a k e no restriction o f x a n d y at all, since any function of to, t 1, t 2 , . . . may be written in the form (12). There is obviously a certain amount of ambiguity involved in eqs. (12). For example if x =
~(t0,
we could have x~ = qJ, X f = ~
" ,
tl,
t2, • • • ) +
q,(t,,
t2, • • • ),
METHOD OF MULTIPLE TIME SCALES III
225
or x s = a~,
xf = ~" + (1 - a)tO, and so on. In practice this ambiguity will cause no problems. We will simply absorb any t0-independent terms into x~ or y~. Substituting eqs. (12) into eqs. (11) we find _(,,-.) = _A(x~,,-1) + X~Cr-1))_ B(y~,-z) + y~O,-z)), ~ 0'. x f(~'-p) + ~ Otp.~ ~
p=0
p=l
(13a)
o.yf~.p'O'-P)+ ~ Otpy(~,-p) =_F(y(,,) + y~,,))_D(y~,,-a) + y~,,-1)) p=O
p=l
--C(x~ °'-1) @X~°'--I)),
(13b)
a7= xs(0) + xf(0),
(13c)
y=ys(0) +yf(0).
(13d)
Of course; eqs. (13) are even more underdetermined than eqs. (11), since we have added the arbitrariness of the division (12) into fast and slow parts. However, we will show that two assumptions are sufficient to make the scheme deterministic: a) We assume that the fast (t0-dependent) parts may be separated from the slow (t0-independent) parts in eqs. (13a) and (13b); and b) we require that the ~ expansions for x s, x~, Ys, and yf each be convergent for all values of t 0, tl, t2,... (The "closure" condition.)
4. Solution for the slow variables
The slow parts of eqs. (13a) and (13b) are
p=l
atp x ('-p) = -Ax~ °-1) - By~°-x) s _
.
_
_
p=l
For or = 0 eq. (14a) gives us no new information. Eq. (14b) gives
(14a)
(14b)
226
D. WYCOFF AND N.L. BALAZS y~0) : 0.
(15)
For o-= 1 eq. (14a) gives (16a)
(E~tl -[- A)X~ 0) = 0 ,
while eq. (14b) gives y(sl) = - F - I C x ~ °~ .
(16b)
For or = 2, use of eq. (16b) in (14a) gives (at2 - B F - I C ) x ~ °) + (at1 + A)x~ 1) = 0 .
(17)
The dependence of x(s°) on the t~ scale is known from eq. (16a) therefore the terms involving X(s°) in eq. (17) may be treated as a known inhomogeneity in an equation determining the t I dependence of x~1). We can then show that eq. (17) has the particular solution ,I,Xs(1),)part. = -- t~1 vt2'~s a _(0) + E [(_)i/(i + 1)!]til+l(a,, + i=0
A)iBF-lCx(sO)
(18)
The requirement
,<,, < < 0 , which is a necessary condition for the convergence of the e expansion for x S, will be violated by eq. (18) for large values of t 1. Therefore we require that the first term on the right-hand side of eq. (18) be canceled by the term in the series which is linear in t~, (a,2 - BF-1e)X(s°~ = 0 .
(19)
BI= 1Cx~°~ in eq. (18) can then be replaced by at2x~ °~. Since t2 and t I are considered to be independent, all other terms in the series in eq. (18) then vanish automatically, since x~°~ satisfies eq. (16a). Substituting eq. (19) into eq. (17) we find (a,, + A)x~ '> = O. The part of eq. (ldb) which is second order in ¢ is
METHOD OF MULTIPLE TIME SCALES III .(1)
.(2)
, (1)
0,2y(~°)+0,1ys = - F y s - D y ~
_(1)
-C*s
227
.
