Phase change between separatrix crossings

Physica D 36 (1989) 287-316 North-Holland, Amsterdam

PHASE CHANGE BETWEEN SEPARATRIX C~OSSINGS John R. CARY Department of Astrophysical. Planetary. and Atmospheric Sciences and Department of Physics. Unit,ersiO, of Colorado. Boulder. CO 80309-0391, USA

Rex T. SKODJE Department of Chemistry and Biochemistr),, University of Colorado, Boulder, CO 80309-0215. USA Received 28 May 1988 Revised manuscript received 3 March 1989 Communicated by J.E Marsden

The change of the crossing parameter (essentially the phase) between separatrix crossings is calculated for Hamiltonian systems with one degree of freedom and slow time dependence. This completes the calculation of the map for an arbitrary. sequence of separatrix crossings. The change of the crossing parameter is used to calculate the retrapping probability. It is found flint, even in the limit of infinitely slow time dependence, correlations persist between the separalrix crossings, and corrections to the usual zero-order adiabatic approximation must be included. Comparison with numerical experiments show that the result derived here is accurate for quite large values of the slowness parameter.

1. Introduction The adiabatic approximation eliminates a degree of freedom from Hamiltonian systems. This may lead to an integrabie system, if only a single degree of freedom remains, but iI at least leads to a system that is computationally more tractable because of the removal of the most rapid time scale trom the problem. A shortcoming of uscal adiabatic theory is that it does not apply near separatrices of the rapid degree of freed,3m. The - --:t u- ua rcason is :hat adiabatic .tllg;Ot L_ _.y reqtiires • the pcl of the rapid degree of freedom to be much shorter man all o'her time scales, but this cannot hold on the separatrix where the period of the "'rapid" degree of freedom is infinite. This can be important even though the region near the separatrix where adiabatic theoD is invalid is small, because

in the course of time the separatrix may sweep across large regions of phase space. The consequences of crossing a separatrix in a general Hamiltonian with one degree of freedom and slow time dependence in the adiabatic limit were analyzed [1, 2] recently. (Special cases were considered previously [3, 4].) There it was found that to lowest order in the slowness parameter e, essentially the inverse of the time scale for significant variation of the Hamiltonian, the new value of the adiabatic invariant depends on only the Yd.lLl~)

UI

ill~

~'lK-tlUllb

¢III.,IUbCU

u.y

Lll~

IL,'UK.3 U 1

tltk.

separatrix at '.he pseudo-crossing time ~,. whic]', is the time prediL.tcd for the crossing assuming adiabatic theory works even in the vicinity of fl~e separattix. To ncx I order it was fotmd that the new value of the adiabatic invariant depends a!so on a crossing parameter, which is related to the

0167-2789/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

288

J.R. Cary and R.T. Skodje/ Phase change betweenseparatrix crossings

initial phase of the particle, as well as on three parameters describing the near-separatrix dynamics. This phase dependence is crucial, because it implies that repeated separatrix crossings in, say, a periodic Hamiltonian could lead to chaotic motion. However, to complete the description of repeated separatrix crossings, one must also know how successive crossing parameters are related. This reduces the dynamics of systems with repeated separatrix crossings to a map. The first component of the map is to calculate the new value Jm of the adiabatic invariant given the initial value Ji and the crossing parameter M for the first crossing. 'The second component is to calculate the crossing parameter M' for the second crossing given M and Jm. Then the dynamics for an arbitrary sequence of crossings may be calculated. In this paper the relation between the crossing parameters is calculated, thereby completing the reduction of the dynamics of separatr:,x crossing to a map. (These results were announced previously [51.) Our relation between the crossing parameters for sequential general separatfix crossings involves certain parameters describing th,: near-separatrix dynamics as well as the integral of the first-order corrected frequency between pseudo-crossing times. This second aspect is a ~urprise because, as has been emphasized by Hannay [6], this integral, which is just the first-order corrected phase difference, is not a coordinate-independent concept in standard adiabatic theory unless a closed path in the parameter space of Hamiltonians is traversed. But we prove here that this quantity is coordinate-independent also ~,nen the integration is from one pseudo-crossing ume to the next even when a closed loop in parameter space is not traversed. By reducing the dynamics of an adiabatically varying system with separatrices to a sequence of maps, the rapid variations are eliminated from the ~y~tem (Thi~ dc~e~ nnt eliminate the rapid degree of freedom; it is still present via the map. However, the integration between crossings has be-n

reduced to quadrature.) Hence, this work in combination with the previous research [1] will allow analyses of a large number of such systems. As examples of such systems we mention the motion in slowly varying waves [7], especially in collisionless plasma [8], the trapping and bunching of coasting particles by slowly varying rf fields or by the effective potential of a free-electron laser [9], and separatrix crossing induced, by e.g. tidal effects, in celestial mechanics [10]. We note also that diffusion in phase space due to separatrix crossings occurs in accelerators because of synchrobetatron resonances [11] and in three-dimensional plasma confinement systems [12, 13] due to changes between trapping states. Furthe :more, an analysis of slow separatrix crossing is crucial to understanding the validity of the numerical results obtained from the adiabatic switching method [14-16], for example, as applied to the semiclassical determination of energy levels [17, 18]. Finally, separatrix crossing analysis is quite useful in characterizing dynamics of chemical re~-tions [19]. In addition to the work on the adiabatic invariant chan~e mentioned previoL~sly, there has been other work on systems with slowly vau,ing separatrices but with Hamiltonian periodic ia time. Numerical work of Menyuk [2q] indicated :Lat in the adiabatic limit the region of phase space swept by the separatrix is ergodic. Wiggins [21] showed that chaotic orbits exist in this region by proving the existence of a transversal intersection of the stable and unstable manifolds. Escande [22] gave the relation between crossing parameters for symmetric separatrix crossings. (That is, our general result in eqs. (37) reduces to his f,~r the symmetric case.) Escande was also able to estimate the size of the stable regions in separatrix-swept phase space and show that the size reduces to zero in the adiabatic limit. Finally, we mention the work of Rom-Kedar and Wiggins [23], who discuss transport in adiabatic systems with separatrices in terms of the lobes of the homoclinic tangle. As the two approaches (this work and antecedents [1-5] versus ref. [23]) both descril, adiabatic systems with separatrices, comparisons of the two approaches,

J.R. Ca~. and R. T. Skodje / Phase change between separatrix crossings

specifically concerning what is predicted and how much each reduces the problem from the original primitive equations of the Hamiltonian system, should be performed. This work we leave for future research. To supply the background for this work we begin in the following section with a review of adiabatic invariance theory through first order in the adiabatieity parameter. In section 3 we review the main results of separatrix crossing theory. Section 4 contains our calculation of the relation between crossing parameters for successive separatrix crossings. For completeness, we repeat the results for the change of the adiabatic invariant, so that the entire map may be constructed. In section 5 we apply these results to a symmetric system. We construct the map, and we discuss the retrapping probability. That is, given an initial ensemble of particles uniformly distributed on an adiabatic invariant ring, the retrapping probability is defined to be the fraction that end up in the original lobe of the separatrix after two crossings. In section 6 we compare om analytic results with numerical integrations. We find that the results derived here for the limit e--, 0 are good at surprisingly large values of the slowness parameter ~. We provide a summary and conclusions in the last section. In the first appendix, we present the error analysis for tbis calculation. Finally, we discuss the coordinate invariance of our result in appendix B.

2. Adiabatic invariance theory Adiabatic invariance theoG ~ concerned with the analysis of Hamiltonian systems governed by slowly varying Hamiltonians. In the present work we are concerned with systems wi_'thone degree of freedom and slow time dependence, so that the Hamiltonian has the form H(q, p, ~ - et). The slow time dependence arises by assuming that the characteristic rate of change, e, of the system is small compared with typical orbit frequencies. Results are obtained by introducing an ordering in

289

the parameter e. In section 2.1 we briefly review the usual adiabatic ,_'nvariance theory, which neglects the effects of separatrices. In section 2.2 we discuss the formulas for the adiabatic invariant near the separatrix. The results reviewed here are needed for the calculation of the change of the crossing parameter. 2.1. Review of adiabatic invariance theory A slowly varying Hamiltonian system has an adiabatic invariant [24, 25], I °° = l ( q ,

p, X ~

+ d ~ ( q , p , h ) +e212(q, p , X ) + ....

(])

which is cc,nserved to all orders in the slowness parameter e (the series is typically only asymptotic, not convergent) provided H is infinitely differentiable in time. (Additional discussions of adiabatic invariance theory are presented in refs. [26, 271.) The lowest-order term in the series is commonly taken to be the action

E, X)dq,

x)

(2)

which is the phase-space area enclosed by the contour of the Hamiltonian H(q, p. ~ ) = E. In thi', expression, the function P(q, E, ~) is that obtained by sob,i~,'; for tile momentum given the coordinate and tbe energy, the value of the Hamiltonian, i.e. it satisfies the identity H( q, P( q, E, h), h) = E. The function P( q, E, ~ ) may have the minor inconvenience of multiple branches. More generally, as discussed in ref. [27]. the lowest-order term in the series (1) can be any function of the action. With eq. (2) used for the action, one period in the conjugate angle variable i~l&

d

ll~.t&ll

/..~IV~

*

i14J r

ulAJtff*

The first step of adiabatic invariance theoLv is the transformation to action-angle vz,.riables (L ~) via the generating function

F(q,I,X) = f/o(I.~) d q ' P ( q ' , H ( l , h ) , , k ) .

(3)

J.R. Ca~.'and R.T. Skodje/ Phase change betweenseparatrix crossings

290

[qo(E, x), Po(E, X)]

(the overdot denotes time derivative), with OH ~0(I, X) - -~- ( t , X),

(5b)

is ~-independent through 0(1). A sequence of higher-order adiabatic invariants is obtained [27] by introducing successive transformations to new variables,

• " - E Ek~k

(6a)

k=0 q

and Fig. 1. Contours of a typical Hamiltonian. The curve [q0(E, h), po(E, h)] cutting all of the contours is more heavily drawn.

I "=-- Y'. eklk .

