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On the extraction of slow motion
Physica 137A (1986) 477-501 North-Holland, Amsterdam
ON THE EXTRACTION OF SLOW MOTION II. NON-LINEAR SYSTEMS
J.A.M. JANSSEN Institute for Theoretical Physics, Princetonplein 5, P. 0. Box 80.006, 3.508 TA Vtrecht, The Netherlands
Received 20 February 1986
There is no general method which tells one how to eliminate fast variables in case of deterministic non-linear equations of motion. We consider the corresponding Liouville equation and search for a suitable projection operator. This approach has two appealing features: first, the problem formally reduces to the linear case and secondly the projection operator is fully determined by the structure of the equations and does not appear as an ad hoc choice. The formalism is applied to so-called rapid phase systems, systems for which a guiding center can be defined. The equations of motion for the guiding center are calculated to second order in the ratio of the two time scales.
1. Introduction In an earlier communication’) we studied the problem of the elimination of fast variables in systems where two separate time scales are present. We restricted ourselves to the case of linear equations of motion and formulated a method which makes use of linear transformations in order to extract the slow part of the motion. In this paper we turn to the general case of non-linear systems. In the context of non-linear problems with two time scales one is faced with the following set of equations: jr=gr(y,z;e),
r=l,...,
R;
v=l,...,N.
(1) The parameter E denotes the ratio of the two time scales, which are assumed to be widely different. Thus E is small and will serve as expansion parameter. The functions g, and h, are assumed to be analytic in E and to be of zeroth order in E. 0378-4371 i86f $03.50 @J Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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The variables y and z are commonly called slow and fast variables. This is because their time derivatives are small (order e”) and large (order 6’) respectively. In general eqs. (1) describe a very complex behavior containing both slowly and fast evolving processes. Elimination of fast variables means that one can find a reduced description in terms of the slow processes only. Our method consists in reducing the non-linear to the linear case by means of a projection method, which is analogous to the one Gardiner used for stochastic systems4’5).
2. Reduction to the linear case We consider a number of systems, each evolving according to (1) but with different initial conditions. This ensemble may be described by a density p( y, z), the evolution of which is given by the Liouville equation
The point of this algebraic transformation of the equations of motion is that (2) is a linear equation. However, one also has to pay a price: instead of finite-dimensional vectors y, and z, one now has to deal with an infinitedimensional function space of densities p( y, z). The formal structure of eq. (2) is given by
P=(iL,-tL,)p. The evolution operator in (3) consists of two parts: (l/e),%, , which describes the fast processes, and a remaining part L, associated with the slow motion. A similar type of equation for the case of stochastic systems was studied by Gardiner4). In the following section we will adapt the projection method for eliminating fast processes so that it may be applied to the determini.stic systems we consider. The slow and fast part of the density p may be separated if a suitable projection operator can be found. This projection operator, denoted by P, is subject to the following four conditions*: i) P2 = P. * Contrary
to Gardiner
be true for dissipative
we do not require systems.
that P&P
= 0 and P = kir exp(.L,t),
which would only
ON THE EXTRACTION OF SLOW MOTION II
479
As a consequence the density p can be decomposed in two parts: p = Pp + Qp, where Q = 1 - P. ii) PL, = 0. The time derivative of Pp is of order lo if this condition is satisfied. Pp will therefore be called the slow part of the density. iii) L, is invertible in the null space of P. This condition will be necessary for the algebra. The inverse will be denoted by L;‘. It is understood that it can only be defined for elements f that obey Pf = 0. In order to achieve the uniqueness of L;’ it is required that PL,‘f = 0. iv) LIP = 0. This fourth condition, which is not strictly necessary to achieve a separation in slow and fast parts, is invoked in order to determine P uniquely3). With the aid of the projection operator we will now rewrite eq. (3) in terms of the two parts Pp and Qp. Let P respectively Q act on (3) and use the properties (i), (ii) and (iv). One then finds the following two equations: PP
= PL,P(Pp)
+ PL,Q( QP) 7
(4) (ip = $QP)
+ QkQ(Qp>
+ QUV’P).
Now one clearly notices that there is a difference in the order of magnitude of the two time derivatives. Eqs. (4) are of the same general form as the equations );=Ay+Bz,
i=!Fz+Dz+Cy, E
which were studied in ref. by Pp andQp, which are Instead of matrices A , . With these substitutions projection operator can previously studied.
(5)
1. The finite-dimensional vectors y and z are replaced elements of an infinite-dimensional function space. . . , F there now appear operators PL,P , . . . , L,. we have reduced (under the assumption that a be found) non-linear systems to the linear case
3. The extraction of slow motion The next question concerns the elimination of the fast part of the motion or in other words the derivation of a reduced equation valid on the slow time scale. The quantity Pp is only slow in the sense that its time derivative is of
480
J.A.M.
JANSSEN
order E’. This property by no means excludes the possibility that Pp contains terms of order E that are fast varying. Hence one may write Pp = r(t) + dqtk)
.
(6)
The fast varying terms, denoted by 0, may be eliminated by time smoothing: averaging over time intervals that are large compared to E and small compared to 1. If time smoothing is denoted by an overbar we have -
Pp=lr.
(7)
The slowly varying quantity rr thus equals the time smoothed part of the projected part of the density. According to (6) the motion of Pp is guided by 7~, Pp differs from r only by a small but rapidly varying term. Therefore rr will be called the guiding part of the density. It can be shown that the guiding part obeys a reduced equation of the following type:
in which Lred is expressed in terms of P, L, and L,. There is, however, no guarantee that Lred is an evolution operator of Liouvillian type, as were L, and L,. The property of being a Liouville operator is necessary in order to establish the connection with the original variables y and z. If there are no characteristics associated with Lrcd, the relation with the original level of description becomes questionable. In order to avoid this difficulty we will consider the evolution of another slowly varying quantity, u, which will be called the “sliding” part of the density. The quantity u and its corresponding fast part 2 are related to Pp and Qp according to the following near-identity transformation:
PP = (QP)
1+ CT,,
(
ET21
eT12 1+ eTz2
(9)
We require that in the new quantities u and 2 fast and slow motion decouple from one another,
(10)
ON THE EXTRACTION OF SLOW MOTION II
It has been shown in ref. 1 that this requirement equations: L, +
lTllLs
= PL,P +
lPL,PT,,
ET,,& = Q&P + lQL$‘Ttt T,,Lf =
lPL,PT,,
+
lPL,QT,,
leads to the following
,
(11)
+ L, T,, + EQL~QT~~ ,
+ PL,Q + EPL,QT,,
L, + cTz2Lf = L, + cL,T,,
481
(12)
,
+ eQL2Q +
(13)
l2QLzQTz2
+ E~QW’T,,
.
(14)
These equations fix only L,, L,, T12 and T,,. Here T,, and T22 are left undetermined; they represent the obvious freedom to transform within the slow and fast subspace respectively. Up to second order one finds for the slow evolution operator (ref. 1; eqs. (23), (25) and (27)) L, = PL,P - ePL,QL;‘QL,P
l[PL,P,
+
+ E~PL~QL;~QL~QL;‘QL~P -
l2Ti;)[PL2P,
Ti;‘] -
+
Ti;‘] - E~PL~QL;~QL~PPL~P
l2[PL2P,
l2[PL2QLi1QL2P,
If it is possible to choose T,, in such a way that the type, there are characterics, which describe a motion. This is the slow motion that we set out comparison of (6) and (9) shows that sliding and resulting L, will be denoted by Lred, L red
=
PL,P - cPL,QL;‘QL,P
T’,;‘] T’,;‘] + O(e3).
(15)
resulting L, is of Liouvillian reduced deterministic slow to find. If T,, is set zero, guiding parts coincide. The
- E~PL,QL;~QL~PPL~P
+ E~PL~QL;~QL~QL;‘QL~P
+ S(e3).
(16)
In general Lred will not be a Liouville operator, which obscures the connection with (1). On the other hand it describes the evolution of the guiding part, which may be an interesting quantity by itself. Especially if the projection operator is such that it integrates out the fast variable z, the guiding part is nothing but the (time smoothed) marginal density of slow variables y. Up to this point the discussion applies to a general non-linear system whose evolution is given by (1). In order to get explicit results and to illustrate some points we will now turn to a special subclass of (1).
482
J.A.M. JANSSEN
4. Rapid phase systems Suppose that eqs. (1) have the following special form:
Y, = G,(Y) + g,(y,
,
i =
r = 1,. . . , R ,
+z(y) + h( y, 2) .
(17)
The functions g,(y, z) and h(y, z) are assumed to be 2n-periodic in z. Furthermore, it is required that their average, denoted by angular brackets, is zero,
(p,)=&rg,(y,z)dz=O,
(h)=+-?z(y,z)dz=O.
0
(18)
0
The function H is assumed to have a positive lower bound independent of E, implying that the variable z gives rise to rapidly oscillating terms in g,( y, z) and h(y, z). We will call systems the evolution of which is described by (17) “rapid phase systems”. The motion of gyroscopes and the motion of charged particles in inhomogeneous strong magnetic fields is governed by eqs. (17). A characteristic feature of rapid phase systems is that the solution of (17) perform small but rapid oscillations around a slowly varying quantity, the so-called “guiding center”‘). The guiding center itself is not a particular solution of (17) but is defined as the time smoothed part of the actual solutions of (17). If we denote the guiding center by Y and the rapid oscillations by 71 this may be summarized into y, = y, + q
9
Y, =
yr.
