Chemical Physics

Chemical Physics 183 ( 1994) 365-373

Rate processes in dissipative systems: scaling in the canonical variational transition state theory

Surjit Singh a, G. Wilse Robinson a*b ’ SubPicosecond and Quantum Radiation L&oratory, Department vf Chemistry, Texas Tech Universiiy. Lubbock, TX 79409-1061, USA

b Department of Physics, Texas Tech Universiry. Lubbock, TX 79409-1051, USA

Received 18 May 1993; in final form 23 November 1993

Abstract

We study the activated barrier crossing (ABC) problem using the canonical variational transition state theory (CVTST) of Pollak et al. with general potentials and memory friction kernels. Our proposed scaling hypothesis is obeyed in the limit of large bath correlation times. In the scaling region, the ABC rates obtained in the CVTST reduce to those of Grote and Hynes (GH) with corrections coming from the finiteness of the barrier height. In the core of the critical region, where a transition takes place from the strong coupling to the weak coupling region, these corrections give an improved bound on the critical amplitude. The results are illustrated in the particular case of a parabolic barrier with a perfectly reflecting wall (PRW) .

1. Introduction

A classic problem in the theory of rate processes is the activated barrier crossing problem studied by Kra-

mers [ 1 ] more than fifty years ago. In this problem, there is a particle caught in a potential well that can escape over a barrier by the process of thermal activa- tion by a heat bath. Kramers solved the problem in the low and the high damping limits using a memory free Markovian bath. Forty years later, the ABC problem was discussed for a bath having a memory by Grote and Hynes [ 21. These authors solved the problem in the low and high damping limits. Carmeli and Nitzan [ 31 independently solved the problem in the low damp- ing limit. More recently, Pollak, Grabert and H5nggi (PGH) [4] obtained a single rate expression for any memory friction and any potential with a parabolic barrier for all dampings within the weak coupling approximation.

Last year, we [5] discovered a remarkable scaling property of the escape rate in the high damping limit.

0301-0104/94/$07.00 8 1994 Elsevier Science B.V. All rights reserved SSDIO301-0104 (94)00027-g

We found that when the system crosses from the weak coupling energy diffusion (ED) regime, to the strong

coupling spatial diffusion (SD) regime, the change in

the power-law exponents can be described by a scaling

function. In analogy with critical phenomena [6,7],

this occurs in the limit of large bath correlation times

or high damping. We have verified that our scaling

hypothesis is applicable both in the GH and the PGH

theories. Meantime, Pollak and coworkers [ 8,9] have

continued to try to improve upon the strictly one-

dimensional PGH theory by proposing a variational

transition state theory (VTST) based on traditional multidimensional TST. They have shown that one can

get improved results for the ABC rate by varying the

dividing surface rather than restricting it to the top of

the barrier as is done for example in the GH and the

PGH theories. They have used this variational approach both in the canonical (CVTST) [ 81 and in the micro-

canonical ( p,VTST) [ 91 ensembles. In this paper, we

will focus on the canonical version of their theory.

366 S. Singh, G. W. Robinson / Chemical Physics 183 (I 994) 365-373

In the present paper, we apply the CVTST to a class of power-law potentials which have varying well fre- quencies and barrier heights. For the bath friction, we use a class of functions with different behaviors at long times. Our purpose is two-fold. First, we want to see if the scaling hypothesis is applicable to the CVTST. Sec- ondly, we wish to determine the dependence of the critical exponents and scaling functions on various par- ameters of the Hamiltonian. The potentials are obtained by adding a term of the form xzn to the purely parabolic potential. For n + CQ we get a parabolic barrier with a perfectly reflecting wall. We solve the problem using the Hamiltonian equivalent to the generalized Langevin equation [lo]. We then study the scaling regime in detail. We find that the CVTST obeys our scaling hypothesis [ 51 and gives results that are similar to the GH theory. A detailed summary of results in the scaling region for the PRW case with the standard exponential friction is given in Table 1. In section 2, we introduce our mode1 and, in section 3, we explain the scaling hypothesis and its results for the GH and the PGH theories following ref. [5]. In section 4, we give a summary of the CVTST, and in section 5, we present the CVTST solution to our mode1 and study it in the scaling limit. The last section contains our summary and concluding remarks.