(6b)

k=0

The function qo(l, ~) is the value of the coordinate on a curve [qo(l, h), p0(l, ~.)] that cuts the Hamiltonian contours as in fig. 1. This curve, which is arbitrary except that it must vary slowly, essentially provides an origin for the angle variable, which changes by unity in one circuit of the contour and increases along the direction of the flow. The function (3) is multivalued phase space without further specification, because after one loop of integration it must equal the action (2). To uniquely specify this quantity the convention is made that all such integrals proceed along the flow from the lower limit to the upper limit. However, the new Hamiltonian, which is obtained from the transformation equation

The transformations are chosen to reduce the angle dependence of the new Hamiltonian, nO

K" =- ~, ekKk.

(6C)

k=O

For example, in n th order, the transformation is chosen such that the angle-dependent piece of the new Hamiltonian is of order e"+1, i.e. OK"/O~" = O(E"+I). As a result the nth-order adiabatic invariant is constant through order e", and the rate of change of the n th-order angle is ~independent through order e", i.e. (7a)

x) +

OF

K(~,I,h)=H(I,h)+~.-~(q,I,X),

(4)

is single valued. The derivative 0 F / a ~ is single valued, because the change of F after one loop of integration is the action, which is held constant when taking this derivative. Because the an~e dependence of the new Hamiltortian K is only in the first-order term EOF/'O~, the rate of change of the actton is O(e), and the rate of change of the angle variable, (5a)

with a

"

al" ~ ekK~'(l"' A).

(7b)

k=O

Moreover, because the rate of change of lhe n thorder adiabatic invariant is an oscillatory function, the change of the adiabatic invariant over a time of order ! / e is also ~(e"+x). For present purposes we need the results of only first-order theo~ [1], which are as follows. The first-order correction to the adiabatic invari-

J.R. Ca~. and R.T. Skodje / PhcL~echange between separatrix crossings

ant is given by

show the dependence on 11 and h.) For the choice (3) made here, ene finds

~I~( q, E, h) =½eOE 8)~

x

291

8KI(I I, )l) = -£p0(11, X)ll(q0(E, )[), E, X)

e_dq'-~(q',E,)~)

'dq" }-X OP , q,, , e , x ) .

fqq

To convert this result to a function of q and p, E = H ( q , p, h) must be substituted. Canonical perturbation theory does not change the functional form of the lowest-order term. Hence,

K~(II, X ) = H ( I t ,

h).

_ep(qo(E,X),E,X)

(8)

Oqo

x), (u)

in which one must substitute E = H ( I ~, 2~). These relations are functional as before. The first-order corrected frequency is given by the derivative of (10) with respect to 11.

(9) 2.2. Adiabatic inuariants near the separatrix

It is important to keep in mind that eq. (9) is a relation between functions, not valt, es. That is, the right side is obtained by calculating the function H ( I , h), then replacing I in this function by 11. The first term in eq. (8) is seen to be proportional to e/Z,o(1, h) times an order unity quantity. This, together with the requirement that successive terms of the achabatic invariant series decrease in magnitude, is one source for the statement that adiabatic theory relies on the typical rate of change of the Hamiltonian being small compared with the instantaneous orbit frequency. The first-order correction to the first-order corrected Hamiltonian is given as usual by averaging the correction term of (4) over the lowest-order angle variable at fixed ! and 2~ and then subst;.tuting 11 for L The averaging over angle yields

For the separatrix crossing theory that follows, the form of the adiabatic invariants and the first-order corrections to the Hamiltonian near the separatrix are needed. The near-separatrix adiabatic invariants were derived in ref. [1]. These are briefly reviewed here. Then these results are used to find the form of the first-order correction to the Hamiltonian. Contours of a Hamiltonian with a typical figure-eight separatrix are shown in fig. 2. The separatrix divides phase space into three regions labeled a, b, and c. The phase-space area enclosed by lobe-a (a = a or b) is denoted by Y,~(~), while Y~-- Y~+ Yb denotes the total separatrix enclosed area. For the analysis of near-separatrix part of the trajectory a particular coordinate system is

Oepl , ,\OFt ' I t,h) = e ~ d q ' -~7~ q , ll, ^ ) ~ - ~ q , = e~,o~dq, or

(q')

!

°l

q" d , I "dP , ,, OH OF(q,,)) Xfqo(t,.x, q ['~-~~q ) + 0X 0E

q

_ ~p~'t, ~ 0q0

(10)

(The notation in the last line does not explicitly

Fig. 2. Contom.- ~f a Hamiltonian wtth a typ2cal figure-8 separatrix, which divides pta,:.c ~nace into the three regions labeled a, b, and c. The separatrix is st~o,~.~ as a darker cu~;e.

292

J.R. Cary and R.T. Skodje / Phase change betweenseparatrix crossings

introduced. In this coordinate system the x-point of the separatrix is at the origin, and the directions of the stable and unstable manifolds are fixed to be the lines p = +q. Thus, the Hamiltonian is given by

vertex-v (v --- a, b, u, or l') are given by

H(q,p,X)=½to(p2-q2)+SH(q,p,X).

(Near the separatrix E is a small parameter.) For vertices in the lobes (a = a or b) the deviations from the value on the separatrix are given by

(12)

(This form assumes without loss of generality that H increases in going from the lobes to region-c.) In the work of Cary et al. [1], it was shown that one could choose coordinates such that BH = O(q3, q2p, qp2, p3). More recently, Henrard [28] has shown that the coord'.':,.ztes can be chosen to make 6H of fourth order in the phase-space coordinates. This allows better estimates of the error of the separatrix crossing analysis. Finally, a change of the temporal variable allows one to keep to, the rate at which trajectories near the x-point separate exponentially, fixed. This coordinate system also has the property that E, the value of the Hamiltonian, vanishes at the x-point. Given this coordinate system it is possible to define a vertex of a trajectory as a point where it crosses the q- or p-axis. There are four types of vertices, which are labeled as in fig. 3, depending on which axis is being crossed. In previous separatrix crossing theory [1], as well as here, the values at the vertices of the various phase-dependent quantities, such as the corrections to the adiabatic invariants, are needed. In ref. [1] it was found that the first-order corrected adiabatic invariants at P

ii

Fig. 3. Phase space near the x-point. The laoels a, b, u, and ?', refer to the vertex label for crossing the axsociated q- or p-axis

I~(E, ~.) =

Yo(X)+SIo(E,X ) +¢(IEI3/Z) + O(rE).

81,,(E,

I

X) = ~- 1 + In -~-

+ eg,,.

(13)

(14a)

While for vertices in region-c the deviations from the value on the separatrix are given by

1

I~.

Eb (14b)

and

T). (14c) The conventions introduced here will be used throughout the paper. The index a will always refer to one of the lobes, either a or b, or the associated vertex. The index v may refer to any vertex a, b, u, or ~'. Finally, the indices ~, rl, and ~" will refer to any of the regions a, b, or c. In eqs. (13) and (14) it is seen that the near-separatrix invariants depend on several quantities that describe the system near the separatrix. In addition to the previously mentioned exponentiation rate ¢, the separatrix actions Y~, and their time derivatives 1~, there are the parameters Et~(~) and &(h). The parameters E~ (denoted h,~ in ref. [1]), which essentially ~'. ~.. :. . . ~tJe " the contributions due to difference of the action from ihat of the ~eparatrix due to the pa::t: of the Hamiltonian contour far from the x-poh~t, are calculated as definite inte-

J. IL Cary and R.T. Skodje / Phase change between separatrix crossings

grals in ref [1]. This calculation yields

for a = a or b, while

In IEbl = In [2t~q2al

V~o= 21n I E d ' E I "

o~

+2cof0q'dq ( i}E

(2¢gE + 602q2) x/2 '

(lSa) where qn is the limit, as the separatrix is approached, of the value of the coordinate of the point on the energy contour half of a period from the vertex. This is discussed in more detail in ref. [1]. An analogous expression yields E a. The quantities g,,-l,,x(e=0,

h)

293

(a=aorb)

(15b)

are found by taking the limit of the first-order corrections to the adiabatic invariant in lobes a or b as the x-point is approached along the q-axis. It is shown in ref. [1] that this limit exists. In fact, the quantities g,, appear in separatrix crossing theory in only the combination G=gb--g,,Yb/lra.

(16)

(lSb)

The quantity E c - - ~ E a E b is the geometric mean of E~ and E b. To find the value of the first-order correction to the adiabatic Hamiltonian, it is net~,ssai-y to define the origin of the angle variable or, equivalently, qo(l, ~). Here qo is defined by assuming that P(qo, E, h ) = 0 for lobes a and b, while q0 = 0 and P(qo, E, h) > 0 for region-c. This implies that the lower limit of the integral in eq. (3) is at either vertex-u, vertex-a, or vertex-b. It also implies that the angle variable vanishes at these vertices. This choice, together with eq. (11), also implies that the first-order correctiou to the new Hamiltonian is given by 1 1 Ko,(~,, X) = - ,'~o(V, X)Io1( n(J'~, X), X)

(19a) for a = a or b, while

' ' X) = -"co(I' , x ) 1 o l ( u ( t ' , X }, X). KoI(*,

(~.9b) The last terms in eqs. (14b) and (14c) come from the first-order correction to the adiabatic invariant at vertex-u. That is, ~I~l-- 2-~ ];'bin T

-f'~ln

T

+o(e~). (17)

The result for lea is obtained by changing the sign of eq. (17). The lowest-order frequencies for trajectories in the various regions are given by tl z inverse of the derivative of the action with respect to the energy. Expressed as a function of the energy, the lowestorder frequency is V"o = In I E" J E I

(lSa)

The process of finding the corrected Hamiltonian near the separatrix is completed by inserting eqs. (15) and (17) into eqs. (19). The frequencies v~0 multiplying the action corrections in eqs. (19) vanish logarithmically as the separatrix is approached. This dependence cancels the logarithmic divergence of the correction Iul. Thus, the corrected Hamiltonian is finite even as the value of the a~Aabatic invariant approaches the value Yo on the separatrix. The value of the Hamiitonian on the separatrix,

K ~ - - K~( L , x), is, therefore, given by

~ =0 K,~x

(:Oa)

J.R. Cary and R.T. Skodje / Phase change between separatrix crossings

294

for a = a or b, while =

- ;,).