(19)
It will be our aim to deduce the equations guiding center,
I’, = F?'(Y)
+ EFj"(Y)
+ 2p(Y)
that govern the motion of the
+ ....
(20)
The slow motion of the guiding center according to (20) comprises the desired reduced description of the rapid phase system (17). The zeroth order term Fp’ is well known, we will be particularly interested in the higher order terms Ft’) and Fy’. A straight forward way to derive (20) is to make an ansatz like (19) about the solutions of (17)) substitute this ansatz into the equations and separate slow and fast terms (Van Kampen3)). Alternatively one may transform the variables and demand that in the new variables the fast variable has no influence on the
ON THE EXTRACTION OF SLOW MOTION II
483
slow ones (Bogoljubov and Mitropolsky*)). Both methods are conceptually transparent but one has to make rather formidable calculations in order to find F(2) r . As contrasted with these two methods we will follow the indirect approach based on the projection operator and derive the reduced equation for the guiding part. The equations describing its characteristics will then be identified with (20). In order to avoid too complicated expressions we will study a simplified version of (17) in which only one slow variable is present.
i=+(y).
j=G(y)+g(y,z),
(21)
The general case will be treated in appendix B. The evolution operator of the Liouville equation associated with (3) consists of a fast and a slow part given by
L, = -H(Y);, As projection
L2
operator
=
--$(Y)
+ g(yt
z)l.
(22)
we define
277
Pf(y,r)=&]
f(r,z’)dz’=(f).
(23)
0
The operator P takes the average of a function over one period. One easily checks that it fullfills the requirements (i) till (iv). The null space of P consists of those functions that have zero average. The inverse of L, in this space, L;‘, will be denoted by -H-‘J’ dz’, in which the integration constant is chosen in such a way that the resulting function has zero average, and therefore lies in the null space of P. The fact that this inverse can be given explicitly is the reason why we restrict ourselves to rapid phase systems. The spectrum of QLIQ is purely imaginary, the eigenvalues are given by i&Z, n E Z, II # 0, implying that the class of rapid phase systems in the context of the elimination of fast variables belongs to the third category3). The present approach may be viewed as an alternative way to handle this category. It should be noted that if P is applied to the density one gets, apart from a factor 27r, the marginal density p,,, of the slow variable 297
P,,,(Y) = I p(y, 2’) 0
dz’ .
(24)
484
J.A.M.
JANSSEN
First we will calculate the different orders of L, according to (15)) at a later stage the evolution of &,, will be studied. According to (15) the zeroth order of L, is given by (25) The characteristics are determined by the equation Y = G(Y), which is identical with the zeroth order of the equation of motion of the guiding center” “). Calculations concerning the first and second order of L, are more complicated and will be treated in appendix A. For the first order one finds
Li” = -&A”‘(y)
- [$
G(y), T;;)] .
The function A(‘) is built up by the functions g and H. Its exact expression will be given presently. Without the commutator La” is already of Liouvillian type, we therefore choose T (0) 11 = 0 . In second order in E, second order differential terms appear: Lc2) S = -aA’2’(y) dY
+ ; -$
B(y)-
(27)
Before giving the expressions for A (*) , Ac2) and B, we first have to introduce some other quantities, Y(Y,
2)= OYMY,
z>> (Y) =o>
T(Y,
z) = Y(Y, z’) dz’ ,
I
(r)=o.
(29)
In terms of y, r, H and G one finds for A(l), Ac2) and B (differentiation respect to the variable y is denoted by a prime)
with
A(‘)=
(28)
-(y’T)H,
Ac2)= ((rT’)‘T)H+ B = (r’)‘G
-2(r2)G’.
(30) (W)G’-
(PT’)G+
((H-‘H’T)‘r)G,
(31) (32)
Inspection of the expression for B shows that an appropriate choice for T’,:’ that causes the commutator to cancel the second order differential term is given by
ON THE EXTRACTION OF SLOW MOTION II
z-g’=
485
&$(T”)( y) .
(33)
The resulting expression for L, is L, = -;[G(y)
lA”‘(y)
+
+
l2Ac2’(y)
+ ;G”(y)(T2)(y),
+
O(e3). (34)
Up to second order the guiding center thus obeys
i’=
G(Y)
+l A"'(Y)
+ l2Ac2)(Y) +
fe2G"(Y)(l-')(Y)
+ C7(e3).
(35)
The same result is found by the other two methods we mentioned2’3).
5. The connection
between u and the guiding center
In this section we will show that the sliding part (+ indeed describes an ensemble of guiding centers. First we recall the connection between solutions of (21) and the corresponding solutions of the Liouville equation defined by (22). Let y(l; y,, zO) and z(t; y,, zO) denote the solutions of (21) with initial conditions y(0) = y, and z(0) = zO. Let p( y, z, t; y,, z,,) be the corresponding solution of the Liouville equation with initial delta distribution p(O) = 8(z zO)S( y - y,,). The relation between these two solutions is given by P(Y,
z, c Y,, zo)= S(Y - y(t; Y,, zo)P(z - z(t; Yo, zo)).
(36)
From (36) one may conclude YP(Y, 2, t; Y,, 20) dy dz
Y(C Yo, 20) = =
Time-smoothing
Y(t; Y,,
zo)
I
Y&Y,
c ~0, zo)dy = {y>(t).
(37)
of this equation yields [see (19)] =
I
Y&,(Y,
c Y,,
zo)
dy = {y>(t).
(38)
This equation tells us that the guiding center can be found as the ensemble average over the time-smoothed marginal density, 5,. In order to avoid
486
J.A.M. JANSSEN
confusion with the average over average will be denoted by curly As already mentioned in (16), the relation between Pp and u is Pp = (1+
lT,,)a
After time-smoothing
+ ET&
one period of the variable z, this ensemble brackets. p, is proportional to Pp. Furthermore, (7), given by
.
(39)
this relation reduces to
&i = (1+ ET,,)~.
(40)
_I$= 0, because all eigenvalues of its evolution imaginary. To second order (40) reads Pm(Y)=
[ 1+- ;+$(T’)(Y)]27rfl(Y)
operator
QL,Q
are purely
(41)
-
(41) implies that (42) This equation proves our assertion that the motion of the guiding center is determined by the characteristics of U.
6. The evolution of p,,,(y): diffusion and anti-diffusion We now study what happens if one does not require L, to be a Liouville operator. In particular we will study the evolution of the guiding part of the density, which in this case equals the time smoothed marginal density &,, of the slow variable. As one may expect the derivation of the equations of motion for the guiding center turns out to be more complicated. The reduced evolution operator can be found by setting T,, = 0 in the expression for L,. Putting T,,(l) = 0 in (27), we find up to second order L red = --$G(y)
+
lA”‘(y)
+ E!A'~'(Y)]
+
;ri$(y).
(43)
We define A(y) = G(y) + eA(l)( y) + e2Ac2’(y)
(44)
ON THE EXTRACTION OF SLOW MOTION II
487
and recall that p,(y) = 2nPp( y) and thus conclude that the evolution of the time-smoothed part of the marginal density of the slow variable is given by 1
-&P,(Y) = -;NY~-~(Y~
*a*
+ -2E ay2B(Yk(Y)
Although it has the required formal structure’), eq. (37) is not a genuine Fokker-Planck equation. The reason is that the coefficient B is not necessarily positive. A counter example is provided if one puts H(y) = 1 and g( y, z) = cos z. In this case B, eq. (32)) is a negative number In general the second order differential term contains both diffusive and anti-diffusive effects and it is not a priori clear which mechanism is dominant. Anti-diffusion might be so strong that it causes a collapse of the solutions after a finite time. It will appear that the collapse does not take place if one takes appropriate initial conditions. We already mentioned, (19), that the slow variable oscillates around its time-smoothed part. The time-smoothed part of these fluctuations does not vanish, (y-y)*=E*~#o.
(46)
Heuristically one may say that these fluctuations are order differential term in (45). The initial distribution due to time-smoothing it is smeared out to a certain appropriate initial conditions we solve (21) for small scale 7 = t/e the equations read
dy
z
=
lG(Y)
dz + lg(y, z) 2 z = H(Y).
The initial conditions are y(0) = y, and z(O) = zO. Substitution of perturbation series y = y(O) + my . . . yields in lowest order dy”’ -= d7
o
dz(‘) = H(y’O’) dr
-
Y (O)= constant ,
’
)
described by the second of p,,, is not a delta peak: extent. In order to find times. On the slow time
(47)
+ . . . , z = z(O) + EZ(~) +
(48)
z(O) = H( y(O))7 + constant .
The constants are chosen in such a way that the initial conditions are satisfied in zeroth order,
488
J.A.M.
Y
(0) -
Yo
7
z(O)= H(
JANSSEN
yo)7 + .zo .
(50)
The first order equation reads
(‘)= WY,) + g(yo> If one requires y”‘(0) Y(‘)(T) = ‘Zyob
+ ~0).
WY,)T
(51)
= 0, the solution is given by +
lr(yo,
WY&
+
z,,)-
20).