2. Model

We use the power-law potentials previously studied by us [5] in the PGH formalism. The potentials are given by

G V(x)=E,-fw;x*+ - xb, ( 1 2n

n=2,3,4, . . . . x-CO, (1)

v(x)=E,-$o;x*, x,0, (2)

where the system coordinatex has unit mass, the barrier frequency is % and its height is L&. The quantity C, is a positive constant. We have restricted the values of II as shown for the following reason. Only for n > 1 does one get both a well-defined well and a barrier. For other values of n, one has infinite barriers and wells. It is easy to verify that the potential has a minimum at

4 ( ) l/(Zn-2)

x1=- - cl ’

(3)

and if the zero energy is chosen at the bottom of the well, the barrier height is given by

wfx: . (4)

The well frequency is given by

%=(2n-2)“*o,. (5)

Clearly for larger n, the well becomes narrower. The limiting case of the PRW arises if the limit n -+ 03 is taken for a fixed E,,. In general, the potential is deter- mined by the parameters n, a+, and E,, (or C,,) .

The bath memory friction is taken to be of the general form [5]

5(r) = (Y/r)g(t/r) 9 (6)

where r is a relaxation time such that for long times the memory friction becomes very small, becoming zero as T-) m. The damping rate y is the overall friction and is given by the integral Jr dt l(t). The relaxation time T and the damping rate y are related by the equation r= ay. It is convenient to study the problem for fixed y and (Y. The parameter a determines the overall strength of coupling between the system particle and the heat bath in different regimes. Large (Y corresponds to weak coupling and small (Y to strong coupling. As seen before [ 51 there is a critical value of (Y in between these two extremes, which corresponds to critical cou- pling. As shown by PGH, strong coupling always occurs in the SD regime, but weak coupling may be present in either the SD or the ED regimes. The GH and the CVTST theories have only an SD regime, but their large Q! regimes do correspond to weak coupling.

As we know from our previous work [5], in the relevant scaling region, one requires an expansion of the memory friction kernel g(t/r) for large r. We assume that

g(n)=g(0)-wIUIP9 IUI < 1, (7)

where the expansion is characterized by the parameters g(O), w and p. For p a positive integer, w is a positive constant that is related to the pth derivative of g( u) at u = 0. For all other p values, w is simply an amplitude. For the standard exponential case given by

S. Singh, G. W. Robinson/Chemical Physics 183 (1994) 365-373 367

g(u)=exp(-u), we get g(O)=l, w=l andp=l. Another interesting case is the Lee-Robinson [ 1 I] friction,

2 sinu g(u)= --

lr u ’

where g(0) =2/q w = 1/31r and p = 2. This general form of the friction allows us to explore the dependence of the critical exponents on various parameters of the bath friction namely g( 0) , w and p.

3. Scaling hypothesis

Scaling is expected to be valid for large T, which is the case either for large y or large (Y. ‘Ihe second pos- sibility is somewhat trivial, so here we concentrate on the case of large damping. It is well-known that for a large fixed y, when one goes from small to large a, the behavior of the escape rate changes qualitatively. We have shown [ 51 that the changes are dramatic close to cu,, where one sees critical phenomena and scaling. The critical point is located at (Y = oC, y - ’ = 0 and EC ’ =

0. Clearly, if E,, is truly infinite, the rate would drop to zero because of the Arrhenius factor. So the scaling hypothesis is formulated for the reduced rate, which is given by the rate divided by the TST rate. For simplicity and for the purpose of introducing scaling ideas, let us first discuss the GH case.