Using eqs. (15), (16a), and (16b) we find y(2) s
=
_F-ICx~I) + (F-IDF-ZC _ F-2CA)x~O)
°
(20)
The expansion may be continued in this way to any order desired. It is clear from the structure of eqs. (14) and the results we have just obtained that (to o',, (tr) _ ( o ' - 1) order e ) Ys will be determined as a linear functional of x(s°), J~S _(1) ' " " " ' 'XS and x~ p) will be determined on the time scales to, t l , t 2 , . . . , G-p- It remains to be seen how the fast parts of x and y evolve, and what the initial data are for the slow and fast variables.
5. Solution for the fast variables
The fast parts of eqs. (13a) and (13b) are
p=O
0 X (°'-p) : tp f
--AX~ °'-1)
--
ay[ ~-1)
O y(O--p) = _Fy~O) _ D y ~ - l ) _ Cx{~-l)
p=O 9 f
(21a)
(21b)
Since eqs. (21) have the same form as eqs. (11) it may not be obvious what we have gained by making the separation into slow and fast parts, this will become clear when we apply the "closure" condition to the first order solutions of eqs. (21). For cr = 0 eq. (21a) becomes
OtoX~O) =
0
.
x~ ° ) is independent of t o, so we can take x~°) = 0 . Eq. (21b) for or = 0 gives (0,0 + F)y[ °) = 0 . For tr = 1 eq. (21a) becomes
(22)
228
D. WYCOFF AND N.L. BALAZS
0t0X~ 1) = - a y ~ °) .
(23)
Since y~0) is determined on the t o scale by eq. (22), eq. (23) may be integrated on the t o scale to give
x~')
= BF-ly[O)
.
(24)
We have set the homogeneous part of the solution to eq. (23) equal to zero, since it is independent of t o and can be absorbed into x~~). The first order part of eq. (21b) is (25)
(0t0 + F)y~ 1) + (0tl -]- D)y~ °) = 0
y~0) is known on the t o scale, so the terms involving y~0) in eq. (25) may be treated as an inhomogeneity in an equation determining the t o dependence of y~l). Integration of eq. (25) on the t o scale then gives the particular solution
Yf(1)~)part. =
L [(-)i/(i + 1)!]to+a(0to _tOOtly(1)f i=0
+ F)iDY~ °).
(26)
In order to insure
eye1) < y~O) ,
(27)
for all values of t 0, we require that the first term in eq. (26) be cancelled by the term in the series which is linear in t o. Thus (0q + D)y~°) = 0,
(28a)
(O,o + F)y~ ') = 0.
(28b)
Now we can understand the importance of separating eqs. (13) into fast and slow parts. Had we not done so, there would have been an extra term on the right-hand side of eq. (25), namely
-Ax~ °) . The particular solution (26) f o r y(1) would then have had the extra term
-F-1Ax~ °) .
(29)
METHOD OF MULTIPLE TIME SCALES III
229
Since y(O) decays as e - Ft°
on the t o scale (while x (°) has no t o dependence), the term (29) would be much larger than the terms in eq. (26) for large to; in fact it would be much larger than y(O). Thus the condition (27) cannot be fullfilled for large to, unless we set Ax~ °) = O.
Similar arguments give the second order solution to eqs. (21). For o" = 2 eq. (21a) gives (0,1 + A)x~ ') + O,oX~z)= -By~ ') .
(30)
Using eqs. (24) and (28) we find that eq. (30) may be integrated for x~2) on the t o scale, x~ 2) = B F - l y ~ ') + ( A B F - z - B F - ' D F - 1 ) y ~ ° ) .
Once again we have set the homogeneous part of the solution equal to zero, since it can be absorbed into the slow part of x. The second order part of eq. (21b) is
(0t0 "~ F)y~2) + (0,1 + D)y~ O + (0,5 + CBF-1)y~ °) = 0.