~r(yo,
(52)
For the density we thus find for small times E Q t 4 1 P(Y, 2, c Y,, zo> = 6 z - WY,): (
- zo 6 Y - Y, - Go H
Yo,H(Yo)~+zo
+ ~r(yo,
>> .
zo>
(53)
From this exact expression (to order E) for the density one may conclude that the initial distribution of 5, is given by &(YA=~~@
( y-yo-G(y,)t+~r(~o,~o)-~r
i ~o>H(y,)~+zo
>> .
(54) This time average of the delta function cannot be evaluated in general, but one may look at first and second moment,
= yo + G(yo)f
- •r(~o,
20).
(55)
The term r( y,, H( yO)tIe + zo) disappears because it is rapidly oscillating. The initial value of the first moment is thus given by {Ywa
= Yo -
lYo7
20).
(56)
For the second moment one finds
11 2
{Y’>
= [ Y, + Go
=
[Y, + Go
- cT(yo, - E~(Y,,
20) + or ( Y,, WY,) zo)l” + E~(O(Y,)
i + zo
3
(57)
ON THE EXTRACTION OF SLOW MOTION II
489
2
where we have used the equality r (y,, H( y,,)tIe + zO) = ( r2)( y,,). From (55) and (57) follows the initial width of P,: {{Y2H(0)
=
yJ
{Y'>
-
{Y>'
= E2V2NY,)
(58)
*
In the same way one may find the initial values of higher order moments of i&. For our purpose, however, we do not need them. The result (58) is sufficient to see that in (45) only a &, can occur which has a width at least equal to (58). Under this condition the equation (45) does not give rise to difficulties in spite of the anti-diffusion, as we shall now show.
7. Small fluctuation expansion In this section we will solve eq. (45) subject to the initial condition (54). Advantage will be taken of the fact that the fluctuations are of order E by making a small fluctuation expansion6). First we transform to the new variable 5, which measures the fluctuations Y =
444 + 4 .
(59)
The function 4, which depends explicitly on time, will be determined during the calculation. The new density L!( 5) is related to the old density b,,,(y) according to P,(Y, t) = &(4(t)
+ l Z, t) = f n( 590 .
(60)
From (60) follows
a _
1 a17 and -a[
‘dyPm=e
1 an --=E
at
a& at
+=cj. 2 at
(61)
We next
an 1 an . ---E ag c$ = - :;A(+ at
+ &)Il+
f-$B(c$ +l-f)II
.
(62)
Expansion, of the functions A and B around C$yields all
-
at
-
I[$- A(+)] g =-A'(+) 5 E
@I-
f d”(4)
$
[*II + . . .
(63)
J.A.M. JANSSEN
490
In order to cancel terms of order E -I, the function 4 is required
to obey
4 = A(4). Consequently, $5)
;
{ (‘I=
(64) the equations for the first and second moment read + 1l A”(+){52)
= A’(4)(5)
244~){5~~
+ W2)
+ EA”W{~~I
7
+ B(4) + W4){
(65)
S> + W2)
.
(f-9
The equations for the moments are coupled. An approximate solution can be found if one solves (66) in zeroth order and substitutes the result in (65)6). In terms of the variance {{S’}} = { 5’) - { 5)’ the equations become $5) $W%
= A’(9)(5)
+ t
= ~W#J)US~H
lA”WW2
+ f eA”(+)W2H
+ Q(e2) 7
+ B(4) + o(e).
(67) (@9
If one uses the expressions for B and the zeroth order of A, (32) and (44), the equation for the variance can be written as ;{152H
=2G’(MM2H
+ (r’)‘(NG(+)
-2(r2)(+)G’(4)
+ o(e). (69)
Because I$ = G(4) + B(E) a particular
solution of (69) is given by
W”H(t>= (r’)w))~
(70)
The solution of the homogeneous W’)>
part of (69) is
= aG2(+).
(71)
The constant (Y is arbitrary. solution of (69) &E’)> = V2)(+)
Combining
(70) and (71) we find as general
+ QG’(+).
In order to obey (58) the constant
(72) a has to be positive,
implying that the
491
ON THE EXTRACTION OF SLOW MOTION II
width remains positive for all times. If we had not restricted initial conditions by (58), a collapse might indeed occur in case of a negative (Y. The guiding center can be found if one chooses the minimum width solution corresponding to (Y= 0. Positive (Ywould correspond to an ensemble of guiding centers. Substitution of (72) with (Y= 0 in (67) gives
${5}=A'(9)(5) +1 d’(4){0*
+ t lA”(#d(r*)(+) + @*).
(73)
According to (38) and (59) the relation between the average fluctuation the guiding center is given by y=
{Y> =
4 + E(5).
and
(74)
From (73) and (64) we may then derive
E’= A(4) + lA'(+){
5) + +*A”@){
6)’ +
$e*A"(@(
I-*)@) + 0(c3) . (75)
Replacing the argument
I’=
A(Y)
+
4 by Y we find to the same order
+ 0(e3).
$e2A"(Y)(l-*)(Y)
(76)
To the same order this equation is equivalent with (35). As the solution of (21) is of the form (19) one may conclude (77) Expansion b,(y)
to second order around y - Y yields = S(y - Y) -
As { = 0 comparison
ljS’(y
-
Y) + &%yy
- Y) .
with (41) shows that
7’(t)= (~“HW)). Inspection of (72) leads one to the conclusion equal to these time fluctuations, CC’>> = 7 .
(78)
(79) that the fluctuations
of 6 are
030)
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J.A.M. JANSSEN
8. Discussion
In the first part of this article we develop an abstract formalism by which the elimination of fast variables can be performed in the case of non-linear systems. Two steps are essential: first, the transformation of the equations of motion to the corresponding Liouville equation and secondly, the introduction of an appropriate projection operator by which the problem formally can be reduced to the linear case. In many cases, when projection operator techniques are used, the projected part of the density contains all relevant information of the system. In fact this is the way in which the projection operator is often defined. In the context of our two time scale systems we find that there is no freedom to define the projection operator, it is fully determined by the structure of the equations. Furthermore, for our purpose, the projected part of the density is not the most relevant object as it is contaminated with fast terms. It is the so-called sliding part of the density which is best suited to describe the reduced (slow) motion. It can be obtained by making a transformation which decouples fast and slow motion. The usefulness of the sliding part is demonstrated in the second part, where we illustrate our formalism in the case of rapid phase systems. In spite of the fact that our approach follows and indirect route to derive the equations that govern the motion of the guiding center, the calculations seem to be less formidable compared to straight forward methods2’3). In the third part we investigate how one should pursue calculations to derive the equation for the guiding center in case one does not introduce the sliding part. The main points are 1) The evolution of the time-smoothed marginal density p,(y) is given by (45), which, although it has the required structure, is not a genuine FokkerPlanck equation. 2) The average {y} is equal to the guiding center Y. 3) The ensemble fluctuations {(y - {y .
Acknowledgements
The author wishes to thank Professor N.G. van Kampen for stimulating discussions and critical reading of the manuscript. This investigation is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)” which is financially supported by the “Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (Z.W.O.)“.
ON THE EXTRACTION
OF SLOW MOTION II
493
Appendix A we will give the
that
needed
eqs. (26) till
(32). that
parts of
Liouville operator
The expression for the slow evolution operator,
which
are given
need to evaluate,
and calculate the
L:"=PL,P=y&G(Y),
(A.3
where we used (g) = 0. Next we calculate some quantities which are needed in order to evaluate the first order, L;‘QL,p=
L H
a
ey
64.3)
g(y,z’)dz’=
64)
-
pL,L;‘QL,p=
rH + $y$rH l2ayGH _!t ay > c
>
=-a; g $ (r)H++-&-$rH) =$
ww--$(($)T)H.
(A-5)
as (r) =0. At this point it is convenient to make some remarks. The first remark concerns the notation: in order to reduce the writing derivatives with respect to the variable y will be denoted by a prime. A superscript comma will denote a differentiation with respect to z. The second remark is about averaging. Taking the average is nothing but an integration over a period of the angle variable z. Therefore we can by partial integration deduce an identity, which will be very useful,
494
J.A.M. JANSSEN
(A’B)
=A
r dz tj+ (y, z)B(y,
z>
0
=
&
[A(y,
z)B(y,
z)lp-
&
jdz
A(y,z);
B(y,z).
(A.6)
0
Due to the periodicity of the function A and B one finds the following identity: (A’B)
= - (AB’)
64.7)
The definitions of y and r, (28) and (29), imply ( yr> = (r’r)
= $ ((r*)‘) = 0 .
64.8)
Therefore the second order derivative in (A.5) vanishes and one finds for the first order L(l) 5
=
-a ay
[ $ G,
WQH-
T;‘?]