In this case, the term containing C, can be neglected in the potential ( 1). As shown by GH [ 21 the problem can then be solved exactly. The reduced rate r* = rGH/ r,,, is independent of Eb and is given by the GH equa- tion [2]

r*= 1

r*+&r*&l% ’ (8)

where g(s) is the Laplace transform of t(t) . Using the expansion (7), it is easy to show that for Cr*=Cyw2 - b - g( 0) and large y, we have three different behaviors.Forcu*>g(O) and y*=y/q,+m,weget

r* = [ 1 -g(O)/a*] 1’2. (9)

On the other hand, for (Y* <g(O) and y* + ~0, we get r* = 0. Clearly, the reduced rate can not be analytic at the point cy* = g( 0) and ( r*) - ’ = 0. We identify this singular point with a critical point and the phenomena

associated with it as critical phenomena. By expanding the GH rate near the critical point (Y,* =g(O) and ( y*) - ’ = 0, one can easily show [ 51 that the GH rate obeys the scaling equation

r*(o*, r*) =:Ai(y*) -“‘f(y) , (10)

where the scaled distance from the critical point, y, is given by

~=A~d(y*)“~, A=l- 2 (11) a

and the scaling functionf(y) is given by the transcen- dental equation [ 5 ]

V(Y)12-Y-Lf(Y)l-P=0.

The various exponents and amplitudes are

(12)

a, =pl(p+2) , u2 =2u, ;

Al = (,,,*) “(P+2), A2 ,A;2 ;

W *=wT(p+l)/((Y,*)p+’ ; (13)

where r(z) is the standard gamma function. The scaling description is valid asymptotically near

the critical point for any real value of the scaling param- eter y. Different values of y represent different paths to the critical point. For example the three main approaches to the critical point give us the following results. For fixed (Y < cu, and y * + CQ, y + - ~0, we have from ( 12), f(y) = ] y 1 -“j’. Substituting this in ( 10) and using ( 11) and ( 13), we get for the rate

r*~(~*)“~(~*)-‘~~~-“~~ (14)

This is the typical strong coupling Smoluchowski behavior. Similarly, for fixed a> cu, and y* + m, y + m, we have f(y) = y “2. This corresponds to the approach to the critical point from the weak coupling regime and we get for the rate

,*,A”2 , (15)

as seen above in (9). Finally, the pointy = 0 represents the limit when (Y has already been set equal to cu, and y becomes very high. We havef( 0) = 1 and the rate is given by

r*,(w*)ll(P+2)(y*) --Pl(p+21. (16)

Other values of y represent other approaches to the critical point.

368

Table 1

S. Singh, G. W. Robinson/Chemical Physics 183 (1994) 365-373

Comparison of the rate in the scaling limit, for the PRW potential and the exponential friction, with various theories for the case of moderate scaled barrier height, z. For high z, all theories reduce to the GH theory in the scaling limit. For explanation of symbols, see text

Different regimes

strong coupling A < 0

critical coupling A = 0 weak coupling A > 0

CVTST theory GH theory PGH theory

(r*)-‘I Al-’ (Y*)-‘IAl-’ (Y*)-‘IAl-’

(Y*)-“’ 3E:(y*) -’ A”2 2E:(y*) --I

It is seen above that all the exponents, whether they

To go beyond the GH theory one has to include the

occur in the definition of the scaled variables or in

determining the behavior of the reduced rate in various

term containing C,, in the potential ( 1) , which intro-

regimes, depend only on p. These exponents do not depend on w or g(0). This is an illustration of the well-

duces a finite well. Unfortunately the problem can no

known phenomenon of the universality [ 6,7] of critical exponents.

longer be solved exactly. We have studied [5] this problem in the PGH formalism. Guided by our calcu- lations, we proposed that for (Y near cu,, and for high ‘y, the exact reduced rate should obey the scaling hypoth- esis [5]

R*(a*, r*) =&(y*) -Q’Rl(~, z) , (17)

where y is the scaled deviation of (Y from its critical value cu,, as before, and z is the scaled barrier height,

z=A3E;( r*) -w, (18)

with EC = Eb/kB T. The details are given in ref. [ 51. Here we give only a summary of these results. In addi- tion to the exponents and amplitudes defined above in the GH case, we have

2Pn u3 = (p+2)(n- 1) ’