(20b)

Eqs. (20) show that the fu'st-order corrected Hamiltonian is not continuous across the separatrix. Lack of continuity across the separatrix of the Hamiltonian as a function of the action-angle variables is not surprising. It nearly always occurs, because the action-angle variables themselves are discontinuous across the separatrix. (Indeed, the angle variable is singular at the separatrix.) Lack of continuity does not affect our later calculations, where we require only that K be well defined through first order. We summarize the results of this section as follows. The first two terms of the adiabatic invariant series (6b) are given by the action (2) and the first-order corrections (8). The first two terms of the corrected Hamiltonian as a function of the corrected adiabatic invariant are given by eqs. (9) and (10). The terms of the adiabatic invariant (13) near the separate, at a vertex are given by eqs. .." :r~ ~:~.,:~ -,~ the near-separatrix quar, tities E~ ,naci g, which are given by eqs. (15). Finally, the first-order corrected Hamiltonian is given by eqs. (20).

3. Separatrix crossi~ng theory To calculate the relation between successive crossing parameters, it is first necessary to understand the basic separatrix crossing theory which is reviewed here very briefly. This discussion is also presented in order to introduce the concepts and notation of the separatrix crossing calculation which will be used again here in the calculation of the phase change. As mentioned in tb_~ last section, a separatrix such a! m rig. 2 ¢aiviJ:s phase space into three regions. For an adiabatically varying Hamiltonian, a trajectory initially in any one of the re~ions and away from the separatrix has a well-conserved adiabatic invariant, according to the theory of the

last section. This implies that as the particle moves across the contours of the Hamiltonian, it nevertheless always stays close to that contour containing an amount of phase-space area equal to the initial value of the adiabatic invariant. However, if a particle is in region-c, for example, and the area Y~ enclosed by the separatrix is growing in time, then there eventually comes a time when the trajectory crosses the separatrix. This critical time, after which a contour of that area no longer exists, is called the pseudo-crossing time. More specifically, for a particle initially in region-~ (~ = a, b or c) having first-order corrected adiabatic invariant of value Ji, i.e. Ji = I11~, the pseudo-crossing time tx is defined to be the time at which

Y~( hx = etx) = Ji

(21)

holds, i.e. the area enclosed by the separatrix equals the initial value of the first-order corrected adiabatic invariant of the particle. (The fact that 11 is important and neither higher nor lower order is because the calculation is carried out through ~:~,t order only.) Beyond that time, adiabatic theory does not state what will happen to the trajectory. Indeed it cannot, because there no longer exists a contour of the Hamiltonian containing that amount of area. However, separatrix crossing theory [1] does yield the new value of the adiabatic invariant for a trajectory ultimately in region-~ given that it was initially in region-~. As discussed in ref. [1], three distinct types of transitions can occu,'. The trajectory can leave region-c and enter one of the lobes, it can leave one of the lobes and enter regi,on-c, or it can leave one of the lobes and enter the other. In tiffs last case, the trajecto D, actually crosses the ~nstantanecus separatrix twice in quick succession. However, ~t enters and exits the separatrix region, the region where adiabaticity breaks down, only once, so this is also known as a single separamx crossing. The lowest-order expected result is that the final value of the adiabatic invariant for a particle

295

Jr.R. CaO, and R.T. Skodje / Phase change between separetrix crossings

ultimately in region-,t (~ = a, b, or c) will be close to the area enclosed by the separatrix for the appropriate regaon at the pseudo-crossing time. That is, one expects the final value of the adiabatic invariant ~0 be given by (22) with 8./t,~ small. Separatnx crossing theory shows this to be the case. The deviation is found in ref. [1] to be of order ~ln [El and to depend on the phase of the particle. (We give the explicit expression for the deviation (22) in section 4, eqs. (39), where we group the results for the change of the adiabatic invariant and the phase.) Because this deviation is phase-dependent, it leads to spreading of particles in phase space as discussed in ref. [1]. Separati~, ~rossing is analyzed by the method of matchec, as,pmptotic expansions [29, 31]. Far from the scparatfix, the trajectory is well described by adiabatic invariance theory. Near the separatrix, the approximation that the energy is close to that on the separatrix is made. By convention the separatrLx energy was chosen in the pre~5ous section to vanish, so the near-separatrix approximation is effected by an ordering in E. At intermediate parts of the trajectory, both theories apply, so that matching is possible. The near-separatrix part of the trajectory is analyzed in a particular coordinate system described in the last section. One part of the calculation is to determine the change of the energy, A E,, caused by encircling lobe-a from one vertex to the next. This portion of the trajectory is called a step. N e a r the separatrix it is found that the step changes of the energy are nearly constant and are given by

aEo = - r'; (:,x) -: o( '-in

+ tv(E max { IEil, lEd}),

{ levi, le l }1)

(23a)

in which E i aad Ef are respectively the initial and final values of the energy for this step. Tht. change of the energy due to encircling both lobes is given

by addition, AE e = - I~c(X~) + ¢9(E2In [rain {IEi[, lEvi}I) + tV(emax { lEd, led}).

(23b)

(In fact, eq. (23) breaks down for an exponentially small class of trajectories. This is discussed in detail in ref. Ill.) The critical vertex is defined to be the first, last (or only) vertex of the orbit in region-c. (As fig. 2 of ref. [1] shows, there can be only one such vertex.) The value of the energy at this vertex is denoted by E o. The crossing parameters are defined by

(24)

-e0/ dXx)

for ~ = a, b, or c. As these definitions show, the various crossing parameters describing a single crossing are all related,

= Mj',,( X,) /

Xx).

(25)

Tile a~,c,,~,~ -" .... -' ;ailgcs of the crossing paramcters can be determined from the allowed range of the parameter E 0. As an example we consider the range of M a for a trajectory leaving lobe-a. The minimum value of E 0 is zero, since the critical vertex is on the p-axis and (12) holds. The maximum value of E o for a trajectory leaving lobe-a is - 1:'~, because the energy one step previous, E 0 + I?a, must be negative. Thus, 0 < Ma < 1 for a trajectory leaving lobe-a. More generally, 0 < I n d < 1 for a trajectory entering or leaving region-~. The crossing parameter M~ is positive if the particle is leaving lobe-a or Iobe-b or entering region-c. This crossing parameter is negative if the particle is entering lobe-a or lobe-b or leaving region-c. These facts are summarized in table i. (The parameter Mu in lable I will be introduced in the next section.) _The crossing parameters are important because they de~ermine the final value of the ddl,~ba~ic invarian::; the deviation 8Jr, is a function of M,, which was derived in ref. 11], and is repeated here

296

J.R. Car),and R.T. Skodj,./ Phase change betweenseparatrix crossings

Table I Allowed ranges of crossing parameters for the various possible separatrix crossings

Process

Range of crossingparameters

Leavinglobe-a Enteringlobe-a Enteringregion-c Leavingregion-c

O
in eqs. (39). (The general formulas also depend on other parameters that describe the adiabatic invariant near the separatrix.) Moreover, as shown in ref. [1], the distribution in the crossing parameter is uniform tor an ensemble that initially is uniform in the angle conjugate to the adiabatic invariant. (This corresponds to lowest order to a microcanonical ensemble within ,one of the three distinct phase space regions at energy H(Ji, ~).) This implies that the initial phase of the particle is linearly related to the crossing parameters. From this fact we may deduce the probability of entering a particular region of phase space for the distribution that is uniform in the adiabatic phase. We now turn to this question. Because there are three regions of phase space, for any crossing there is a unique region, or state. Either trajectories arc leaving two of the regions and entering the thrd, or else trajectories are leaving one of the regions and entering the other two. Using the terminology of ref. [13], we call the unique exit state or entrance state, as the case may be, the majority state. For example, when AE a and 8 E b are negative and, hence, so is AEc - h E a + 8 E b, trajectories are leaving region-c and entering regions a and b. In this case c is the majority state. We define oa = AE~, % - AE b. and o~ = -AE¢. It follows from AE¢ -- AE~ + AE b that % + o b + t~c = O.

(26)

Hence, for one of the states ~ ba~ a unique sign. A case by case analysis shows that the majority state ( 4 - m) is that state with the unique sign. Moxeover, eq. (26) implies that the majority state

is that state, m, for which om and, hence, A E m are the largest in magnitude of the o¢ and AE~. We must now consider two cases. The easier case is when trajectories are entering the majority state. This case corresponds to om < 0. By definition it follows that the probability of entering state-m is un;ty, and the probability of entering any other state is zero. Suppose instead the trajectories are leaving the majority state and enterh-B the states/~ and ~1. Then the crossing trajectories have values of E 0 within a range 0 < IE01 < IAE ml of length IAE.d. Of these, those with values of E 0 within a range 0 < IE01 < IAE~I of length IAE~I enter region-~. Since, as mentioned above, the distribution is uniform in the parameter Eo, the probability of entering region-f is given by IAEJAEml = I This probability is less than unity because of the aforementioned fact that AE m is the largest in magnitude of the

aE .

4. Relation between successive crossing parameters

The review of the last section poir, ted out that the process of separatrix crossing is governed by the crossing paras'rioter, which is essentially a phase. Given this parameter and the old value of the adiabatic invariant, one can calculate the new value of the acliab~.tic invariant. If one can also determine the next crossing parameter as a function of the previous crossing parameter and the intermediate value of the adiabatic invariant, one has sufficient information to determine the consequences of a sequence of separatrix crossings. In this section we derive the relation between successive separatrix crossing parameters. The particular process considered is shown in fig. 4..'. trajecto~, initially in lobe-a, enters region-c. It does so with some par:;.cular value of the crossing parameter, which determines the new value, J~, of the adiabatic invadant in region-c according to the theory of refs. [1-4]. At some later time, the lobe shrinks, so the trajectory is forced out of region-c and into one of the lobes. In this section we calculate the lowest-order crossing

J.R. Ca~. and R.T. Skodje / Phase change between separatrix crossings

A)

from Me, which is calculated with the energy, E0, of the last crossing of either the positive (upper) or negative (lower) p-axis: M u - - E o / ~ ¢. Table II shows the relations between the crossing parameters M,, and the M~ (,~ = a or b) for a single separatrix crossing. In this table, the definition

C

8)

Fig. 4. (A) Detrapping from lobe-a. (B) Trapping into lobe-b. The figure is schematic. More precisely, the trajectory remains close to a contour of the Hamiltonian containing a constant amount of area, while the area enclosed by the separatrix changes.