(A.9)
Thus (26) and (30) are justified. We proceed with the second order,
&L,L;‘QL,,P=
_a G d rH ay H ay
--$y$TH+; (A.lO)
(L,‘QL,)*P
= ;
$
;
-& H j dz’ r
+H-(,$rH)},
PL,(L;‘QL,)*P
=
-( -&y -$ ; $
Hi
(A.ll)
dz’ r) 0
(A.12) This expression can be simplified if one uses (A.7) and r’ = y,
ON THE EXTRACTION OF SLOW MOTION II
PL,(L;1QL,)2P
= ($
r -$
;
-$ TH) + (-$
r -$ y ;
495
TH) . (A. 13)
There is another term in L,(2) which we need to calculate, L;‘QL,P=
L a Hay
L,‘QL,P=--L
Hr
(A. 14)
’
?H H2 dy
(A. 15)
PL2L;‘QL2P=+-;(y $HjdiT)=--$
i(T-$H+
(A.16) where we again used (A.7). The results (A.13) and (A.16) imply that the second order of L,, (15), is given by
(A.17) Consider the first term in this expression,
(iip$
Y
$ rH ) = -$ (yr’>H- -$
((ryyr)fz- -$ (rryr)fz
+ $ ((r’y)‘r)H. The third order differential vanishes because ( yr”) = 5 ( (r3)‘) the second order differential disappears,
((ryyr+r+r) = ((yr2y)= $ ((P)y=o.
(A. 18) = 0. Similarly
(A. 19)
One is left with (A.20) Next we write out the following two terms in (A.17):
496
J.A.M. JANSSEN
-!$(_(cE)’
rH
- l-II-G +(r*)'G +(i)’ rHG>
+-$(q)‘rH-(;)‘I’HG)
(A.21) With the help of (A.20) and (A.21), (A.17) may be reduced to
5{(r*yG-2(r*)Gy+
p=
+((y)'r)G-(rr')G
$
((my
Gr
+(rfy)fr)~}-[~G,T::)]. (A.22)
Eq. (A.22) proves the correctness
Appendix
of (27), (31) and (32).
B
We will show that the results of section 4 may be generalized more slow variables. Consider the following equations:
Yi = G,(Y) + gi(Y, 2)
i= t
H(y)+h(y,z).
7
to the case of
(B-1) (33.2)
It is our aim to derive the equations of motion for the guiding center using the projection method. The Liouville operators associated with (A.l) and (A.2) are given by
4 = -H(Y)
$3
(J3.3)
497
ON THE EXTRACTION OF SLOW MOTION II
L2= -$h(y,
z) -
The projection P=(
operator
$
‘
[G,(y) + gi(y,z)],
(B-4)
again is given by
(B.5)
).
Differentiation with respect to z is again denoted by a superscript comma, while an upper index i denotes a differentiation with respect to yi. Throughout the summation convention will be used. We start to evaluate the slow evolution operator, given by (15), by calculating the zeroth order, L;” = PL,P=
-(h’)
- (a,G,)
where we used (18). Before we proceed we introduce ‘vi(Y?z, = &)
&(YPZ)P
- (a&
= -diGi,
some useful quantities:
(X)=0,
(B.7)
(r;l) = 0.
(B.8)
Z &(y, z) =
s
~(y, z’) dz’ ,
The first order is calculated in a number of steps. QL,P = -a,h L,‘QL,P
-
aig, )
= hH-’
- PL,L,‘QL,P
+ H-’
(B.9) aicH,
= (a,r,(h
+ ai&H))
(B.lO)
=aj(Yih)
-a,(~f~)H+aiai(~i~)a.
(Bell)
The last term in this expression equals zero, aiaj(yjq)H= Consequently,
+ (y,q)]H=
faiaj[(yjc)
$a,a,((I&)‘)
H=O
(B.12)
we are left with
Li” = a,( yjh) - aj( rjq)
H - [aiGi, z$)]
For obvious reasons we choose TIT’ = 0.
.
(B.13)
498
J.A.M. JANSSEN
For the calculation needed, QL,L;‘QL,P
of the second order a lot more intermediate
= -ajGjHpl
steps are
aiqH - ajGjhHml - a,h2Hm1 - a,hH-’
-ajyj a,&H + (ajyj a,l;lH) - ajyjh + (a,r,h)
,
ai&H
(B.14)
Z (L;‘QL2)2P
= H-’
ajGjH-’
dz’ {a,cH
I
+ h} + (h2 - ( h2))H-2 I
+ hHe2 ai&H - (hH-*
a,rl,H) + H-’
I
dz’ {ajyj aicH
f
+ H-l J dz’ {ajyjh - (ajyjh)}
- (ajyj aiIp)}
Bearing in mind that ((L ;‘QL2)2P)
= 0 the following expression
.
(B.15)
results
I
PL2(L,‘QL2)2P
= -(a,y, -
a,G,H-’
dz’ {ai&H + h})
(akyk(h2 - (h2))H-‘)
- (akyk(hHel
Application
I
a,qH
- (hH-’
a,q)))
1
-
(akyk dz’ {ajyj aiqH - (ajyj a,qH)>>
-
(akYk f dz’ {ajyjh -
(ajyjh)})
(B.16)
.
of (A.7) with r; = yk yields
PL2(L,‘QL2)2P
= (a&
ajGjH-’
- a,( ykh2) H-’ + (a,r,
a,cH)
+ (a&
- (a,y,hH-’
ajyj a,rp)
+ (a,r,
ajGjhH-‘) aicH) a,y,h) .
(B.17)
If one takes the fourth and sixth term in this equation together one finds - a,( y,hH-’
a,l;.H> + a,&.
a,y,h) = a,((y,hH-‘)jqH)
=ak((y&-;yj)h)+ak(yk(hH-‘)jq)H.
- a,(r’,yjh)
(B.18)
ON THE EXTRACTION OF SLOW MOTION II
499
Next we write out the fifth term in (B.17), a,(~kaj~jaj~)H=aiaja,(r’,‘,yj~)H-aja,((~kyj)i~)H - ai a,(T’,yjq)H The third order differentials aiaja,(~rk(l;)‘)H=
vanish,
= -$aj
(B.20)
f aiaja,((
The second order differentials - aj aJ((&rj)‘C
(B.19)
+ a,((r’,rj>i&)H.
+
also vanish,
(‘ixq)lH
a,((r,r,)‘q
+~q + r:yiT;
+ rjy,r,)H
=-z * aja,((T:&T;)‘+(rjcrk)V)H=O.
(B.21)
Only the last term of (B.19) survives. If one writes out the first and second term in (B. 17) and uses (B. 18) till (B.22) it is found that Pz&;1QL,)2P
- aj a,((I”GjH-l)i~H
= ai aj a,(cGjr’) - ai ak(r’,q)Gj
+ a,((r’,G,H-‘)‘q)H
+aja,(r’Gjh)H-1-a,(r’,Gjh)H-‘-a,(ykh2)H-1 + a,((r’,r;
+ a,(y,(hH-‘)$)H
- r’,yj)h)
+ ak ( (riyj)il;I)
H .
(B.22)
Eq. (B.lO) yields z L,2QL2P=-H-2
I
dz’h-H-2a.H
I
I
‘dr’r
z PL,LL2QL2P
= (ajyjH-’
which by partial integration
I
(B.23)
I9 I
dz’ h) + (ajyjH-’
can be reduced to
a,H
dz’ r;l) ,
(B.24)
500
J.A.M.
PL,LL2QLzP
Consequently,
JANSSEN
- aj( qh) He1 - a,( I;.H-’
aiHq)
(B.25)
.
using (B .6)
- PL,L L2QL,PL2P
= -aj( qh) H-’
akGk - aj( ‘;H-’
aiHc)a,Gk
.
(B.26) This expression may be written out and one finds - PL2Li2QL2PL2P=
-ai aj a,(I&)G, + ai aj( 45)&G,
+ aj a,((T;.H-‘)‘HCG,)
- aj a,( qh) H-‘G,
- aj((T;H-l)iH&)kG,
+ aj(qhH-l)kGk.
(B.27)
In order to find L, (2) ( B .22) and (B.27) have to be summed. The third order differentials then cancel, as do some second order ones. One is left with Lz2’ = $ aj ak{(r;rk)jGi
+ a,( (ri,yj)‘q)
- (cr,)Gj H + ak ((riq
- ak(Kh2)Hp1
- (&q)G’,} - r’,y,)h)
+ a,(I”(hH-‘)j)Gj
- a,((r,H-‘)‘H&)‘G,
- [aiGi, I!$)] + ak ( y,(hH-‘)‘T;)
H
+ a,((T’,GjH-l)i~)H (B.28)
.
It can easily be checked that La” can be made a.first order differential operator if one puts
T’,:) = 1 ai a,(cl;)
(B.29)
.
The resulting expression for Li2’ is given by Li2) = a,{-$
(I;I”)Gy
- ( (r’,yj - y$)h) + ((TjGjH-l)kTk)H
+ ((riyj)krk)H + ( y,(hH-')'T;.)
+ (q(hH-‘)‘)Gj H - ( yih2) H-’
- ((&H-l)‘Hq)kG,}.
(B.30)
The characteristics of the operator Lie’ + EL i” + e2L i2’ yield the desired equations for the guiding center to second order in E.
ON THE EXTRACTION
OF SLOW MOTION II
501
References 1) J.A.M. Janssen, Physica 133A (1985) 497. 2) N.N. Bogoljubov and Y.A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations (Hindustan Publ. Corp., Delhi, 1961). 3) N.G. van Kampen, Phys. Rep. 124 (1985). 4) C.W. Gardiner, Phys. Rev. A 29 (1984) 2814. 5) C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer, Berlin 1983) chap. 6. 6) R.F. Rodriguez and N.G. van Kampen, Physica 85A (1976) 347. 7) T.G. Northrop, The Adiabatic Motion of Charged Particles (Interscience, New York, 1963). 8) N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).