(31)
The t o dependence of y~O) and y[X) is known from eqs. (22) and (28) so the terms involving y~0) and y~X) in eq. (31) may be treated as known inhomogeneities in an equation determining the t o dependence of y~O). If we are to have E- ( 2 ) / . . (1)
yf ~- yf
,
for all values of t 0, a repetition of the argument following eq. (26) shows that eq. (31) should be split into two parts, (0,0 + F)y~2) = 0 , (Oil +
D ) Y ~ 1)
(32)
+ (0, z + CBF-1)Y~ °) = 0.
The t 1 dependence of y~O) is known from eq. (28a), so eq. (32) may be integrated for y~l) on the t I scale. To insure e" (i) I - . (o) l,f -~.yf ,
230
D. W Y C O F F A N D N.L. B A L A Z S
for all values of t l, eq. (32) must be further split into two parts,
(Oq + D ) y l
l)=0,
(Ot2+CBF-1)yl°)=0.
As was the case for the slow variables, the expansion for the fast variables may be continued in this way to any order desired. It is apparent from the structure of eqs. (21) and the results obtained through second order in • that at order • ~ we will find xl ~) determined as a linear functional of y~0), y~l) , . . . , Y f .(~-1) , a n d y~P) determined on the time scales to, t l , . . . , G p .
6. Summary and conclusions Collecting our results and identifying the time scales as in eqs. (9) and (10) we find
x(t) = xs(t) + [eBF ~ + E2(ABF -2 - B F - ~ D F -~) + ©~(E3)]yf(t),
y(t) : y f ( t )
(33) + [-•F
1C+ E 2 ( F - 1 D F - 1 C - F eCA)
+ C(E3)]Xs(t) ,
J?s = [ - E A + •2BF-1C + •(E3)]Xs , (34) .vf = - [ F + •D + • 2 C B F - ' + ~ ( e 3 ) ] y f . This is precisely the form (5), (6) assumed by GTF; the fast part of x is given as a linear functional of the fast part of y, the slow part of y is given as a linear functional of the slow part of x and the fast and slow parts evolve independently. Since the initial data for eqs. (34) may be found by inverting eqs. (33) at t = 0, it will also agree with the results of GTF. The multiple time scales technique gives the same results as the matrix methods of GTF. For practical calculations the matrix methods are much easier to apply, but the multiple time scales approach gives us a fundamental physical insight into how the functional ansatz (5), (6) arises as a consequence of the existence of the different time scales.
Acknowledgements Both authors thank Professor Max Dresden for discussions, and professors G. Brown and A. Jackson for the use of their computing facilities. NLB thanks the National Science Foundation for their support; D W was supported in part by a Mumford Fellowship.
METHOD OF MULTIPLE TIME SCALES III
231
Appendix Brownian motion in a strong magnetic field In this appendix we will apply the general results of this paper to a specific physical problem: Brownian motion of a charged particle in a strong external magnetic field. We will use the functional approach of GTF, as discussed in section 2, since it gives the same results as the multiple time scales method, but provides a more convenient tool for calculation. We take as our starting point the Kramers-Chandrasekhar 6'7) equation for a single Brownian particle in the presence of an external magnetic field B(x) and non-magnetic force F(x),
Off + v. Vxf + [m-iF + (e/mc)v × BI" V~f = OK" (v + (kT/m)V~)f . (A.1)
f ( x , v , t) is the probability of the Brownian particle having position x and velocity v at time t. f is positive and normalized to unity and is specified initially as f(x, v, o) =
v).
It is convenient to expand the velocity dependence of f in a series of three-dimensional Hermite polynomials2), m
]3/2
f(x, v, t) = \~--~-~/
Z ~ e-'~2/2kr(aN(X, t ) ) i , . . , iN N = 0 {i}
X
(A.2)
VA/i 1 . . . iN ~
where .2 = (--1)N e
(nN[u]),l...iN
0
d
OUil OUi2"'"
0 OUi'~N e -"2
•
Hn is totally symmetric, so a N may also be taken to be so. a N is given in terms of the distribution function, f, by (BN(X, t))il..