parameter relation. As th2s is the most complicated process, it can easily be specialized to treat the case of a particle entering and then leaving one of the lobe~. W~ ~cleg'.-tt~ the clio, atral~is to appendix A. The necessity of referring to a fixed origin of angle requires us to define flew crossing parameters M~ and M~, which are the crossing parameters for region-c calculated using the energies, E~0 and E~0, at the first or last time that the trajectory crosses the (upper) positive p-axis: M u =- - E u o / ~ " ~. This crossing parameter may differ

for a = a or b has been used. To provide further explanation, we consider the first line of this table, which, as the entry of the first column indicates, applies to trajectories leaving lobe-a and entering region-c, First, A E a > 0, because trajectories are leaving lobe-a. Because the critical vertex is in region-c, E o > 0. Furthermore, E 0 - AE, < 0, because the orbit one step previous was in lobe-a, and E o + AE b > 0, because the lot~e remains in region-c after the next encircling of lobe-b. These three inequalities, upon division by A E, are expressed in the second column o; the f, si ,L,, ,,i ,~,,,: ii. Since :.he ,~rzt vertex in region-c is an upper vertex, E~0 = E,~. Hence, M u = E , , o / A E ~ = E o / A E ~ = ( E o / A E a ) x ( A E J A E ¢ ) , which gives the last entry in this row. The remaining rows of this table are found by similar analyses. As in separatrix crossing theory, close to the separatrix, adiabatic invariance breaks down. However, for tlmes after t N, the time the trajectory is N steps after the first u-vertex for the first crossing, and before t'. s, the time the trajectory is N steps before the last u-vertex for the second crossing, adiabatic theory applies. T.h]s means that

Table II For crossing parameter in the range indicated by the first column the transition indicated by the second column is made. The third column gives the crossing parameter relevant to the new region. This is essentially d~e new phase upon entering f.he new r e , o n

Transiuon

Range of crossing parameter

From From From From From

max (0, - R b / R a) < M~ < 1 max(O, -- R a / R b) < M b < I - I < M u < m i n ( O , - R b) R b / R ~ < M~ < rain(O, R~) O< M~ < rain(l, - R o / R ~ )

lobe-a to region-c Iobe-b to region-c region-c to Iobe-a region-c to lobe-b lobe-e to lobe-/]

297

Crossing p=~-amelcr rcJevant to new region

M . = MbRb + R~ Mb = M J R b M~ = ,% R o / Rt,

J.R. Ca~ and R. 7". Skodje/ Phase change betweenseparatrix crossings

298

the accumulated phase during this intermediate interval (tN, t'_ N) must be an integer. For region-a the angle variable is chosen to vanish at the vertex. Thus, through order unity, the accumulated phase for this trajectory, which has value of the adiabatic invariant equal to Jm in region-c, must satisfy

f t'N~,xc(jm, 3 , ) d t = n,

(27)

tN

in which n must be an integer. (For this integral to be accurate through order unity, the integrand must be calculated through first order because the range of integration is 0(l/E).) In eq. (27) the assumption being made implicitly is that even though N is large, it is still true that t'_ N > t N. We will return to this assumption at the end of this section, where we discuss the error of the complete map.

In these expressions t~ and t" are the first and second pseudo-crossing times, that is Jm = Yc(hO = Y~(X'). Because the integral on the left side of eq. (29) is at fixed Jm, so must be the integrals on the right side, even though the adiabatic theory breaks down in the vicinity of the separatrix. This fact allows the integration variable to be changed to E z = Kt(Jm, k), the value of the first-order corrected Hamiltonian. This is accomplished via differentiating the relation

This relation is taken to be exact in the differentiation, because, as just mentioned, the integrals of eq. (29) are at fixed arm. Since Jm is constant, ~E 1 d E 1= - h - - ~ d l =

- [~'c(h,,) or Y¢(hx)]dl

It is also known that

(31)

Jm = Ic1(E N, tu) = I~(E'_. N, t'_N),

(28)

with an error due to the neglected terms that is 0(e2I¢2). Eqs. (27) and (28) give three relations, which can be used to eliminate tN and t'_ N and then to find E ' N in terms of E N and Jm" The remainder of th~s section is devoted to this analysis. The first step is to rewrite the left side of eq. (27) as follows:

f"~r,2(Jm.X)dt= ~ c ( J m ) tN

for times near the first or second separatrix crossing respectively. With this change of the integration variable, the integral is seen to be given by

XN) f'~(J_.,X)dt= K2(J~,X,)-K2(Jm, f "X~) lx

(32a) and

ft'['~(Jm,h)dt

ftlN~xc(Jm, X ) d t

",'_, t

- /'" ~ ( Jm, h) dt, %" .v

(29)

--- --Klc(Jm'h'x)-K~(Jm'h'-N) ro(.V~) (32b)

The quantity K~(J m, XN) in the above expression must be kept through first order,

where f : ; - l ( J m , X) dt

•'~2(Sm, XN)

l~

----no(Sin, aN) -- ~K2I( Sm, aN )

1 /~", =i = r¢(J~, X) dX

= Hc(S.,, ~ N ) - . . c O ( J m , X~.)

(30)

Xlut(qou(Jm, XN), Hc(Jm,XN).XN).

(33)

J.R. Cary and R.T. Skodje / Phase change between separatrix crossings

because the denominator on the right side of eq. (32b) is ¢(e). The last line in (33) follows from eqs. (19). The right side of eq. (33) is evaluated through use of eq. (28), which implies that

J . = t¢(eN, X,,) + a,,,(eN, x,,).

(34)

By inserting this into (32), Taylor-expanding in ~, and using the identity

he(to(e,,, x,,), x,,) -- e,,, cne obtains the result ', r~(-m, XN) = EN = Eo- N ~ ( X , )

(35)

through O(e). The last equality of (35) relies on eq. (23). Combining eqs. (29), (32), and (35) yields the general result M u = frac( M u + tb,(J~) + K~l~(X x)/1;" ( h , )

+ KXl,(h'~)/~'¢(h')) - 1,

(36)

with frac being the function that takes the fractional part of a real number, i.e. the number minus the greatest integer less than that number. The subtraction of unity at the end of this equation puts M" in the proper interval ( - 1, 0). However, in practice this is not needed, because the adiabatic invariant deviation 8Jr,7 is periodic in the crossing parameter with unit period. Eq. (36) may be further simplified by inserting the results (20) for the near-separatrix Hamiltonian. This gives M,," = fr~," _- [\a~d ,u ~- ~c[' Jm) + I [ R b ~ X , ) -

- - R b ( V ) + R , ( h ' ) ] ) - 1.

Ra(~kx)

299

Thus,

M" = frac ( M, + ~,,(Jm)),

(37b)

for a = a or b. In analogy with eq. (30), the phase function in eq. (37b) is defined by ~ o ( j ) _ ft;, ~x ( j , 2 ~ ) d t - a , , ( J ) / e . tx

(38)

Eqs. (37), which give the general relation between the crossing parameters for successive separatrix crossings, are the central results of this paper. It can be combined with the results of refs. [1-4] to determine the dynamics of an arbitrary sequence of separatrix crossings in an adiabatic system. For completeness, we now show how to obtain the entire map by combining the results of ref. [1] and the present rcsults. The trajectory is assumed to begin in lobe-a, lobe-b, or region-c with phase such that the crossing parameter for the ~ext crossing out of region-~ is M~, Mr,, or M~. According to ref. [1], the region entered is given by the second column of table II. The new value of the adiabatic invariant is given by eq. (22) with 8Jr, as follows. If the transition is from lobe-~ to region-c or from region-c to lob,'-.-.', the new value of the adiabatic invariant in region-r/ is given by

~J,.= c + (~#,o)[In IV(-IM~l)r(1 + n.l~t.I) × r ( 1 - R= + R,Im~l)/(2"~)3/z[ +InlEJE~I + (½ + IM=I)InlE,/('~I -(!-R..+

2R,IMJ)inlEJ~'~I].

(39a)

If the transition i: from lobe-a to lobe-,& the new value of the adiabatic invariant in Iobe-fi is given by

(37a)

The result for particles entering and leaving the lobes is found by eliminating the term in brackets in eq. (37a), because the first-order correction to the Hamiltonian for the adiabatic invariant vanishes at the separatrix according to eq. (20a).

×[lair(1 + M~)r/1- ,~~ , , ) :,", 2 ~ 1 1 - ~ In IM, MaI + M,, In IZ,~/f:,,I (39b)

300

J.R. Car), and R.T. Skodje / Phase change between separatrix crossings

(The quantities G and E, were discussed in section 2.) The crossing parameter relevant to the new region (essentially the new phase upon entering the new region) is given by the third column of table II. To complete the map, one needs to know the crossing parameter for the next separatrix crossing. This is given by eq. (37a) if the trajectory has entered region-e. It is given by eq. (37b) if the trajectory has entered one of the lobes. Our results imply that correlations between separatrix crossings may not, generally, be neglected. To illustrate this, we consider a transition from region-c to lobe-a back to region-c. The initial value of the adiabatic invariant, denoted by J~, determines the pseudo-crossing time b y J,~= }~(h~). The lowest-order value for the new value of the adiabatic invariant, denoted by J=a, is given by Jma = Ya(hx) • Eqs. (37) and (39) and table II imply that the crossing parameter for the second separatrix crossing is given by M~--- frae( M~ + O~(Jma + 8Jr,)).

(40)

Taylor expansion of this expression gives the result M~ = frac ( M a +

d~b~,

)

~a(Sma) + 8Jfa(Mu)--~(Jma ) ,

(41) which is correct to order ¢(1) as is eq. (40). The last term in eq. (41) carries tke effects of phase mixing. If it is large, then phase mixing is extensive because the initial unit interval of Ma is wrapped many times on the unit interval of M~. If instead the last term is small, phase mixing is negligible, and the second crossing parameter is simply a constant displacement of the first. This second case is ',,nown as the primitive adiabatic limit, because it ignores the nonadiabaticity at the separatrix crossing, i.e. it is equivalent to setting ~iJfa to zero. In general, neither the completely phase mixed nor the primitive adiabatic limit is approached as goes to zero. As indicated by eq. (30), O~(Jma ) as

well as the factor dOa/dJ in the argument of the fight side of eq. (41) are of order 1/e. As discussed in ref. [1] and indicated by eqs. (39), the quantity 8Jta is of order EIn lel in general and of order e in the symmetric case (which is treated explicitly in the next section). This implies that the phase mixing term has at most a weak In lel scaling in the general asymmetric problem and actually no scaling with t in the symmetric case. Thus, it neither vanishes nor diverges strongly in the limit e ----~0.