ON THE EXTRACTION OF SLOW MOTION II. NON-LINEAR SYSTEMS
J.A.M. JANSSEN Institute for Theoretical Physics, Princetonplein 5, P. 0. Box 80.006, 3.508 TA Vtrecht, The Netherlands
Received 20 February 1986
There is no general method which tells one how to eliminate fast variables in case of deterministic non-linear equations of motion. We consider the corresponding Liouville equation and search for a suitable projection operator. This approach has two appealing features: first, the problem formally reduces to the linear case and secondly the projection operator is fully determined by the structure of the equations and does not appear as an ad hoc choice. The formalism is applied to so-called rapid phase systems, systems for which a guiding center can be defined. The equations of motion for the guiding center are calculated to second order in the ratio of the two time scales.
1. Introduction In an earlier communication’) we studied the problem of the elimination of fast variables in systems where two separate time scales are present. We restricted ourselves to the case of linear equations of motion and formulated a method which makes use of linear transformations in order to extract the slow part of the motion. In this paper we turn to the general case of non-linear systems. In the context of non-linear problems with two time scales one is faced with the following set of equations: jr=gr(y,z;e),
r=l,...,
R;
v=l,...,N.
(1) The parameter E denotes the ratio of the two time scales, which are assumed to be widely different. Thus E is small and will serve as expansion parameter. The functions g, and h, are assumed to be analytic in E and to be of zeroth order in E. 0378-4371 i86f $03.50 @J Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
478
J.A.M.
JANSSEN
The variables y and z are commonly called slow and fast variables. This is because their time derivatives are small (order e”) and large (order 6’) respectively. In general eqs. (1) describe a very complex behavior containing both slowly and fast evolving processes. Elimination of fast variables means that one can find a reduced description in terms of the slow processes only. Our method consists in reducing the non-linear to the linear case by means of a projection method, which is analogous to the one Gardiner used for stochastic systems4’5).
2. Reduction to the linear case We consider a number of systems, each evolving according to (1) but with different initial conditions. This ensemble may be described by a density p( y, z), the evolution of which is given by the Liouville equation
The point of this algebraic transformation of the equations of motion is that (2) is a linear equation. However, one also has to pay a price: instead of finite-dimensional vectors y, and z, one now has to deal with an infinitedimensional function space of densities p( y, z). The formal structure of eq. (2) is given by
P=(iL,-tL,)p. The evolution operator in (3) consists of two parts: (l/e),%, , which describes the fast processes, and a remaining part L, associated with the slow motion. A similar type of equation for the case of stochastic systems was studied by Gardiner4). In the following section we will adapt the projection method for eliminating fast processes so that it may be applied to the determini.stic systems we consider. The slow and fast part of the density p may be separated if a suitable projection operator can be found. This projection operator, denoted by P, is subject to the following four conditions*: i) P2 = P. * Contrary
to Gardiner
be true for dissipative
we do not require systems.
that P&P
= 0 and P = kir exp(.L,t),
which would only
ON THE EXTRACTION OF SLOW MOTION II
479
As a consequence the density p can be decomposed in two parts: p = Pp + Qp, where Q = 1 - P. ii) PL, = 0. The time derivative of Pp is of order lo if this condition is satisfied. Pp will therefore be called the slow part of the density. iii) L, is invertible in the null space of P. This condition will be necessary for the algebra. The inverse will be denoted by L;‘. It is understood that it can only be defined for elements f that obey Pf = 0. In order to achieve the uniqueness of L;’ it is required that PL,‘f = 0. iv) LIP = 0. This fourth condition, which is not strictly necessary to achieve a separation in slow and fast parts, is invoked in order to determine P uniquely3). With the aid of the projection operator we will now rewrite eq. (3) in terms of the two parts Pp and Qp. Let P respectively Q act on (3) and use the properties (i), (ii) and (iv). One then finds the following two equations: PP
= PL,P(Pp)
+ PL,Q( QP) 7
(4) (ip = $QP)
+ QkQ(Qp>
+ QUV’P).
Now one clearly notices that there is a difference in the order of magnitude of the two time derivatives. Eqs. (4) are of the same general form as the equations );=Ay+Bz,
i=!Fz+Dz+Cy, E
which were studied in ref. by Pp andQp, which are Instead of matrices A , . With these substitutions projection operator can previously studied.
(5)
1. The finite-dimensional vectors y and z are replaced elements of an infinite-dimensional function space. . . , F there now appear operators PL,P , . . . , L,. we have reduced (under the assumption that a be found) non-linear systems to the linear case
3. The extraction of slow motion The next question concerns the elimination of the fast part of the motion or in other words the derivation of a reduced equation valid on the slow time scale. The quantity Pp is only slow in the sense that its time derivative is of
480
J.A.M.
JANSSEN
order E’. This property by no means excludes the possibility that Pp contains terms of order E that are fast varying. Hence one may write Pp = r(t) + dqtk)
.
(6)
The fast varying terms, denoted by 0, may be eliminated by time smoothing: averaging over time intervals that are large compared to E and small compared to 1. If time smoothing is denoted by an overbar we have -
Pp=lr.
(7)
The slowly varying quantity rr thus equals the time smoothed part of the projected part of the density. According to (6) the motion of Pp is guided by 7~, Pp differs from r only by a small but rapidly varying term. Therefore rr will be called the guiding part of the density. It can be shown that the guiding part obeys a reduced equation of the following type:
in which Lred is expressed in terms of P, L, and L,. There is, however, no guarantee that Lred is an evolution operator of Liouvillian type, as were L, and L,. The property of being a Liouville operator is necessary in order to establish the connection with the original variables y and z. If there are no characteristics associated with Lrcd, the relation with the original level of description becomes questionable. In order to avoid this difficulty we will consider the evolution of another slowly varying quantity, u, which will be called the “sliding” part of the density. The quantity u and its corresponding fast part 2 are related to Pp and Qp according to the following near-identity transformation:
PP = (QP)
1+ CT,,
(
ET21
eT12 1+ eTz2
(9)
We require that in the new quantities u and 2 fast and slow motion decouple from one another,
(10)
ON THE EXTRACTION OF SLOW MOTION II
It has been shown in ref. 1 that this requirement equations: L, +
lTllLs
= PL,P +
lPL,PT,,
ET,,& = Q&P + lQL$‘Ttt T,,Lf =
lPL,PT,,
+
lPL,QT,,
leads to the following
,
(11)
+ L, T,, + EQL~QT~~ ,
+ PL,Q + EPL,QT,,
L, + cTz2Lf = L, + cL,T,,
481
(12)
,
+ eQL2Q +
(13)
l2QLzQTz2
+ E~QW’T,,
.
(14)
These equations fix only L,, L,, T12 and T,,. Here T,, and T22 are left undetermined; they represent the obvious freedom to transform within the slow and fast subspace respectively. Up to second order one finds for the slow evolution operator (ref. 1; eqs. (23), (25) and (27)) L, = PL,P - ePL,QL;‘QL,P
l[PL,P,
+
+ E~PL~QL;~QL~QL;‘QL~P -
l2Ti;)[PL2P,
Ti;‘] -
+
Ti;‘] - E~PL~QL;~QL~PPL~P
l2[PL2P,
l2[PL2QLi1QL2P,
If it is possible to choose T,, in such a way that the type, there are characterics, which describe a motion. This is the slow motion that we set out comparison of (6) and (9) shows that sliding and resulting L, will be denoted by Lred, L red
=
PL,P - cPL,QL;‘QL,P
T’,;‘] T’,;‘] + O(e3).
(15)
resulting L, is of Liouvillian reduced deterministic slow to find. If T,, is set zero, guiding parts coincide. The
- E~PL,QL;~QL~PPL~P
+ E~PL~QL;~QL~QL;‘QL~P
+ S(e3).
(16)
In general Lred will not be a Liouville operator, which obscures the connection with (1). On the other hand it describes the evolution of the guiding part, which may be an interesting quantity by itself. Especially if the projection operator is such that it integrates out the fast variable z, the guiding part is nothing but the (time smoothed) marginal density of slow variables y. Up to this point the discussion applies to a general non-linear system whose evolution is given by (1). In order to get explicit results and to illustrate some points we will now turn to a special subclass of (1).
482
J.A.M. JANSSEN
4. Rapid phase systems Suppose that eqs. (1) have the following special form:
Y, = G,(Y) + g,(y,
,
i =
r = 1,. . . , R ,
+z(y) + h( y, 2) .
(17)
The functions g,(y, z) and h(y, z) are assumed to be 2n-periodic in z. Furthermore, it is required that their average, denoted by angular brackets, is zero,
(p,)=&rg,(y,z)dz=O,
(h)=+-?z(y,z)dz=O.
0
(18)
0
The function H is assumed to have a positive lower bound independent of E, implying that the variable z gives rise to rapidly oscillating terms in g,( y, z) and h(y, z). We will call systems the evolution of which is described by (17) “rapid phase systems”. The motion of gyroscopes and the motion of charged particles in inhomogeneous strong magnetic fields is governed by eqs. (17). A characteristic feature of rapid phase systems is that the solution of (17) perform small but rapid oscillations around a slowly varying quantity, the so-called “guiding center”‘). The guiding center itself is not a particular solution of (17) but is defined as the time smoothed part of the actual solutions of (17). If we denote the guiding center by Y and the rapid oscillations by 71 this may be summarized into y, = y, + q
9
Y, =
yr.