A, =A?““-” (19)

The scaling function R, (y, z) is given by

R,(Y, z) =f(y) Me(a) , (20)

where Me(6) is the Mel’nikov-Meshkov function

[5,121

Xln{l-exp[ -6(l+s’)/4]) . 1

(21)

Here 6 is the dimensionless energy loss of the PGH theory,

qy, z) = ipD,zf(y) -2-p+*n’(n-‘)

x1+ p ( 1

n/(n- I)

v(Y)p+* ’ (22)

where D,, is a complicated number depending on n [ 51. By studying these equations in various cases, one can obtain the asymptotic behavior of the rate. As stated previously, this was done in ref. [ 51. The results are summarized in Table 1 for the PRW case with expo- nential friction, where one can compare the scaling results with other calculations. We will discuss these results later when comparison with the CVTST is made.

Once again, as expected from the universality hypothesis, the critical exponents in ( 13) and ( 19) depend on very few parameters of the potential given by (1) and(2) andthefiictiongivenby (6) and(7). These exponents are functions of II, which describes the shape of the well, and p, which determines the behavior of the friction for long T. The exponents are identical for all systems with different E,,, a+,, w and g (0) as long as they have the same values of n and p.

4. Canonical variational transition state theory

The starting point of the CVTST [ 81 is the usual Hamiltonian approach to the escape rate. The Hamil- tonian of the system is given by

H=fp2+V(x)+ c *pi + il,[L * +(@i%- zr]p

(23)

where the bath coordinatesx, with conjugate momenta pi, are harmonic oscillators with frequencies wi. For convenience, we have chosen the usual mass-weighted coordinates. The coupling between the system coordi-

S. Singh, G. W. Robinson/Chemical Physics 183 (1994) 365-373 369

nate x and the bath coordinates xi is given by the coefficients cti The memory friction kernel 4((t) is determined by the coupling coefficients:

l(f) = ic $ COS(Uit) . L

(24)

By assuming the existence of a dividing surface F(xi, x) between the reactants and the products, and making the usual TST assumptions, one can write a formal expression for the rate. In general, a variational estimate of the rate will be obtained by varying the surface.

To obtain manageable analytical results, Pollak et al. [8] first defined a collective bath variable s and its momentum ps by the relations

s=%x+ i< aixi 3 Ps=aOP+ 2 aiPi9 (25) i=l

where the coefficients are normalized,

ai+ iaf=l. (26) i=l

Secondly, they assumed that the dividing surface is of the form

F(Xi, X) =X-D(S) 9 (27)

where D(s) is quite genera1 at this point. With these assumptions, they show that the multidimensional TST estimate for the rate can be obtained from an effective Hamiltonian involving only two degrees of freedom, namely, the original system coordinate x and the col- lective bath coordinate s. They then show how to vary the parameters in general for any potential to get the CVTST rate. Since our purpose is to show that the scaling hypothesis is applicable in their theory, it is not

necessary for us to take the most general approach. Instead we will focus attention on a restricted version of their theory. In this version, it is assumed that the potential is given by the expression

V(x)=E,-$o;x*+V,(x), (28)

where V, (x) is the nonparabolic part of the potential. Their second assumption is for the form of the divid-

ing surface. They assume that the quantity D(s) in eq. (27) is proportional to s. This choice is the optima1 one in the case of the purely parabolic potential. It is not the variationally optima1 choice for the full potential. However, for reasons given above, we will use this

choice. With these two assumptions, they find that the CVTST estimate for the reduced rate is [ 81

m

with

d.rexp[ -s*- PV,W)l , (29)

(30)

where /3= 1 lk,T and uc,, is a matrix element of the transformation that diagonalizes the purely parabolic part of the Hamiltonian. In the next section, we apply this theory to our potential given by ( 1) and (2).