To determine the error of the complete mapping we must combine the results of this work with the previous work. In the previous work [1] it was shown that the relative error in the calculation of the adiabatic invariant change is O(et/s). Henrard [2~] has given the improved estimate that this relative error is ~ ( ~ / 3 1 n l l / ~ l ) in the general case. We have found in appendix A that the relative error of the phase mapping is O(d/3/lr3/3 I1/el). Thus, for the complete mapping the relative error found by Henrard, 0 (~t/3 In I1/~1), is dominant. The error for successive composed mappings is less easily estimated. It seems pessimistic to assume that the errors add, because this neglects the possible cancellation of errors. Such an estimate implies that the accuracy retained after N m maps is O(Nmel/3/ln ll/el). However, this estimate neglects error amplification due to orbit divergence, which essentially limits any approximation, even numerical integration on a computer (which necessarily has finite accuracy), to a few Lyapunov times. We leave the calculation of the Lyapunov time for this system, as well as a rigorous discussion of error propagation in this system, to future work. For an ensemble of trajectories, one may expect better results. For example, as we will see in section 6, the error in the retrapping probability seen in numerical experiments is much smaller than that indicated by the previous paragraph. More~,ver, for the features of an ensemble to be well described even for long times, the map need be a curate for only a correlation time of the

J.R. Ca~y.and R.T. Skodje/ Phase change betweenseparatrix crossings

dynamical system, after which small errors are irrelevant. We note that in a related study [30] it was found that in the system of a slowly modulated wave, correlations died away after of the order of five separatrix crossings (independent of e) for small e. Another limitation to be kept in mind is the inaccuracy of the map for those trajectories that pass extremely close to the x-point of the frozen Hamiltonian system. For these trajectories, IE01 < ¢ 9 ( e x p ( - 1 / e ) / e ) . These trajectories are close to the stable and unstable manifolds, which are the trajectories that asymptote to the unstable fixed point. It was noted in ref. [1] that separatrix crossing theory breaks down for such trajectories. Nevertheless, Escande [22] uses separatrix crossing theory to discuss the homoclinic tangle of the stable and unstable manifolds. He does so by first noting that the manifolds cannot be in the region of phase space where the separatfix crossint; map is applicable. Thus, he deduces that the stable and unstable manifolds are in the small, O(exp ( - 1 / e ) / e ) , region where separatrix crossing theory is inapplicable. Finally, we note that the map does not describe trajectories that are not sufficiently separated in time. For eq. (27) to be applicable, a region of adiabatic motion must exist, i.e. it is necessary that t N be less than t'~.. This is possible only when the pseudo-crossing times t x and t~, are well separated compared with the time t N - t x = (t s - to) + (t o - t~). The separation of the pseudo-crossing times is given by t~ - t" = 0 ( l / e ) . In eq. (61) of ref. [1l it is shown that (t o - & ) = 0 ( 1 / l n I1/el). To estimate t N - to for a trajectory in, for example, lobe-a, we use the result from separatrix crossing theory that the time may be estimated by summing the periods for the individ,..,.,,.

S,.,..,t., S ,

tN - lx = - -¢0

n~O

In

Eo+nAEa[.

A rough approximation for this sum, applicable

301

for large N, is given by the integral t~. _

=

+NA&In

Ea

= NAEa (I + In[ Eo + NAEal)

Therefore, for all but the exponentially small class of trajectories mentioned earlier, t , v - t~= O(NElnI1/Nel), which must be less than ¢(1/e). In fact, appendix B gives N = O ( 1 / ( e l n lel)l/3). Thus, the pseudo-crossing times are well separated for typical orbits. However, for those (atypical) trajectories which cross at a time very near where a lobe changes form growing to shrinking, for example, so that the pseudo-crossing times are not sufficiently separated, the analysis breaks down.

5. Application to symmetric systems Symmetric systems are those for which the separatrix crossing parameters and the phase functions (38) of the two lobes of the separatrix are identical. For such systems the formal separatrix crossing theory is considerably simpler than that of the general case. In this section we discuss the dynamics of symmetric cases to illustrate the application of the results of the last section. An example of a symmetric system is that with Hamiltonian having a quartic potential and the usual kinetic energy,

H( q, p, h) = ~p2 _ !,o2q2 + bq4.

(42}

As the parameters co and b are varied the depths of t~e wells on both sides of q = 0 change, and, hence, so does the separatrix area. This causes a trajectory t.o enter and leave the wells repealc;.'!y One complete cycle of this process is shown in fig. 5. To illustrate the use of the mapping of the last section, a trajectory initially in lobe-a with initial

J.R. Cu D' and R. T. Skodje / Phase change between separatrix crossings

302

to lowest order. Through first order, the new value of the adiabatic invariant is given by

(1

J¢o = 2 J~0 - ~ In 12 sin t 'rrM~0) I.

(45)

The correction term, the last term in eq. (45), is obtained from eq. (39a) by setting R~ = R b = ½, E c = E ~, ga = gb, and lye-- 2~"a. The next step in obtaining the map is to find the new crossing p~ameter for the next separatrix crossing. According to table II, the crossing parameter relevant to region-c is M,0 = M.,o/2. Tiffs can be inserted into eq. (37a) to obtain the crossing parameter Muo for the next separatrix crossing. We note that the term in the brackets in ~q. (37a) vanishes for a symmetric system. Thus, we have

b

C

L; d

M~o= frac (Mao/2 + ~¢(41))-

(46)

The phase-advance function ~ ( J ¢ l ) must be calculated numerically. Through order unity this relation may be written as

e

M~0 = frac ( p ( Ma0, Ja0)),

(47)

with Fig. 5. A sequence of two separatrix crossings for a particle in a time-dependent double-well potential. The trajectory suggested by the straight lines in the poter',ial wells is reactive in that it is originally trapped in the left well mad ends up finally trapped in the right well.

value of first-order corrected adiabatic iuvariant, I] = J.o, is considered. The crossing parameter for the first separatrix crossing [s assumed to be Ma0. For such a trajectory the first crossing occur¢ .t pseudo-crossing time Qo S~vcn by

Therefore, the new value of the adiabatic invariant in region-c is given by

Y~(et,o ) --- 2J~o

(44)

p( M~, 4 ) ~- M J 2 +

E

I dq~c. . dye ~0 -d--J (2J~) ~ lnl2sin('nM~)l" (48) Eqs. (47) and (48) imply that the crossing parameter relation for the symmetric case is, oeriodic i~ 1/e for smaii e. This shows once again that neither the primitive adiabatic l!mit (neglect of tl-e last term of eq. ta.~V~, ._,, nor tt. . . . . . h, u, ,,vat+ .... pbase mixing (assuming that the last term of eq. (48) is very large) is valid for small e. Instead, the relation is periodic in 1/e with Feried !,/5~. TI:a'. is, M~0 is the same for ~ and e' whenever i/',~ = 1/e' + I/~,.

J.R. Ca O, and R.T. Skodje/ Phase change between separatrix crossings

The value of the subsequent crossing parameter M~o determines the trapping state at the next crossing. According to table I1, if M,~0 is less than one-half, the trajectory becomes trapped, in lobe-b, and if M~o > ½, the trajectory becomes trapped in lobe-a. We consider momentarily an ensemble of trajectories, all having I~ = Ja, but with uniformly distributed conjugate angles. As shown in ref. [11, the crossing parameters M~ of ti':e ~.rajectofies are then uniformly distributed. This and the fact that the new crossing t~arameter aiven by ca. (4~) must be between ½ and 1 for a tra.,::etory to become trapped in lobe-a, imply that the retrapping fraction is given by / = me.as { M.10 < M a < 1, n
(49)

< n + 12, n is an integer}.

Analogous formulas for the general case may be derived from the phase relation (37a), the deviation formula (39a), and table II. The calculation of f is shown graphically in fig. 6. The calculation is one of summing up the intervals of M a such that the new crossing parameter is between ½ and 1. The form of eq. (49) implies that the trapping fraction is also periodic in 1 / A for small ~. It does not asymptote to the primitive adiabatic limit, in which case the frac2 --

tion becomes

f~=

2 frac (A J e )

for frac ( A J e ) < i,.

2[1 - f r a c ( . 4 J e ) ]

for frac ( A J e ) > ½,

(50) a sawtooth function. It also does not asymptote to the value ½, the result assuming complete phase mixing. The nature of this calculation implies that this trapping probability is for times significantly after the second pseudo-crossing time, so that the bulk of the ensemble has made the second transition into one lobe or the other. As shown in eq. (61) of ref. [1], the spread in the time t o at which the trajectories of such an ensemble cross the u-vertex is 0(ln I1/el) for all but the exponentially small class of trajectories for which separatrix crossing theory breaks down. The instantaneous separatrix is crossed before the time, to+(ln[EJEol + In [EJE l [)/(2~), i.e., before one additional oscillation has occurred. For most trajectories, E o and E t are O(e). There/ore, t.his calcu!atien of the trapping probability is for times later than the pseudo-crossing time by an amount that is O(ln I!,/~I) Of course, this time is small compared with the time 0(1/~) between separatrix crossings. The next part of constructing the mapping is to determine the new value of the adiabatic invariant after entering the lobes. This is found by applying the symmetry conditions to eq. (38a1. The result ."or the new value in either of the lobes is

-- 2

-/ - I

~

-L

303

lnl2sin(2

Md0)l.

(51)

To complete the mapping, one must use the equa,.';~,lt.', i~: i ; l e ,-~¢w ~..ttr-,..,i.5 pmai-ftci.2f NI 0

---~.

A_

t

-4

Fig. 6. Plot cf p(Ma, Ji) vs. , ~ The values of M, corresponding to trajectories t~at r~.trap in Iobe-a are those portions of the M~-axis that are hea~dly drawn.

The complete mapping is defined by eqs. (45). (46), (51), and (52). This map may be iterated repeatedly, wi:hin the accuracy restrictions mentioned in the ins, section, to determine the trajectory. [teratio.n may also be used to find ensemble

304

J.R. Ca~. and R.T. Skodje / Phase change betweenseparatrix crossings

averages of, for example, the retrapping probability after several crossings. The particular lobe entered by the particle at the second separatrix crossingis largely irrelevant to the subsequent dynamics. The equation analogous to (45) appropriate for leaving lobe-b is obtained by simply making the replacement a --* b. The relations in table II indicate that M'0 as given by eq. (46) will change by the addition or subtraction of ½, but this is not important because the relation (51) is periodic in M" 0 with a period of ½. Since this periodicity is also present in eq. (52), the mapping is the same regardless of which lobe is entered. We chose this potential, in part, because the application of the map is particularly easy. Each crossing is a symmetric crossing. This implies that the crossings are from the lobes to region-c or vice versa. Transitions between the lobes do not occur. This implies that when the wells shrink it is known a priori that the trajectory will re-enter region-c. However, application of the map to any system is in principle the same. After each crossing, one must use the new value of the adiabatic invariant to determine the next pseudo-crossing time. Then the new phase determines the region into which the trajectory enters and the next value of the adiabatic invariant.