'
iN ~-"
(2Na!b!c!) -1 f d3v f(x, v, t) (HN[ ( m ]1/2
232
D. WYCOFF AND N.L. BALAZS
where a, b, and c are the number of times the indices one, two, and three respectively appear in the set {i 1, i 2 , . . . , iN}. The coefficients in the expansion (A.2) have simple physical interpretations: a o is the configuration space density, a 1 is proportional to the current density, and a 2 is proportional to the deviation of the kinetic energy density from the value it has when the velocities have a Maxwellian distribution. In general, a N is a linear combination of the velocity moments of f. We substitute the expansion (A2) into eq. (A.1) and use the rescaling x--Ly, ~'= f l t , e = (2kT/m)l/2(flL)
1,
where L is the scale length of the external non-magnetic force. The coefficients a:~(y, t) then satisfy*
o~.a:v+NaN+e[(N+ l)V'alV+l-{J®a:v_~}-N{O.a:v}]=O.
(A.4)
H e r e V is the gradient with respect to y, J is the Smoluchowski current operator, j = _ 1 (v + (kr)-l(vu)),
where U is the potential of the external force, and O is an antisymmetric 3 × 3 matrix defined in terms of the magnetic field by Oi: = ( e L / m c ) ( m / 2 k T ) l / 2 % k B
k .
We use an index-free notation for the direct product of two tensors,
(M ~ N)il... i s J l . .
"
jp ==--M i l . . " iN~1/p
,
and the symmetrization of a tensor, (M}i,i2...
i, ~
(l/s!) ~ M~.l~p,..i.,
where the sum is over all permutations P of the numbers 1, 2, 3 . . . . . , s. The initial data for a N is given by evaluating eq. (A.3) at 1" = 0, and will be denoted * For details, see ref. 2.
METHOD OF MULTIPLE TIME SCALES III
aN(y, 0) = ~i,,(y).
233
(A.5)
The initial value problem defined by eqs. (A.4) and (A.5) was studied in ref. 2 using the method of multiple time scales. There it was shown that the slowly decaying part of a o provides a reduced causal description of the system for t>>/3 -1, provided E is small and the magnetic field is weak in the sense that ojc//3
-
• ,
where coc = ( e / m c ) l B I
is the (position-dependent) cyclotron frequency of the external magnetic field and/3 is the Stokes friction coefficient. We can study the effect of a stronger field by rescaling B with a factor of • -1, B -= • - 1 ~
.
Eqs. (A.4) then become O,a N + N a N - N { I ~ . aN) + • [ ( N + 1)V. aN+ 1 - ( J ~ aN_l} ] = 0,
(A.6)
a s ( y , O) = a N ( Y ) ,
where 12ij =-- •ijk CrPk=-- ( e L / m c ) ( m / 2 k T ) l / 2 • i j k ~
k .
Now we may consider fields for which the magnetic term in eq. (A.6) is of order • 0:
I@1-1, or
(2)C ~
/3 °
The magnetic force on the particle is of the same order of magnitude as tht Stokes force. Eqs. (A.6) are of the same form as eqs. (1) if we identify
234
D. WYCOFF AND N.L. BALAZS
x = a0 ,
y =
(A.7a)
3 ,
(A.7b)
A = 0,
(A.7c)
a N T = (~N1 V" T ,
(A.7d)
C u T = - ~ul {J @ T } ,
(A.7e)
DMNT= ( m + a)~$M+I,N ~7"
T-
~M_I,N{J@
T},
FMNT= M6MN(T- {11. T } ) ,
(A.Yf) (A.7g)
where T is an arbitrary tensor. Notice that for this case x = a 0 is a scalar while the c o m p o n e n t s of y are tensors of differing ranks, as being a symmetric tensor of rank S. F -1 does exist for F given by eq. (A.7g), (F-I)MN-=
M-16MUA-I
,
where A-1 is the inverse of a matrix A defined by A - r-= T - {11- T } , for a symmetric tensor T. If T has c o m p o n e n t s Ta,,2 . . . . , in the frame where 11 is diagonal, then ({1"~ • T } ) a,a2 . . . . ~ = s - l ( (j)a, -}- (-Oa 2 " J r - ' ' ' -~- O.)as) r a la2 . . . . , ,
where % are the eigenvalues of 1~. A-1 is then given in this frame by
l I has eigenvalues +ilcP I and zero, so A -1 is guaranteed to exist. For the special case that T is of rank one the symmetrization brackets are unneeded and we have
A. T= ( I - ~ ) - T ,
235
M E T H O D O F M U L T I P L E T I M E S C A L E S III
where I is the unit matrix, f~ has the special property ~ 3 ---~ -- (~ 2~'~ ,
(A.9)
where ~2_
~.