5. The CVTST study

To apply the CVTST to our model, we see from ( 1) and (2) that the nonparabolic part V, (x) is given by

V,(x)= 2x%, x<o,

V,(x)=O, X&O. (31)

Using this in the CVTST rate expression (29)) we find that the rate is given by

R*=r*[MP”) +t1 9 (32)

with

0

L(Pn) = L_ c(

d.rexp( -sz-p,s”) , (33) lr -m

where

PG 2n pn=x(P . (34)

From (32), we can see that the rate is the GH reduced rate r* multiplied by a correction factor caused by the finiteness of the barrier height which enters through C,,. Because of our choice of the dividing surface to be the optima1 one for the purely parabolic case, the correction factor depends only on the parameters of that case.

The rate (32) is for general damping. Our object is to study it in the scaling (high damping) limit. In that limit we can use the results for the GH case from section 3. The GH rate r * in (32) can be written in the scaling

310 S. Singh, G. W. Robinson /Chemical Physics 183 (1994) 365-373

form ( 10) with the exponents and the amplitudes given in ( 13). As in the PGH case [ 51, the correction factor will involve another scaled parameter, the scaled bar- rier height z. Remarkably, it turns out that in the CVTST rate (32), one needs exactly the same z as for the PGH case. This z has already been given in ( 18) and ( 19). Therefore, using the relation (4) connecting Eb and C,, and the definition of z, one can express p. of eq. (34) in scaled form. To express rp of eq. (30) in the scaled form, one needs to know an additional result proved in ref. [ 51. This result is

( 1 -1

u&= 1+ p 2f(Y)P+2 .

(35)

Using all this, pn can be expressed as a function of y and z alone,

P”(Y,Z)= (n-l)“-‘f(y)-2”z*--n, n”

( 1 112

f(Y) =f(y) 1+ 2f(y;p+2 , (36)

where we have introduced another function f for later convenience. Therefore we see that the CVTST rate obeys the scaling hypothesis in the scaling limit with the scaling function given by

R, (Y, z) =f(v) 1 t +L(P, (Y, z)) 1 9 (37)

with the same exponents and amplitudes as the PGH case [ 51. Of course, the scaling function is quite dif- ferent. Compare the PGH scaling function given by (20)-( 22) with the CVTST scaling function given by (37), (33) and (36).

To obtain the CVTST rate in the whole of the scaling region, one has to carry out the integration in (33). Unfortunately, the integral Z,,(p,) cannot be performed analytically for general n except for n = 2 and n = tQ. In these two cases one finds

Z2(P2) = l -exp(l/8~2)K,,,(l/8~2) I 46

Z_(B) = f erf(j) , ji= lim (p;“‘“) , n+m

(38)

(39)

where Kl14(x) and erf(x) are the modified Bessel func- tion of the second kind and the error function, respec- tively. It can be shown using (36) that p’(y, z) = ?(y)z”2.

To study the behavior of the rate in different regimes, we do not need to evaluate the integral in general. We only need the behavior of the integral in the limits of small and largep,. This can be obtained using standard analysis. For small p., we simply expand the second exponential in (33) and integrate term by term using a tabulated integral (ref. [ 131, p. 337, eq. (3.461.2) ). We get

Z,(p,)=i (

1+ 5 (-l)‘(Znr-l)!!-& ) r=l )

Pn+O. (40)

Here the double factorial symbol stands for 1 x3x5x... X (2nr- l), as usual. For large p,,, we first make the substitution pns2* = f, expand the first exponential and again integrate term by term using another standard integral (ref. [ 131, p. 317, eq. (3.381.4)). Weobtain

UP”) = 1 m ( - l)‘r((2r+ 1)/2n)

c 2n&p;‘2” r=O r! pi’” ,

P”+m* (41)

Of course, for n = 2 and m, the corresponding results agree with (38) and (39).

With the above asymptotic results, it is straightfor- ward to study the CVTST rate in all cases. Of course, it should be kept in mind that all of the following results are strictly true only in the scaling limit. Our strategy in obtaining the behavior of the rate in various cases is as follows. We first decide how we are approaching the critical point, that is, what is kept fixed and what is varying. Then, we calculate how the scaling variables y and z behave in that limit, by using ( 11) and ( 18), respectively. Finally, using ( 12) and (36), we deter- mine how pn behaves so that we can look up the appro- priate result (40) or (41) to use in the case at hand. This result substituted in (37) tells us how the rate behaves in that case.