6. Numerical results for a model problem In the preceding sections, a general theory of correlations between successive slow separatrix crossings was developed. In this section, a numerical study of correlations for the symmetric Hamiltonian (42) is presented. This study serves both to determine the limits of validity of the theory and to provide a r,hysically intuitive characterization of the dynamics of repeated separatrix crossings. The Hamiltonian (42) is adiabatically varied by changing ~he parameter ~0. The variation is given by

,,,= [2as(X )l '/',

(53)

for which the switching function s ( h ) is defined by [lSa]

f 1 - x + sin (2~rh)/2~r s ( a ) = i A - 1 - sin (2~rh)/2"rr

for 0 < h < 1, for 1 < A < 2 . (54)

This switcrang function is a smooth function of time through its second derivative and so nonadiabaticity in regions away from the separatrix are small, scaling to zero [15a, 27] as 0(e3). As usual, h = et, so a switching time is defined as

r-

(55)

The time-dependent double-well system presented above is a useful model for a variety of physical problems. For example, in chemical reactions such as I + HI -o IH + I the light hydrogen atom moves in an effective double-well potential whose time-development is governed by the slow evolution of the heavy iodine atoms. Thus, a direct reaction consists of two sequential separatrix crossings, such as above, which are preceded (t < 0~ and followed (t > 2T) by the fre, asymptotic motion of the atom-diatom pair which need not be explicitly treated. It has been found that many aspects of the reaction dynamics can be understood and predicted using the phase change theory [19f]. For the choices a = 4 and b = 1, the bander hAghl at t = 0 is AE = 4. The separatrix lies at E = 0 for all t. (At t = T the separatrix is absent, because the two minima coalesce.) Thus, during '.he interval t --- (0, T) the barrier falls smoothly to zero, and during the interval t = (T.2T) the barrier rises again back to its original height. Therefore, as depicted in fig. 5, trajectories trapped in the left well at t---0 eventually untrap as the barrier falls and will subsequent!y oscillate a number of times over the top of the barrier (located at q = 0). As the barrier rises, the tJajectory t~traps in either the right or left well. Hence, all such trajectories must ex0erience two separatrix crossings during the interval (0,2T). The time scale in

J. R. Caty and

R. T. Skodje/

Phase change between sepurarrix

the system is set by the choice of the switching tims, T. The rate of passage through each separatrix scales as E = 1/T, and the difference in pseudo-crossing times for the two separatrix crossings scales as T. The implementation of the adiabntic theory for this system is simpMed by the use of the analytic formulas available for the instantaneous actions and frequencies of the quartic double-we!1 problem. These quantities may be written in terms of the complete elliptic integrals K and E of the first and second kinds, respectively. The convention used here for the argument of the elliptic functions is that of ref. [31], in which the argument of the elliptic function appears squared in the integrand. We define the intermediate parameters

(56)

(57)

in terms of these parameters, (dy= a or b) is

the actioii in lobe-a

I a0 -- w3'~~~1'2[E(p)+(y-1)K(p)].

(5th)

and the frequency in lobe-a is given by

+yy2 2fiK(p) ’

w(l ‘.co=

(58h)

For trajectories outside the sepatattix (in region-c) the action is

[ZE(y-‘)+ (y -

I)K(p-I)],

(59a)

305

and the frequency is given by (59b) Of course, it is not required to have analytic representations for any of these functions in orr.;er to apply the theory. Indeed, in an analysis [19(1 of the chemical reaction I f HI + IH + I all of the theory has been imp!emented numerically. For application of separatrix crossing theory, the near-separatrix crossing parameters must be known. These are found by using the asymptotic expressions for the elliptic functions that are valid near the separatti, i.e. y close to unity. The separatrix actions are Y, = w3/3i,

@a)

for the lobes, and Y, = 2Y,

and

crossings

(60b)

for region-c. The exponentiation rate w of s?paratrix crossing theory that appears In eq. (45) is identical to th? one defined by eq. (42). A typical trajectory of this sysiem is plotted in fig. 7 in three diKerent representations. This trajecLx> ha3 initial energy Ei = - 2. initial action i, = 3.3325?, XX! initia! vibatiena! period pi = 1.781 (Since the first derivative of the switching function (54) is zero at t = 0, we need only initially specify the zero-order action.) The switching time is chosen to be T = 50. In panel (a) the coordinate LJif, plotted versus time. As expected, the detrppping of the irajectory from the left well takes place near the first Fseudo-crossing time, I, = 23.01. and the retrapping into the right well occurs near the .V.-.r\nApse~d+crsssigg !i_r??eof t; = 76.99. The JCG”.IU crossing of the seyarahi,i is most easi!y seen in panel (b) where the orbit is plotted in Ghase S~ZCC. ((I, p). In panel (c) is plotied the continuous dction 1. This quantity is defined to be the action in the lobes, but half of the action in region-c. As defined. I is a continuous function of phase space. The continuous action, I, is seen to oscillate abm

306

J.R. Ca~ and R.T. Skodje / Phase change between separatrix crossings 20

1.0

q

0.0

-1.0

|

-2.0

I

I

0.0

25.0

I

I

50.0

75.0

100.0

L 4.0

2.0

P

C

i,,.._

ii

-'d°ll

0.0

y -2.0

b •

-4.0

-2.0

I

I -I0

'

I

* oo

,I

I 1.0

7.0

q .3.7

3.5

3.3

J 0

25 0

50 0

75.0

100 0

Fig. 7. One trajectory for the time-dependent quartic double-well problem plotted in three representations. In (a) the q-dependence is plotted versus time. The pseudo-crossing times are t~ = 23.01 and t~ = 76.99. In (b) the orbit is plotted in phase space, (q, p). In (cJ the zero-order action is plotted against t,.'me.

J.R. Cam.'and R. T. Skodje / Phase change between separatrix crossings

an approximately constant mean value except near the pseudo-crossing times, at which there are non-negligible at, tion changes. Much of the oscillation of I would be suppressed in the first-order corrected invafiant It. However, the net action change from ".he separatrix crossing cannot be eliminated by the use of the corrected invariant, We now consider the predictions for an ensemble of trajectories. The trajectories of the ensemble are initially in the left well with initial action I i = 3.3257. The trajectories are initially distributed uniformly in angle. Thus, the ensemble of trajectories forms a loop in phase space. At t = 0, the time derivative of the switching function vanishes. Therefore, the theory of section 2 indicates that the first-order corrections to the adiabatic variable vanish at t = 0. This implies that our ensemble is one of constant I 1 = li and uniform distribution in ~ at t = 0. The loop of trajectories evolves according to primitive adiabatic theory up until the first separatrix crossing. At :his time the orbits acquire a spread in adiabatic invariant, since the change of the adiabatic invariant depends on the initial phase through the crossing parameter, M~ The new value of the adi ,batic invariant after this crossing is found from eq. (45) by inserting the derivative of eqs. (60). This gives

2.50 - -

1.25

p 0.0 -1.25

-2.50 -2.50

2ca d5 b dA lnlasin('rrM~)l"

(61)

During the time interval between separatrix crossings, the trajectories accumulate different amounts of phase, because the frequency is a function of the action. This causes the phase loop to become convoluted. This is illustrated in fig. 8(a), where an ensemble of two hundred trajectories is shown at the halfway time t = T. Most of the trajectories lie in the vicinity of the expected invariant curve, shown with a solid line, but phase mixing is clearly evident. After two crossings, t = 2T, depicted in fig. 8(b), the trajectories are distributed between the two separatrix lobes and still mostly lie in the x4cinity of one or the other expected invariant

I

I

-1.25

0.0

I

I

1.25

2.50

q

2.50

1

~.25

p

0.0 -1 •25 -

-2.50 [ -2.50

J . ' = 21i

307

!

I

-1.25

I

I

0.0

I

I

1.25

2.50

q Fig. 8. An ensemble of 2.90 trajectories, initially evenly spaced on the adiabatic invariant 6ng I, = 3.325"/in lobe-a, is shown at two times during the process depicted in fig. 5. In (a) the trajectories, shown as individual dots, are plotted at the halfway point t = T. In (b) the trajectories are shown at the final time t = 2T. The solid lines in (a) and (b) are the curves of constant action corresponding to the zero-order expected actions related to J, = 3.3257.

curves. The phase mixing has progressed even further by this point. Note, however, that the phase mixing is far from complete since that would imply that the trajectories should ~ecome uniformly distributed in angle. Th.s is definitely :~,~¢ the case.