ff~ ,
SO
A -1
=
(I
-
n ) -1 = E n' ---t + (1 + , 2 ) - 1 [ n + n 2 ] .
When T is of rank two or higher the more general form of eq. (A.8) or its equivalent in a non-diagonal frame, must be used. Using eqs. (A.7) in eq. (7) we find that the matrix f satisfies Nf N
-
N(I~ "fN} = ¢[fNV" fl + {J®fN-1} -- (N + 1)V. fN+l] + ~SNIJ. (A.10)
Solving eq. (A.10) order by order in ~ we find f(0) = 0 N
f (Nl ) ~. (~NI(I -- l - ~ ) - I . j
{(I - 1~). f~)} = ( 1 / 2 ) S m { J ® [(I - 1~) -1- J ] } . For any tensor T, s satisfies NSN( T - I I . T} = ~N1 V" T + e [ - - N S N _ 1 V" "r + S N + I { J @
T}
+ $1{J~$NT}]
,
(A.11) SO
s N( o=) 0 s2 ) T = (~N1 ~'° s (N' ) { r -
(l -- ,~)-i
.
T,
1~- T} = --SNEV" (I--1~) -1 "V" T
In principle s and f may be found to any order in E from eqs. (A.10) and (A.11), but the complicated form of F -~ will make the solution quite difficult except for special cases.
236
D. WYCOFF AND N.L. BALAZS
If we write a 0 = b + ~ SM'OM , N=1,2,3,...
8N = 'ON -JC-I N b ,
the initial value problem (A.6) is equivalent to the separate initial value problems for the slow and fast variables, 19rb --It- E2V , [] @ (1
+ ~2)-1(O + 02)] .Jb
+ ~(4)
__~0 ,
b(y, O) = ~o - elY. [i + (1 + q~:)-~(O + 02)] • a, + ~ ( 2 ) ,
(A.11a) (A.11b)
and Cg'ON + N'ON - { N i l " ,8N} - e[(N + 1)V. 'ON+I -- { J (~ 'ON-1 }]
q- aNl[--~2{J~)(V • (]-I- (1 + @ 2 ) - ' ( n + 012) • 'Ol)} + ~(E4)] ~- 0, 'ON(Y, O) = aN -- e6Nl[I + (1 + ~ 2 ) - 1 ( O + f12)] "Jao + 0~(e2) •
Though they are rapidly oscillating (with the cyclotron frequency of the external field) the fast variables 'ON are also quickly damped out, so that for r >> 1 (t>> 13-1) the slow part of the density gives a reduced causal description of the system, with eqs. (A. 11) as the reduced initial value problem and aN(y, t)--+e6u~[l+ (1 + @ 2 ) - ' ( O + 0 2 ) ] ' J b ( y ,
r ) + G(e2).