(19)
It will be our aim to deduce the equations guiding center,
I’, = F?'(Y)
+ EFj"(Y)
+ 2p(Y)
that govern the motion of the
+ ....
(20)
The slow motion of the guiding center according to (20) comprises the desired reduced description of the rapid phase system (17). The zeroth order term Fp’ is well known, we will be particularly interested in the higher order terms Ft’) and Fy’. A straight forward way to derive (20) is to make an ansatz like (19) about the solutions of (17)) substitute this ansatz into the equations and separate slow and fast terms (Van Kampen3)). Alternatively one may transform the variables and demand that in the new variables the fast variable has no influence on the
ON THE EXTRACTION OF SLOW MOTION II
483
slow ones (Bogoljubov and Mitropolsky*)). Both methods are conceptually transparent but one has to make rather formidable calculations in order to find F(2) r . As contrasted with these two methods we will follow the indirect approach based on the projection operator and derive the reduced equation for the guiding part. The equations describing its characteristics will then be identified with (20). In order to avoid too complicated expressions we will study a simplified version of (17) in which only one slow variable is present.
i=+(y).
j=G(y)+g(y,z),
(21)
The general case will be treated in appendix B. The evolution operator of the Liouville equation associated with (3) consists of a fast and a slow part given by
L, = -H(Y);, As projection
L2
operator
=
--$(Y)
+ g(yt
z)l.
(22)
we define
277
Pf(y,r)=&]
f(r,z’)dz’=(f).
(23)
0
The operator P takes the average of a function over one period. One easily checks that it fullfills the requirements (i) till (iv). The null space of P consists of those functions that have zero average. The inverse of L, in this space, L;‘, will be denoted by -H-‘J’ dz’, in which the integration constant is chosen in such a way that the resulting function has zero average, and therefore lies in the null space of P. The fact that this inverse can be given explicitly is the reason why we restrict ourselves to rapid phase systems. The spectrum of QLIQ is purely imaginary, the eigenvalues are given by i&Z, n E Z, II # 0, implying that the class of rapid phase systems in the context of the elimination of fast variables belongs to the third category3). The present approach may be viewed as an alternative way to handle this category. It should be noted that if P is applied to the density one gets, apart from a factor 27r, the marginal density p,,, of the slow variable 297
P,,,(Y) = I p(y, 2’) 0
dz’ .
(24)
484
J.A.M.
JANSSEN
First we will calculate the different orders of L, according to (15)) at a later stage the evolution of &,, will be studied. According to (15) the zeroth order of L, is given by (25) The characteristics are determined by the equation Y = G(Y), which is identical with the zeroth order of the equation of motion of the guiding center” “). Calculations concerning the first and second order of L, are more complicated and will be treated in appendix A. For the first order one finds
Li” = -&A”‘(y)
- [$
G(y), T;;)] .
The function A(‘) is built up by the functions g and H. Its exact expression will be given presently. Without the commutator La” is already of Liouvillian type, we therefore choose T (0) 11 = 0 . In second order in E, second order differential terms appear: Lc2) S = -aA’2’(y) dY
+ ; -$
B(y)-
(27)
Before giving the expressions for A (*) , Ac2) and B, we first have to introduce some other quantities, Y(Y,
2)= OYMY,
z>> (Y) =o>
T(Y,
z) = Y(Y, z’) dz’ ,
I
(r)=o.
(29)
In terms of y, r, H and G one finds for A(l), Ac2) and B (differentiation respect to the variable y is denoted by a prime)
with
A(‘)=
(28)
-(y’T)H,
Ac2)= ((rT’)‘T)H+ B = (r’)‘G
-2(r2)G’.
(30) (W)G’-
(PT’)G+
((H-‘H’T)‘r)G,
(31) (32)
Inspection of the expression for B shows that an appropriate choice for T’,:’ that causes the commutator to cancel the second order differential term is given by
ON THE EXTRACTION OF SLOW MOTION II
z-g’=
485
&$(T”)( y) .
(33)
The resulting expression for L, is L, = -;[G(y)
lA”‘(y)
+
+
l2Ac2’(y)
+ ;G”(y)(T2)(y),
+
O(e3). (34)
Up to second order the guiding center thus obeys
i’=
G(Y)
+l A"'(Y)
+ l2Ac2)(Y) +
fe2G"(Y)(l-')(Y)
+ C7(e3).
(35)
The same result is found by the other two methods we mentioned2’3).
5. The connection
between u and the guiding center
In this section we will show that the sliding part (+ indeed describes an ensemble of guiding centers. First we recall the connection between solutions of (21) and the corresponding solutions of the Liouville equation defined by (22). Let y(l; y,, zO) and z(t; y,, zO) denote the solutions of (21) with initial conditions y(0) = y, and z(0) = zO. Let p( y, z, t; y,, z,,) be the corresponding solution of the Liouville equation with initial delta distribution p(O) = 8(z zO)S( y - y,,). The relation between these two solutions is given by P(Y,
z, c Y,, zo)= S(Y - y(t; Y,, zo)P(z - z(t; Yo, zo)).
(36)
From (36) one may conclude YP(Y, 2, t; Y,, 20) dy dz
Y(C Yo, 20) = =
Time-smoothing
Y(t; Y,,
zo)
I
Y&Y,
c ~0, zo)dy = {y>(t).
(37)
of this equation yields [see (19)] =
I
Y&,(Y,
c Y,,
zo)
dy = {y>(t).
(38)
This equation tells us that the guiding center can be found as the ensemble average over the time-smoothed marginal density, 5,. In order to avoid
486
J.A.M. JANSSEN
confusion with the average over average will be denoted by curly As already mentioned in (16), the relation between Pp and u is Pp = (1+
lT,,)a
After time-smoothing
+ ET&
one period of the variable z, this ensemble brackets. p, is proportional to Pp. Furthermore, (7), given by
.
(39)
this relation reduces to
&i = (1+ ET,,)~.
(40)
_I$= 0, because all eigenvalues of its evolution imaginary. To second order (40) reads Pm(Y)=
[ 1+- ;+$(T’)(Y)]27rfl(Y)
operator
QL,Q
are purely
(41)
-
(41) implies that (42) This equation proves our assertion that the motion of the guiding center is determined by the characteristics of U.
6. The evolution of p,,,(y): diffusion and anti-diffusion We now study what happens if one does not require L, to be a Liouville operator. In particular we will study the evolution of the guiding part of the density, which in this case equals the time smoothed marginal density &,, of the slow variable. As one may expect the derivation of the equations of motion for the guiding center turns out to be more complicated. The reduced evolution operator can be found by setting T,, = 0 in the expression for L,. Putting T,,(l) = 0 in (27), we find up to second order L red = --$G(y)
+
lA”‘(y)
+ E!A'~'(Y)]
+
;ri$(y).
(43)
We define A(y) = G(y) + eA(l)( y) + e2Ac2’(y)
(44)
ON THE EXTRACTION OF SLOW MOTION II
487
and recall that p,(y) = 2nPp( y) and thus conclude that the evolution of the time-smoothed part of the marginal density of the slow variable is given by 1
-&P,(Y) = -;NY~-~(Y~
*a*
+ -2E ay2B(Yk(Y)
Although it has the required formal structure’), eq. (37) is not a genuine Fokker-Planck equation. The reason is that the coefficient B is not necessarily positive. A counter example is provided if one puts H(y) = 1 and g( y, z) = cos z. In this case B, eq. (32)) is a negative number In general the second order differential term contains both diffusive and anti-diffusive effects and it is not a priori clear which mechanism is dominant. Anti-diffusion might be so strong that it causes a collapse of the solutions after a finite time. It will appear that the collapse does not take place if one takes appropriate initial conditions. We already mentioned, (19), that the slow variable oscillates around its time-smoothed part. The time-smoothed part of these fluctuations does not vanish, (y-y)*=E*~#o.
(46)
Heuristically one may say that these fluctuations are order differential term in (45). The initial distribution due to time-smoothing it is smeared out to a certain appropriate initial conditions we solve (21) for small scale 7 = t/e the equations read
dy
z
=
lG(Y)
dz + lg(y, z) 2 z = H(Y).
The initial conditions are y(0) = y, and z(O) = zO. Substitution of perturbation series y = y(O) + my . . . yields in lowest order dy”’ -= d7
o
dz(‘) = H(y’O’) dr
-
Y (O)= constant ,
’
)
described by the second of p,,, is not a delta peak: extent. In order to find times. On the slow time
(47)
+ . . . , z = z(O) + EZ(~) +
(48)
z(O) = H( y(O))7 + constant .
The constants are chosen in such a way that the initial conditions are satisfied in zeroth order,
488
J.A.M.
Y
(0) -
Yo
7
z(O)= H(
JANSSEN
yo)7 + .zo .
(50)
The first order equation reads
(‘)= WY,) + g(yo> If one requires y”‘(0) Y(‘)(T) = ‘Zyob
+ ~0).
WY,)T
(51)
= 0, the solution is given by +
lr(yo,
WY&
+
z,,)-
20).