308

J.R. Cary and R.T. S.~odje/ Phase change betwcen separatrix crossings

To determine the consequences of two .~equential crossings, the phase (30) must be kn.own. In the present case, the first-order correctio~ dae to

Ucl, rx. "¢1t, ,J.,, h)dh, '1' 1 - Jx,

J'= li + -6 -d~ In sin(2~Mg) "

vanishes [9] because the path in the parameter space of Hamiltonians is a degenerate loop, i.e. H ( h ) = H(1 - },). Therefore, to lowest order, O(Jm) can be computed using the zero-order frequency, (59b). This implies that eq. (30) may be significantly simplified, 1

r~'~

.

q~c(Jm) --- ~ J;~ Vc0(Jm' h)d2k"

(62)

The differences in accumulated phase between crossings is due to the dependence of the phase (62) on the intermediate value of the adiabatic invariant. To lowest order in e, tiffs difference in accumulated phase is given by the derivative term d O c / d J m. The contribution to the derivative of (62) of the end points, i.e. the term

o0(g ,

-

oo(g ,

vanishes because the orbit frequency vanishes on the separatrix. This implies that the phase mixing, the last term ip eq. (48), is due to the instantaneous anharmonicity,

dj m

=

7ix

eq. (61) and the substitutions J,q =Jr, J¢o = 2Jm, M a = 2Mu0, and M" o = M'. Through order unity, the facto,- multiplying the logarithms in eqs. (51) and (61) are eqaal in magnitude but opposite in sign. Hence,

h) dh-

(63)

Finally, near t~, the trajectories retrap in either the right or left well with their crossing parameter M~ determined as a function of the first crossing parameter M~ via eq. (46). Due to the correlations discussed above the phase mixing will not be complete and the final trajectories will not be uniformly distributed in angle at the end of the switching process. To calculate the final value of tile adiabatic invarian:, eq. (51) may be used x~:,th

(64)

In fig. 9 the results of the analytical theory are compared with those of numerical integrations for switching time T= 50. In panel (a) the action change for a single separatrix crossing is shown as a function of the crossing parameter M a. The solid curve is the prediction from eq. (61), and the crosses are the actions of the exact trajectory measured at the midpoint of the process, t = T. This comparison shows that the separatrix crossing formula (61) is accurate for T = 50. The singulaxity at Ma = 0 corresponds to the orbit that is on the stab;" manifold of the x-point. This orbit asymptotes to the fixed point. This causes a large change of the adiabatic invariant. Fig. 9(b) is a plot of the final action after two separatrix crossings versus the initial crossing parameter, M a. The solid curve is the prediction of k:e theory (64), and the crosses are the results of the numerical integrations. To use e q (64), one must calculate the final crossing parameter with eq. (46). The final action plot shows a series of peaks as a function of M a. The origin of this structure is brought out in panel (c), where we plot the second crossing parameter 2M" versus the initial crossing parameter, Ma. It is seen that at the position of each spike in Jr, 2M(~ passes through an integer value. Thus, the upward spikes in Jf occur for orbits on the stable manifold of the second separatrix crossing. These orbits have 2M" = 0. Clearly, the more convoluted (i.e., phase mixed) the ensemble becomes during the period t~ < t < t~, the more structure will emerge in Jr. This is the reason that the most intricate structure in fig. 9(b) is observed in the vicinity of M a = 0, where lhe change of the adiabatic invariant for the first crossing and, hence, the accumulated phase

J.R. CaD, and R.T. Skodje / Phase change between separatrix crossings

-

(1

69 dm

6.3 5.7

34 df

3A

2.8 i

i

i

L

a

C ,,.. 0 . 7 5 'o o E

0.50

O4

0.=25

/

29.0

N 27.0t| r 25.0 L

1

L- i ~ -~/

L -0.5 0.5

..... O 1.0

0.5 t.5

Mo Fig. 9. A comparison of the theory predictions and exact numerical trajectory results for the quartic double well with T = 50 for trajectories originating on the adiabatic invariant ring I i = 3.3257. In (a) and (b) the intermediate action, J,,, and the final action, Jr, respectively, are plotted versus Ma with the theory results shown as solid lines whi;e the exac trajectories a*e depicted with +'s. Note the origin of M~ has been shifted relative to fig. 6 for aesthetic reasons. In the trajectory calculations, M a = g~, the initial angle variable ~ t h the angle origin taken at the critical trajectory. The "singularities" in Jr are seen to lie at the points where 2M~(M~) = 0, which is plotted in (c). The number of crossings, N, of the barrier, q = 0, by the exac: trajectories is shown in (d), which illustrates the ranges of M a which yield final trapping in the fight well (viz., when N odd).

309

O¢ are most rapidly varying. Even though some accuracy of the map is lost in the vicinity of the spikes due to the large action change, the predicted results seem to be quite good compared to the exact results. Fig. 9(d) shows the numerical result for the number of times, N, that the barrier position at q = 0 is passed by the trajectory as a function of the initial crossing parameter, M a. If N is even, the orbit will be trapped in the left well. if N is odd, the orbit ultimately is trapped in the right well. The values of the initial crossing pa:'ameter, M,, at the boundaries between !eft and right trapped are seen to correspond exactly to the zeros of M u' and the spikes of ,It. This is consistent, since the boundary orbits must asymptote to the fixed point at the barrier maximum. The theory of the last section showed that as T increases, the persistence of the correlation between Mo and M" is the result of a ~ncellation of two competing effects. First, the action change at the first separatrix crossing scales to zero as 1/T. On the other hand, the time between separatrix crossings increases linearly with T. Therefore, as T increases, the narrowing of the distribution in instantaneous frequency is compensated by the increase of time available for spreading of the phase. For the case of symmetric separatrix crossings the two scale factors exactly cancel, so that in the limit of large T, the periodic function (47) is obtained. Extensive numerical simulations have been carried out tO investigate the accuracy of the analytic theory as the switching time, T, is varied. The switching time was varied through the interval 3 < T < 70. For each value of T, an ensemble of 200 trajectories was propagated. The trajectories in the ensemble were evenly ul~u,~u~,.u .......... a phase. All of the trajectories had the same initial energy, E i = - 2, and, hence, the same initial value of the adiabatic invariant. The results for three quantities the mean final action (Jr), the RMS spread in final action, and the reaction probability, are presented.

310

J.R. Cary and R.T. Skodje/ Phase change betweenseparatrix crassmgs

this range. The horizontal line on the figure is the initial action value, I i = 3.3257, which is the expected result for (J~) in both the phase mixed and primitive adiabatic limits. The predicted RMS spread in final action, 8./'~m~ts, is obtained from eq. (64) by taking the square root of the integral of ( J r - li) 2 over the interval [0, ½l. This shows that 8 J ~ s oscillates with period ,~/2 as a function of T, but now both the mean value and the ampfitude scale to zero as e. The analytic result is shown as a solid line in fig. 10(b), and the crosses are the results from the numerical simulations. Once again the theoretical results and exact results agree to within statistical error above T-- 10. The dashed curve in fig. 10(b)

The final action after two separatrix crossings is given by eq. (64). The predicted mean value, (Jr), obtained by integrating M u over the interval [0, ½], oscillates as a function of T around the average value, Ji, with period 2 / A c and amplitude that scales to zero as ~. In fig. 10(a) the integral of eq. (64) is plotted as a solid line. The results of the numerical calculations are shown as crosses. The agreement between the analytic results and the numerical calculations is seen to be excellent for T > 10. In this range, the difference between the analytic results and the numerical results is within the sampling errors of the numerical calculations. For T < 10 there is some deviation. This indicates that the tLme scale approximations are not valid in 3.9

0 3.6

A V

27i

. 5.0

.

. 12.5

.

.

. 20.0

.

. 27.5

35.0

2.0

~"[ t[ I

1.5 tr,..

b

1.0

¢O

0.5

O 50

12.5

20.0

27.5

35.0

T Fig. 10. A comparison of theory predictions and numerical trajectory results for the quartic double-well problem as a function of switching time. In (a) the mean final action (J,) for an ensemble initially evenly spaced on the adiabatic invariant ring / i = 3.3257 is plotted versus T. The theoretical result is the solid line, the trajectory results from ensembles of 200 initially evenly spaced trajectories are shown as + 's, and the dashed line is the primitive adiabatic (and the complete phase mixed) prediction of (Jr) = I i. In (b) the fin',d action spread, 8JrRMs, for the same initial ensemble as in (a) is plott~ ,ersus T. Again the solid line is theory and the + 's are the trajectory simulations. The dashed line is the prediction assuming complete phase mixing and hence uncorrelated separatrix crossings.

J.R. Ca~. and R.T. Skodje/ Phase change between separatrix crossings

311

aoparently reached its T---, ~ limiting periodic form by the point T = 10. PR 0.50 ~o.2s 0.0

,

',

,

,

,

',

,

,

t

I

i

I

I

t

t

I

I

!

!

I

I

',

,

',

,

,

1

i

t

,

,~

',

I

,

t,

t

10.0

0.0

,

.

t

20.0

I

t

,

',

, I

I

',, , 'I

?

,,

300

I

,,

i

i.

I

t 40.0

T Fig. 11. The reaction probability 'DR plotted against switching time T for the quartic double-well problem. The solid line is the analytic result using eq. (49) at.ong with the definition P~ = 1 - f . The dashed fine is the analytic result in the primitive adiabatic limit from eq. (50). The crosses are the results of exact trajectory simulations carried out using ensembles of 200 evenly spaced trajectories at each switching time.

shows the prediction, 8J~,, m~ls, obtained by assuming complete phase mixing between the two separatrix crossings. Through O(O this is simply the RMS deviation for a single sep.~ratrix crossirg multiplied by v~, i.e. 8jcpR~S=

rrea l ds I

¢gb

a--~"

Clearly, the result assuming complete phase mixing is significantly in error for all values of the switching time, 7. The probability, Pr., of trajectories that originate in the left well and retrap in the right well has been computed theoretically using eq. (49) and the definition PR = 1 - f . The formal theory predicts that PR oscillates with period ,~, which is twice the period of oscillation for (Jr) and 8J RMS. The comparison of this result to the numerical simulations is shown in fig. 11. Again one observes excellent agreement between the theoretical and simulation results. The results of the primitive adiabatic approximation is shown as a dashed line. The primitive adiabatic approximation is significantly in error for almost all T. The assumption of complete phase mixing (not depicted on the figure) would yield the constant value of/'R = 0.5. This figure shows that the function has

PR(T)

7. Summary and conclusions

This work has shown how adiabatic invariance theory is generalized in the presence of separatrix crossings. The rapid degree of freedom is reduced to a map. One part of the map, the change of the adiabatic invariant, was calculated previously [1]. The second part of the map, the change of the crossing parameter, or phase, between separatrix crossings was calculated here for general separatrix crossings. Hence, using the present theory one may predict the full dynamics through an arbitrary sequence of slow separatrix crossings. There are a few limitatiorts for this calculation. As discussed in section 4, the small sets of (1) trajectories too close to the stable and unstc,~!e manifolds and (2) trajectories for which the crossings are insufficiently separated in time must be excluded. In addition, one may expect loss of accuracy after many applications of the map. However, the numerical results of section 6 show that analytic theory gives surprisingly accurate predictions for quite large values of the adiabaticity parameter for ensemble-averaged quantities, such as the retrapping probability, the mean final value of the adiabatic invariant, the mean-square deviation of the adiabatic invariant, and the reaction probability. A side result of this work is the proof in appendix B that the phase change between separatrix crossings is a coordinate-independent quantity. This is .~ surprise because, as discussed in the prex4ous work by Hannay [6], the phase change is not coordi:~ate-independent for the standard adiabatic theoT.~ without separatrL,~ crossings unless a dosed curve in Hamiltonian parameter space is traversed by the adiabatic variation. Thus, the present work enlarges the class of known systems for which the phase change is coordinate-independent.