(A.12)
Eq. (A.11a) is thus the Smoluchowski equation for Brownian motion in a strong magnetic field. We see once again that the reduced initial data (A.11b) for this equation differs by terms of order e from the true initial data for the density. For O = 0 eqs. (A.11) reduce to O,b + ~ 2 V ' j b
+
~(E 4) = 0 ,
b(y, O) = a o - , V . al + G(e2),
in agreement with earlier results1'18'19). If 0--+0,
then eqs. (A.11) become
METHOD OF MULTIPLE TIME SCALES III
237
O¢b+ EEv'(I + ,O)'Jb+ ~(,4) = 0, b(y,O) = a o - ' V ' a l
+ ~(2),
which agrees with the results of the multiple time scales expansion of ref. 2. It is interesting to examine the solution of eq. (64a) for some special cases. If the magnetic field is constant and directed along the z-axis,
= (0, 0, ~ ) ,
(A.13)
and there is no external force, eq. (A.11a) becomes
O,b = [DLO~ + DT(O 2 + 0~)lb, where the transverse and longitudinal diffusion coefficients are D T = 1 , 2 ( 1 q_ t~)2)-i ,
DL = 1 2 .
D L has the value it would have in the absence of the field while
DL/D, r = 1 + cO2 > 1, so the transverse variables thermalize more slowly than they do in the absence of a magnetic field. If • is constant and the external force has the Hooke's law form F = -ay,
(A.14)
eq. (A.11a) can be solved exactly. Rather than consider the full exact solution, it is instructive to examine the time evolution of the electric dipole moment of the Brownian particle's charge distribution,
d(r) =- f Yao(y , r) d3y. For ~">> 1 a o can be replaced by b, so
d('r)---> f yb(y, r) d3y = ds(~") . Using eq. (A.11a) and one integration by parts we find d s = ,211 + (1 + cO2)-I(~ .~_ ~ 2 ) ] . f Jb(y, r) d3y
238
D. WYCOFF AND N.L. BALAZS
which for the Hooke's law force becomes
,2( 2 ~o ) [I + (1 + 4 2)
1(~-~+ ~-~2)1 ° ds '
(A.15)
Eq. (A.11b) shows that d~ is given initially by
ds(0) =
f yao(y)d3y+ e[I + ( 1+ 1 ~ 2 ) i ( ~ - ~
_1_ ~-~2)] ,
f al(Y)d3y +
(~,(~.2) ,
(A.16) which is of course in general different from the true initial value of the dipole moment. Eq. (A.15) is easily integrated using the property (A.9) of ~ . If • is given by eq. (A.13) we find
e axt cos tOTt ds(t) = ~ e -ax' sin ~oxt \ 0
0° ) e -av' cos O-~rt 0
• ds(0),
(A.17)
e -aLt
where
AT=
0) ~-
2
(/)2) (1 +
1__
0")2/~ j~ 2 _~. (./)2 ,
~L
2
60 = -"~- ,
2
(.Oc0) 60T = ~/~T -- j~ 2 ..{_ 60 ~ •
We have returned to the dimensional time, t =/3-1r, used the fact that if F is given by eq. (A.14) then a is related to the usual harmonic frequency by
ot = mo)2L 2 and identified
We see from eq. (A.17) that the component of d along the magnetic field decays with the rate it would have in the absence of the field from an initial value d3(0 ) = f Y3ao(y) d3y + E f (a,(y)) 3 d3y, which, although it is different from its true initial value, is the same as it would be in the absence of the magnetic field. The decay of the transverse components of d is affected by the field in two ways: (a) AT is smaller than it would be
M E T H O D O F M U L T I P L E T I M E SCALES III
239
in the absence of the magnetic field, so the decay is slower, and (b) the transverse components of d rotate around the z-axis with the frequency tar as they decay. In addition the initial data (A.16) for the transverse components of d differs by terms of order ~ from the value it would have in the absence of the magnetic field. If there are n non-interacting Brownian particles per unit volume, the total currrent density produced is al(y, 7") d3y.
j = n e ( v ) = ne
If the external force is due to a constant electric field E, F=eE,
then eq. (A.12) shows, for a constant magnetic field, that j has the stationary value j=o"E,
where the conductivity tensor or is given by ne 2
or = mfl [I + (1 + (I)2)-1([I + 02)].