~r(yo,
(52)
For the density we thus find for small times E Q t 4 1 P(Y, 2, c Y,, zo> = 6 z - WY,): (
- zo 6 Y - Y, - Go H
Yo,H(Yo)~+zo
+ ~r(yo,
>> .
zo>
(53)
From this exact expression (to order E) for the density one may conclude that the initial distribution of 5, is given by &(YA=~~@
( y-yo-G(y,)t+~r(~o,~o)-~r
i ~o>H(y,)~+zo
>> .
(54) This time average of the delta function cannot be evaluated in general, but one may look at first and second moment,
= yo + G(yo)f
- •r(~o,
20).
(55)
The term r( y,, H( yO)tIe + zo) disappears because it is rapidly oscillating. The initial value of the first moment is thus given by {Ywa
= Yo -
lYo7
20).
(56)
For the second moment one finds
11 2
{Y’>
= [ Y, + Go
=
[Y, + Go
- cT(yo, - E~(Y,,
20) + or ( Y,, WY,) zo)l” + E~(O(Y,)
i + zo
3
(57)
ON THE EXTRACTION OF SLOW MOTION II
489
2
where we have used the equality r (y,, H( y,,)tIe + zO) = ( r2)( y,,). From (55) and (57) follows the initial width of P,: {{Y2H(0)
=
yJ
{Y'>
-
{Y>'
= E2V2NY,)
(58)
*
In the same way one may find the initial values of higher order moments of i&. For our purpose, however, we do not need them. The result (58) is sufficient to see that in (45) only a &, can occur which has a width at least equal to (58). Under this condition the equation (45) does not give rise to difficulties in spite of the anti-diffusion, as we shall now show.
7. Small fluctuation expansion In this section we will solve eq. (45) subject to the initial condition (54). Advantage will be taken of the fact that the fluctuations are of order E by making a small fluctuation expansion6). First we transform to the new variable 5, which measures the fluctuations Y =
444 + 4 .
(59)
The function 4, which depends explicitly on time, will be determined during the calculation. The new density L!( 5) is related to the old density b,,,(y) according to P,(Y, t) = &(4(t)
+ l Z, t) = f n( 590 .
(60)
From (60) follows
a _
1 a17 and -a[
‘dyPm=e
1 an --=E
at
a& at
+=cj. 2 at
(61)
We next
an 1 an . ---E ag c$ = - :;A(+ at
+ &)Il+
f-$B(c$ +l-f)II
.
(62)
Expansion, of the functions A and B around C$yields all
-
at
-
I[$- A(+)] g =-A'(+) 5 E
@I-
f d”(4)
$
[*II + . . .
(63)
J.A.M. JANSSEN
490
In order to cancel terms of order E -I, the function 4 is required
to obey
4 = A(4). Consequently, $5)
;
{ (‘I=
(64) the equations for the first and second moment read + 1l A”(+){52)
= A’(4)(5)
244~){5~~
+ W2)
+ EA”W{~~I
7
+ B(4) + W4){
(65)
S> + W2)
.
(f-9
The equations for the moments are coupled. An approximate solution can be found if one solves (66) in zeroth order and substitutes the result in (65)6). In terms of the variance {{S’}} = { 5’) - { 5)’ the equations become $5) $W%
= A’(9)(5)
+ t
= ~W#J)US~H
lA”WW2
+ f eA”(+)W2H
+ Q(e2) 7
+ B(4) + o(e).
(67) (@9
If one uses the expressions for B and the zeroth order of A, (32) and (44), the equation for the variance can be written as ;{152H
=2G’(MM2H
+ (r’)‘(NG(+)
-2(r2)(+)G’(4)
+ o(e). (69)
Because I$ = G(4) + B(E) a particular
solution of (69) is given by
W”H(t>= (r’)w))~
(70)
The solution of the homogeneous W’)>
part of (69) is
= aG2(+).
(71)
The constant (Y is arbitrary. solution of (69) &E’)> = V2)(+)
Combining
(70) and (71) we find as general
+ QG’(+).
In order to obey (58) the constant
(72) a has to be positive,
implying that the
491
ON THE EXTRACTION OF SLOW MOTION II
width remains positive for all times. If we had not restricted initial conditions by (58), a collapse might indeed occur in case of a negative (Y. The guiding center can be found if one chooses the minimum width solution corresponding to (Y= 0. Positive (Ywould correspond to an ensemble of guiding centers. Substitution of (72) with (Y= 0 in (67) gives
${5}=A'(9)(5) +1 d’(4){0*
+ t lA”(#d(r*)(+) + @*).
(73)
According to (38) and (59) the relation between the average fluctuation the guiding center is given by y=
{Y> =
4 + E(5).
and
(74)
From (73) and (64) we may then derive
E’= A(4) + lA'(+){
5) + +*A”@){
6)’ +
$e*A"(@(
I-*)@) + 0(c3) . (75)
Replacing the argument
I’=
A(Y)
+
4 by Y we find to the same order
+ 0(e3).
$e2A"(Y)(l-*)(Y)
(76)
To the same order this equation is equivalent with (35). As the solution of (21) is of the form (19) one may conclude (77) Expansion b,(y)
to second order around y - Y yields = S(y - Y) -
As { = 0 comparison
ljS’(y
-
Y) + &%yy
- Y) .
with (41) shows that
7’(t)= (~“HW)). Inspection of (72) leads one to the conclusion equal to these time fluctuations, CC’>> = 7 .
(78)
(79) that the fluctuations
of 6 are
030)
492
J.A.M. JANSSEN
8. Discussion
In the first part of this article we develop an abstract formalism by which the elimination of fast variables can be performed in the case of non-linear systems. Two steps are essential: first, the transformation of the equations of motion to the corresponding Liouville equation and secondly, the introduction of an appropriate projection operator by which the problem formally can be reduced to the linear case. In many cases, when projection operator techniques are used, the projected part of the density contains all relevant information of the system. In fact this is the way in which the projection operator is often defined. In the context of our two time scale systems we find that there is no freedom to define the projection operator, it is fully determined by the structure of the equations. Furthermore, for our purpose, the projected part of the density is not the most relevant object as it is contaminated with fast terms. It is the so-called sliding part of the density which is best suited to describe the reduced (slow) motion. It can be obtained by making a transformation which decouples fast and slow motion. The usefulness of the sliding part is demonstrated in the second part, where we illustrate our formalism in the case of rapid phase systems. In spite of the fact that our approach follows and indirect route to derive the equations that govern the motion of the guiding center, the calculations seem to be less formidable compared to straight forward methods2’3). In the third part we investigate how one should pursue calculations to derive the equation for the guiding center in case one does not introduce the sliding part. The main points are 1) The evolution of the time-smoothed marginal density p,(y) is given by (45), which, although it has the required structure, is not a genuine FokkerPlanck equation. 2) The average {y} is equal to the guiding center Y. 3) The ensemble fluctuations {(y - {y .
Acknowledgements
The author wishes to thank Professor N.G. van Kampen for stimulating discussions and critical reading of the manuscript. This investigation is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)” which is financially supported by the “Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (Z.W.O.)“.
ON THE EXTRACTION
OF SLOW MOTION II
493
Appendix A we will give the
that
needed
eqs. (26) till
(32). that
parts of
Liouville operator
The expression for the slow evolution operator,
which
are given
need to evaluate,
and calculate the
L:"=PL,P=y&G(Y),
(A.3
where we used (g) = 0. Next we calculate some quantities which are needed in order to evaluate the first order, L;‘QL,p=
L H
a
ey
64.3)
g(y,z’)dz’=
64)
-
pL,L;‘QL,p=
rH + $y$rH l2ayGH _!t ay > c
>
=-a; g $ (r)H++-&-$rH) =$
ww--$(($)T)H.
(A-5)
as (r) =0. At this point it is convenient to make some remarks. The first remark concerns the notation: in order to reduce the writing derivatives with respect to the variable y will be denoted by a prime. A superscript comma will denote a differentiation with respect to z. The second remark is about averaging. Taking the average is nothing but an integration over a period of the angle variable z. Therefore we can by partial integration deduce an identity, which will be very useful,
494
J.A.M. JANSSEN
(A’B)
=A
r dz tj+ (y, z)B(y,
z>
0
=
&
[A(y,
z)B(y,
z)lp-
&
jdz
A(y,z);
B(y,z).
(A.6)
0
Due to the periodicity of the function A and B one finds the following identity: (A’B)
= - (AB’)
64.7)
The definitions of y and r, (28) and (29), imply ( yr> = (r’r)
= $ ((r*)‘) = 0 .
64.8)
Therefore the second order derivative in (A.5) vanishes and one finds for the first order L(l) 5
=
-a ay
[ $ G,
WQH-
T;‘?]