.LR. Cary and R.T. Skodje / Phase change between separatrix crossings

312

Acknowledgements The authors are grateful to Penny Teal for assistance with the numerical calculations. The work by RTS was supported by a grant from

the National Science Foundation CHE-8609975, while the work of JRC was supported by the Department of Energy under grant DE-FG0286ER40302.

Appendix A Error analysis Bqs. (37) are the results of the lowest-order theory for the mapping of the separatrix crossing parameter. This analysis required: (1) the trajectory must be close enough to the separatrix at the matching point (EN, tN) for the near-separatrix formulas such as (13) and (19) to be valid, yet (2) the matching point must be sufficiently far from the separatrix for adiabatic theory to be valid for eq. (27). To determine whether this is possible, error analysis is needed. After the errors of the individual steps of the analysis are summed, one can determine whether there exist~ a value of N such that the two just mentioned requirements can be satisfied simultaneously. The relative error e r in eq. (37) arises from the use of the last approximate equality of eq. (31) in the integrations of eqs. (32) and the use of the equalities in eq. (35). (The additional error that arises from using only the first-order frequency in eq. (27) is t~(E2 At) ----0(e), which will be shown to be unimportant.) The relative error associated with using eq. (31) is denoted ej, and the relative error associated with using eq. (35) is denoted at- The total relative error is the sum er = ej + e~. The replacement, ~ dr= d t/( 0 I~ / 0 E ~) = - d E~/ ( e ~ I~ /~ ),), of the integrands on the right side of eq. (29) is exact. The error arises because the additional approximation

(A.I) is made. Reference to eqs. (13) shows that 8i~ has been neglected in this step. This derivative is nonzero due to the variation of E~ and g~. The contribution due to these variations adds terms that are t~(eE) and t~(e 2) to the right side of eq. (A.1). Since the right si¢~,~of eq. (A.1) is 0(8), this gives a relative error in the integrand that is (P(E)+ d~(e). Thus, the integrals (32) neglect the error (A.2) that arise,~ from the use of (31). The last equality in (A.2) was obtained by use of eq. (35). The error of the last equality of eq. (35) arises as follows. A part comes from the neglect of the correction e2I~2 in eq. (34). As discussed in section IIID of ref. [1], this error is 0(e2/EN) = ~P(e/N). The dominant error in the first equality of (35) due to the neglect of this term and the other terms neglected in the Taylor expansion is v~0e2I~2= ¢ ( e / N In [el). There is also the error that arises from repeatedly using (23) to obtain the second equality of eq. (35). This error is ~(e21n[nfin(!ENl)])+C)(eNEN)= 0(e 2 ln[min(lENI)])+ ~(EEN2).These error terms give rise to an error,

.,, = O ( 1 / N In I,l) + O(,ln [rain (IE,,I)I) + ¢ ( , N 2 ) .

(A.3)

J. R. Car)/and R. T. Skodje / Phase change between separatrix crossings

313

in eq. (37) due to the use of (35). The second term of this equation is sig aificant only for an exponentially small class of particles that very closely approach the x-point. This was discussed in more detail in section V of ref, [1]. As these particles were discussed previousl2~, they ar~ not again discussed here. Th,:s, the second term of the fight side of (A.3) can be ignored with this understanding. Addition of the two terms (A.2) and (A.3) gives the error of the result (37) to be

Er= O(1/N In IEI) + O(N2e).

(A.4)

This error is minimized by the choice N = O[(eln result,

lel) -x/3]

for the matching point. Thus, the error of the

(A.5) does vanish as e ~ 0. This validates the analysis for all but the exponentially small class of particles mentioned earlier.

Appendix B Coordinate invariance of the results As discussed by Hannay, the first-order corrected phase change, Aq0~ = j-~l dt, between two times is not coordinate-independent for the usual adiabatic theory. This occurs because the angle variable is not unique; its origin is modified by a change of the starting point qo(L X). P etause q0 may be modified in any way consistent with slow variation, the first-order correction v1 may be changed arbitrarily. Therefore, there exists a coordinate-dependent part of the phase change, Iv I d)~. This phase change is coordinateindependent only when it is calculated between two times at which the Hamiltonian is ~denzical. so that the requirement can be imposed that the angle variable be identical at the two times. If this were also true for the phase (38), it would indicate problems with the theory., because the vertices, which relate to q0 in this theory, are not unique [1], in their specification is an arbitrary one-parame:er family of canonical transformations. Nevertheless, in the previous work [1] it was shown that the eros: ing parameters are coordinate-independent. The quantities R,,, being ratios of areas, are also coordinate-independent. As these quantities appear in eqs. (37) with the phase ~ , this phase must also be coordinateindependent for the theory to be consistent. Yet this seems to be in contradiction with the results of Hannay. In this appendix, it is shown that there is no inconsistency. The phase ~ , the first-order corrected phase change between pseudo-crossing times, is a coordinate-independent quantity. This holds even when .L^ ~._.:,^..~ . . . . . . . . , ; a , ~ , ~ t ; . - ~ t . t the )w,~ fiTaes Thu.~. the oresent work indicates another situation in which the adiabatic phase is well defined. The outline for this discussion is as follows. First it is shown that the difference of the phases oi~tained by calculating in two different coordinate systems is simply related to the movement ef the q0. which determine the angle origins. The difference of the phases is the product of the lowest-order frequency (which vanishes on the separatrix) and several terms which are shown to be finite on the separatrix. Hc,'ace, the phase difference vanishes as the separatrix is approached. The change of the angle origin is shown in fig. 12. At the initial crossing time t~. the coordinate svstems are identical. "lhe trajectory is entering lobe-b. (The analysis is identical for the other regions.) As time t l l g

II4::I.IlIUtII,~,O'IIIGI~IId

~ 1 ~

LLVL

I ~ & ~ L A ~

.

.

.

.

.

.

.

.

.

.

.

.

.

.

,

•

J. R. Cary and R. T. Skodje / Phase change between separatrix crossings

314

D

qo,

/

Po|

\

/

Fig. 12. Coordinate sysiems near the x-point. In the original system the angular origin is the q-axis. In the modified system the origin is the t~-axis.

proceeds, the origins separate, so that ultimately the angular origin (00, Po) for the second coordinate system differs from that for the first. This gives rise to a difference of the first-order correction ~h. The quer, tion is how ,.., calculate the resulting phase difference at the second pseudo-crossing time t~'. First, a simple, intukk. "on'nula for the difference of the frequencies is derived. As discussed in section 2, the first-order correction to the adiabatic Hamiltonian is obtained by averaging the correction on the right side of eq. (4). This correction may be written in the form

"~'~1= vo~dq '8P fq~'dq" ( aP 8HOP ~-E(q') -~-~(q")+ 0h 3S

(q'')

) -P(qo) 8qo 8k"

(B.1)

The dependence of P on H(,, = I x, h) and h has been suppressed in the notation. As in section 2, this is a functional relafien. A similar result for the other coordinate system is obtained by replacing K~x and q0 b y / ~ 1 and qo in eq. (B.I). The difference between the l~'amiitonians for the two systems is obtained by subtraction of eq. (B.1) and ^X the corresponding equation for Kbv In the corresponding manipulations, the fact that the interior integral of (B.1) is always from qo alone the direction of flow to q is critical. This subtraction yields e[K~x-/(~x] =

~P aH aP ) -eP(~o)--~-~+eP(qo) 20o aqo -e j'~, dq, ( -5-~(q')+ Oh ~-ff(a') ~k qo

a fq~Odq,p(q,,H( 3 X),X).

(B.2)

Differentiation of tMs expression with respect to the argument J gives the difference between the first-order corrected frequencies of the two systems:

= -

. ,, dq

,o,

-~(q', H~J, X). a) + P(qo)-ffff - P(qo)

•

(B.3)

J.R. Cary and R.T. Skodje / Phase change between separatrix crossings

315

The portion of the frequency difference due to the first term in the parentheses has a simvle inte'.~tetauon: it is the rate at which the two origins are separating multiplied by the instan~.,tn,~,~usfrequency, i.e. it is the rate of increase of phase difference between qo and /Io as measured in the origin,.l coordinate system. Integration of (B.3) yields the difference of phase for the two systems,

"b - ' b = f i m [ % o ( l , h ) ( f # ° d q ' ~~-~tq,H(J,X),X)+P(Oo)--~ff-P(qo) Pt' i}q° ~q~

~---~) ] ,

(B.4)

in which h(e(t" - 8)) is substituted. Now the behavior of this quantity near the separatrix (8 ---,0) must be considered. Near the x-point, the vertex in the original coordinate system was chosen to be on the q-axis. Therefore, the middle term of the parentheses of eq. (B.4) vanishes identically. The behavior of the last term is obtained by noting that near the x-point, the curve [~o(I, h),/30(I, h)] obtained by varying I is well approximated by a straight line at some constant angle to the q-axis: /~o/qo- tan0. This condition and that obtained from eq. (:2), qo), can be solved simultaneously. The result is E = ~0=(

- 2 E ]l/: ~0cos20 ] cos0.

(B.5)

Eq. (10) implies that P = ~/2E~/~o+ q2.

(B.6)

Differentiation of eq. (B.5) and substitution of eq. (B.5) into eq. (B.6) show that the last quantity, P(qo) aqo/aE, in the parentheses of eq. (B.4) is finite. Since the faclor in front of the parentheses, the instantaneous frequency of the orbit, vanishes on the separatrix, the last factor also give no contribution to the right side of eq. (B.4). The contribution to the fight side ef eq. (B.4) due to the term containing the integral is analyzed similarly. Differentiation of (B.6), insertion into the integral, and integration to the limit (B.5) shows that tl,e integral inside the parentheses of the fight side of (B.4) is finite. Thus, it also gives no contribution to the right side of eq. (B.4) because the factor ~'bo in front vanishes as the separ~trix is approached. In summary, the fight side of eq. (B.4) vanishes at the pseudo-crossing ti-~ae. This proves that the phase change from one pseudo-crossing time to the next is a coordinate-independent quantity. This analysis did not require that the Hamiltonian be identical at the two end times, only that they both be pseudo-crossing times: Y b ( ~ x ) = J = Yb(h'x). Thus, the first-order phase change is a well-defined concept in cases other than those considered by Hannay [6].

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