(A.18)
If the magnetic field is directed along the z-axis, eq. (A.18) becomes 1
0
09c
or=~--~
1+
fl 0
1
0to 2 .
(A.19)
0
The conductivity (A.19) displays both magneto-resistance and Hall effect terms. Eq. (A.11a) may also be written in two other convenient forms: O,b + e 2 V ' [ J + (l + q b 2 ) - ~ ( - ~ x J + ~ x ( ~ x g ) ] b = O ,
(A.20)
O.~b + e2V- [(1 + ~ 2 ) - ~ ( j _ ~ x J + @ ( ~ . J ) ) l b = 0 .
(A.21)
or
240
D. WYCOFF AND N.L. BALAZS
Using any of the forms (A.11a), (A.20) or (A.21) it can be shown that eq. (Alla) has an H-theorem. Define
f
=- b(y, r) ln[eU(Y)/krb(y, r)] d3y.
"F"(r)
" F " is like a free energy for the coordinates y:
kT"F"= "E" - kT"S", "E"
= f U(y)b(y, r) day,
"S" =
- f b(y, r)ln[b(y, r)] day.
For b determined by eq. (A.11a) we find t"
e-U(y)/kT
" P " ( r ) = - 2 ~ 2 J - -b
(1 + q,2)-l[IJbl2 + 1~ "Jbl2].
(A.22)
We see that
"k"(~) ~<0, "P"(r) = 0
iffJb(y, r) = O,
so all solutions of eq. (14.11a) go to the equilibrium solution
e-V(y)/krd3y
bequilibrium(Y ) =
for any initial data
e-V(y)/~T
b(y, 0). If we write
Jb(y, r) = JL + J T , where JL is directed along the magnetic field and JT is orthogonal to it, eq. (A.22) becomes "/~"'(7) = - - 2 ~ 2
b(y, r) [J2L+ f e-U(y)/kT
(1
+ ~)2) 1j2].
Thus the approach to "parallel equilibrium", Jt=0,
METHOD OF MULTIPLE TIME SCALES III
241
is faster than the approach to "transverse equilibrium",
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
D. Wycoff and N.L. Balazs, Physica 146A (1987) 175, this volume. D. Wycoff and N.L. Balazs, Physica 146A (1987) 201, this volume. E.A. Frieman, J. Math. Phys. 4 (1963) 410. G. Sandri, Ann. Phys. 24 (1963) 332. A.H. Nayfeh, Perturbation Methods (Wiley, New York, 1973) chap. 6. H.A. Kramers, Physica 7 (1940) 284. S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. M.V. Smoluchowski, Ann. Physik 48 (1915) 1103. U. Geigenmuller, U.M. Titulaer and B.U. Felderhof, Physica lI9A (1983) 41. U. Geigenmuller, U.M. Titulaer and B.U. Felderhof, Physica llgA (1983) 53. U. Geigenmuller, B.U. Felderhof and U.M. Titulaer, Physica 120A (1983) 635. L. Onsager, Phys. Rev. 37 (1931) 405. L. Onsager, Phys. Rev. 38 (1931) 2265. H.G.B. Casimir, Rev. Mod. Phys, 17 (1945) 343. N.N. Bogoliubov, Studies in Statistical Mechanics, vol. 1, de Boer, ed. (North-Holland, Amsterdam, 1952) p. 1. G.E. Uhlenbeck, Boulder Lectures, (1951). S.T. Choh and G.E. Uhlenbeck, The Kinetic Theory of Dense Gases, Univ. of Michigan (1958). G. Wilemski, J. Stat. Phys. 14(1976) 2. U.M. Titulaer, Physica 91A (1978) 321.