We first consider the GH limit in which the barrier height becomes extremely large while everything else is fixed. This makes z > 1 so that, by following the procedure outlined above, p,, = 0. The rate is given by

R *sr* (42)

S. Singh, G. W. Robinson / Chemical Physics 183 (I 994) 365-373 371

Here r * is, of course, the reduced GH rate. The negative corrections come into play when the barrier is not too high and the system senses the presence of the well. If, at this stage, one fixes (Y at a value < , > or = a,, while y* Z+ 1, one will get the GH rates (14), (15) and ( 16) in the strong, weak and critical coupling regimes, respectively. So in all of the scaling regimes, one gets the GH limit if the barrier goes to infinity before any other limit is taken. This behavior in the scaling region is to be contrasted with the usual behavior seen outside this region. For example, consider the PGH rate in the case y * * 1, which is clearly far away from the cri t- ical point. In this case the GH rate r * = 1. But we know that in this limit, the true result is given by the ED result where the rate is proportional to y*. Therefore, the GH result is never recovered in this ED limit.

Having disposed of the infinite z limit, now let us discuss the case of a finite but fixed z. We have three subcases depending on the sign of A, which is essen- tially the deviation of cy from its critical value cu,. In the strong coupling regime, where a! < 4, A is negative and y * X+ 1 making y < 0 and of a very large magni- tude. By following the strategy outlined above, we see that p,, = 0. This gives for the rate

R *=(,,,*)l~~lAl-i/~(~*) -1

(43)

which is again the GH rate with corrections. We know that in this regime, even far away from the critical point the rate approaches the GH rate. Therefore it is not surprising that it also does so in the scaling limit. In the weak coupling regime, where cy > a=, A is positive and y* Z+ 1 which makes y > 0 and very large. Again, we find thatp, -K 1 and we need to use (40). For the rate, we get

R*= lAl”2[l-O((E~~:-,)], (4)

which is once again the GH rate with corrections. Notice how the GH rate goes to a constant independent of the damping rate in this “frozen solvent” limit. As before, the corrections arise from the finiteness of the barrier height.

In the very core of the critical region, where cy = (Ye, A = 0 and y * Z+ 1 which makes y = 0. Here everything becomes independent of y to the leading order. For

example,p, a z --n. Two subcases can be discussed. We can fix y* and let Ez go to infinity, which will get us back to the GH limit as before in (42). The more interesting case arises if we fix Ez and let y* go to infinity. Then z < 1, pn > 1 and we have to use (4 1) .

In this case, we find

R* = (,+,*) 1/(P+2) (Y*) - p/(rJ+2)

(E3(4)/2n

( y*)P&J+2) ’ (45)

for fixed Ez. Thus, the exponent for this rate is the same as that for the GH result but the CVTST amplitude is half of the GH amplitude. Evidently, the leading exponent depends only on the form of the friction and not on the form of the potential. The exponent l/3 derived in ref. [ 81 is a special case of our result for p = 1, The fact that we have chosen a more general potential than Pollak et al. [ 8 1, who employed a quartic potential, is irrelevant for the value of the exponent. The GH theory is based on the parabolic approxima- tion, which gives an upper limit to the true rate. Since the CVTST takes into account the nonparabolic part of the potential, it is not surprising that it gives an improved estimate for the amplitude. By improving the variational dividing surface, it should be possible to get an even better amplitude, provided that the exponent in (45) is exact. In summary, one can see from (42)- (45 ) that the CVTST rate is similar to the GH rate with corrections caused by the finiteness of the barrier. It is only in the corrections that one sees the dependence of the CVTST rate on n, the parameter that determines the shape of the well.