(A.9)
Thus (26) and (30) are justified. We proceed with the second order,
&L,L;‘QL,,P=
_a G d rH ay H ay
--$y$TH+; (A.lO)
(L,‘QL,)*P
= ;
$
;
-& H j dz’ r
+H-(,$rH)},
PL,(L;‘QL,)*P
=
-( -&y -$ ; $
Hi
(A.ll)
dz’ r) 0
(A.12) This expression can be simplified if one uses (A.7) and r’ = y,
ON THE EXTRACTION OF SLOW MOTION II
PL,(L;1QL,)2P
= ($
r -$
;
-$ TH) + (-$
r -$ y ;
495
TH) . (A. 13)
There is another term in L,(2) which we need to calculate, L;‘QL,P=
L a Hay
L,‘QL,P=--L
Hr
(A. 14)
’
?H H2 dy
(A. 15)
PL2L;‘QL2P=+-;(y $HjdiT)=--$
i(T-$H+
(A.16) where we again used (A.7). The results (A.13) and (A.16) imply that the second order of L,, (15), is given by
(A.17) Consider the first term in this expression,
(iip$
Y
$ rH ) = -$ (yr’>H- -$
((ryyr)fz- -$ (rryr)fz
+ $ ((r’y)‘r)H. The third order differential vanishes because ( yr”) = 5 ( (r3)‘) the second order differential disappears,
((ryyr+r+r) = ((yr2y)= $ ((P)y=o.
(A. 18) = 0. Similarly
(A. 19)
One is left with (A.20) Next we write out the following two terms in (A.17):
496
J.A.M. JANSSEN
-!$(_(cE)’
rH
- l-II-G +(r*)'G +(i)’ rHG>
+-$(q)‘rH-(;)‘I’HG)
(A.21) With the help of (A.20) and (A.21), (A.17) may be reduced to
5{(r*yG-2(r*)Gy+
p=
+((y)'r)G-(rr')G
$
((my
Gr
+(rfy)fr)~}-[~G,T::)]. (A.22)
Eq. (A.22) proves the correctness
Appendix
of (27), (31) and (32).
B
We will show that the results of section 4 may be generalized more slow variables. Consider the following equations:
Yi = G,(Y) + gi(Y, 2)
i= t
H(y)+h(y,z).
7
to the case of
(B-1) (33.2)
It is our aim to derive the equations of motion for the guiding center using the projection method. The Liouville operators associated with (A.l) and (A.2) are given by
4 = -H(Y)
$3
(J3.3)
497
ON THE EXTRACTION OF SLOW MOTION II
L2= -$h(y,
z) -
The projection P=(
operator
$
‘
[G,(y) + gi(y,z)],
(B-4)
again is given by
(B.5)
).
Differentiation with respect to z is again denoted by a superscript comma, while an upper index i denotes a differentiation with respect to yi. Throughout the summation convention will be used. We start to evaluate the slow evolution operator, given by (15), by calculating the zeroth order, L;” = PL,P=
-(h’)
- (a,G,)
where we used (18). Before we proceed we introduce ‘vi(Y?z, = &)
&(YPZ)P
- (a&
= -diGi,
some useful quantities:
(X)=0,
(B.7)
(r;l) = 0.
(B.8)
Z &(y, z) =
s
~(y, z’) dz’ ,
The first order is calculated in a number of steps. QL,P = -a,h L,‘QL,P
-
aig, )
= hH-’
- PL,L,‘QL,P
+ H-’
(B.9) aicH,
= (a,r,(h
+ ai&H))
(B.lO)
=aj(Yih)
-a,(~f~)H+aiai(~i~)a.
(Bell)
The last term in this expression equals zero, aiaj(yjq)H= Consequently,
+ (y,q)]H=
faiaj[(yjc)
$a,a,((I&)‘)
H=O
(B.12)
we are left with
Li” = a,( yjh) - aj( rjq)
H - [aiGi, z$)]
For obvious reasons we choose TIT’ = 0.
.
(B.13)
498
J.A.M. JANSSEN
For the calculation needed, QL,L;‘QL,P
of the second order a lot more intermediate
= -ajGjHpl
steps are
aiqH - ajGjhHml - a,h2Hm1 - a,hH-’
-ajyj a,&H + (ajyj a,l;lH) - ajyjh + (a,r,h)
,
ai&H
(B.14)
Z (L;‘QL2)2P
= H-’
ajGjH-’
dz’ {a,cH
I
+ h} + (h2 - ( h2))H-2 I
+ hHe2 ai&H - (hH-*
a,rl,H) + H-’
I
dz’ {ajyj aicH
f
+ H-l J dz’ {ajyjh - (ajyjh)}
- (ajyj aiIp)}
Bearing in mind that ((L ;‘QL2)2P)
= 0 the following expression
.
(B.15)
results
I
PL2(L,‘QL2)2P
= -(a,y, -
a,G,H-’
dz’ {ai&H + h})
(akyk(h2 - (h2))H-‘)
- (akyk(hHel
Application
I
a,qH
- (hH-’
a,q)))
1
-
(akyk dz’ {ajyj aiqH - (ajyj a,qH)>>
-
(akYk f dz’ {ajyjh -
(ajyjh)})
(B.16)
.
of (A.7) with r; = yk yields
PL2(L,‘QL2)2P
= (a&
ajGjH-’
- a,( ykh2) H-’ + (a,r,
a,cH)
+ (a&
- (a,y,hH-’
ajyj a,rp)
+ (a,r,
ajGjhH-‘) aicH) a,y,h) .
(B.17)
If one takes the fourth and sixth term in this equation together one finds - a,( y,hH-’
a,l;.H> + a,&.
a,y,h) = a,((y,hH-‘)jqH)
=ak((y&-;yj)h)+ak(yk(hH-‘)jq)H.
- a,(r’,yjh)
(B.18)
ON THE EXTRACTION OF SLOW MOTION II
499
Next we write out the fifth term in (B.17), a,(~kaj~jaj~)H=aiaja,(r’,‘,yj~)H-aja,((~kyj)i~)H - ai a,(T’,yjq)H The third order differentials aiaja,(~rk(l;)‘)H=
vanish,
= -$aj
(B.20)
f aiaja,((
The second order differentials - aj aJ((&rj)‘C
(B.19)
+ a,((r’,rj>i&)H.
+
also vanish,
(‘ixq)lH
a,((r,r,)‘q
+
+ rjy,r,)H
=-z * aja,((T:&T;)‘+(rjcrk)V)H=O.
(B.21)
Only the last term of (B.19) survives. If one writes out the first and second term in (B. 17) and uses (B. 18) till (B.22) it is found that Pz&;1QL,)2P
- aj a,((I”GjH-l)i~H
= ai aj a,(cGjr’) - ai ak(r’,q)Gj
+ a,((r’,G,H-‘)‘q)H
+aja,(r’Gjh)H-1-a,(r’,Gjh)H-‘-a,(ykh2)H-1 + a,((r’,r;
+ a,(y,(hH-‘)$)H
- r’,yj)h)
+ ak ( (riyj)il;I)
H .
(B.22)
Eq. (B.lO) yields z L,2QL2P=-H-2
I
dz’h-H-2a.H
I
I
‘dr’r
z PL,LL2QL2P
= (ajyjH-’
which by partial integration
I
(B.23)
I9 I
dz’ h) + (ajyjH-’
can be reduced to
a,H
dz’ r;l) ,
(B.24)
500
J.A.M.
PL,LL2QLzP
Consequently,
JANSSEN
- aj( qh) He1 - a,( I;.H-’
aiHq)
(B.25)
.
using (B .6)
- PL,L L2QL,PL2P
= -aj( qh) H-’
akGk - aj( ‘;H-’
aiHc)a,Gk
.
(B.26) This expression may be written out and one finds - PL2Li2QL2PL2P=
-ai aj a,(I&)G, + ai aj( 45)&G,
+ aj a,((T;.H-‘)‘HCG,)
- aj a,( qh) H-‘G,
- aj((T;H-l)iH&)kG,
+ aj(qhH-l)kGk.
(B.27)
In order to find L, (2) ( B .22) and (B.27) have to be summed. The third order differentials then cancel, as do some second order ones. One is left with Lz2’ = $ aj ak{(r;rk)jGi
+ a,( (ri,yj)‘q)
- (cr,)Gj H + ak ((riq
- ak(Kh2)Hp1
- (&q)G’,} - r’,y,)h)
+ a,(I”(hH-‘)j)Gj
- a,((r,H-‘)‘H&)‘G,
- [aiGi, I!$)] + ak ( y,(hH-‘)‘T;)
H
+ a,((T’,GjH-l)i~)H (B.28)
.
It can easily be checked that La” can be made a.first order differential operator if one puts
T’,:) = 1 ai a,(cl;)
(B.29)
.
The resulting expression for Li2’ is given by Li2) = a,{-$
(I;I”)Gy
- ( (r’,yj - y$)h) + ((TjGjH-l)kTk)H
+ ((riyj)krk)H + ( y,(hH-')'T;.)
+ (q(hH-‘)‘)Gj H - ( yih2) H-’
- ((&H-l)‘Hq)kG,}.
(B.30)
The characteristics of the operator Lie’ + EL i” + e2L i2’ yield the desired equations for the guiding center to second order in E.
ON THE EXTRACTION
OF SLOW MOTION II
501
References 1) J.A.M. Janssen, Physica 133A (1985) 497. 2) N.N. Bogoljubov and Y.A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations (Hindustan Publ. Corp., Delhi, 1961). 3) N.G. van Kampen, Phys. Rep. 124 (1985). 4) C.W. Gardiner, Phys. Rev. A 29 (1984) 2814. 5) C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer, Berlin 1983) chap. 6. 6) R.F. Rodriguez and N.G. van Kampen, Physica 85A (1976) 347. 7) T.G. Northrop, The Adiabatic Motion of Charged Particles (Interscience, New York, 1963). 8) N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).