To illustrate these results and to compare them with other theories, we choose, for simplicity, the case n + 03, the PRW case. In addition, we choose an expo- nential type of memory friction fp= 1). In this case, g(0) =w=p= 1. Therefore, ai = l/3, n2=2/3, a3=2/3;A, =A2=A3= 1. Theresultsaredisplayedin Table 1. Again, on looking at these results, one should keep in mind that they are valid only in the scaling region. We can see that in the SD or the strong coupling regime, all theories give the same result. This is because in this case only the barrier dynamics plays a significant role, and this is treated correctly by all of the theories from GH onward. In the weak coupling regime, the well dynamics plays a dominant role, so that, only the PGH theory treats this region correctly. However, in

312 S. Singh, G. W. Robinson / Chemical Physics 183 (1994) 365-373

the core regime, the PGH theory gives both an exponent and an amplitude that are in disagreement with the other two theories. In contrast with the PGH theory, the GH and the CVTST theories give the same exponent in this regime. However, they give different amplitudes. We note that the amplitude decreases from 1 in the GH case and 0.5 in CVTST. We therefore believe that the true rate in the scaling limit is likely to agree with the GH and the CVTST theories as far as the exponent is con- cerned, but should’ give an amplitude lower than 0.5, the CVTST amplitude.

6. Summary and concluding remarks

In this paper we have shown that the scaling hypoth- esis is valid for the rate estimate obtained from the CVTST theory for general xzn potentials and general memory frictions. We have already shown the hypoth- esis to be applicable to the GH and PGH theories [ 51. We believe that the hypothesis will be applicable in more general cases, such as space-dependent friction [ 141, models including quantum effects [ 151, low bar- rier problems [ 161, anisotropic friction [ 171, etc. Of course, in analogy with critical phenomena, we expect that each new relevant parameter will have to be scaled with a new exponent, and that each new parameter brings in its own crossover effects [ 6,7] in the scaling region. The following scenario is typical of such cross- over effects. Far away from the critical point, 1 / y * = 0, (Y= CX~, the system behaves as if the new parameter is absent and one sees the old exponents and amplitudes. As one approaches the critical region, one begins to see a change in exponents or amplitudes or both. Finally, very close to the critical point, the behavior of the system crosses over completely to the one governed by the new parameter. The width of this cross- over region is not universal and is determined by the quantities associated with the new parameter. In this connection, we might mention that we have already begun work [ 181 on the quantum version of the ABC problem following the paper of Rips and Pollak [ 151. Our preliminary findings are that (i) the Planck’s con- stant scales with an exponent pl(p+ 2), (ii) new crossover effects are present when quantum corrections can not be neglected.

We believe that the time is now ripe for experimen- talists to look for critical exponents in their rates and

see whether the scaling hypothesis is applicable to real physical systems. Normally, comparison between the- ory and experiment is very difficult because the theo- retical models, by their very nature, present a very restricted and approximate picture of the real world. But the principle of universality in critical phenomena can come to the rescue here. According to this principle,

the critical exponents for a given physical system depend on very few relevant parameters. For example, critical behavior in liquids, liquid mixtures, binary alloys, uniaxial ferromagnets is the same as that of the nearest-neighbor scalar spin Ising model. All of these systems pose complicated many-body problems and their behaviors are vastly different away from the crit- ical or the scaling region. However, the critical expo- nents and scaling functions, which are applicable in the scaling region, are all identical. This has been amply verified by numerous experiments in all these different

systems [ 61. In the ABC problem, we have demonstrated the

validity of universality. Our analytical calculations, using general potentials of the form ( 1) and (2)) and memory frictions of the form (6), based on the GH [ 51, PGH [ 51, and the CVTST results described in the present paper have shown that the exponents depend only on two parameters, n and p. The first one deter- mines the shape of the well; the second one determines the short time behavior of the memory friction kernel. A somewhat similar conclusion has been reached by Tucker [ 191 in a recent numerical study of the escape rate using the PGH formalism. She used a cubic poten- tial and a variety of frictions. Her conclusion was that the ABC rate depends primarily on the damping rate and the memory friction time. Unfortunately, the mem- ory friction time used in her paper was not long enough to see scaling and universality. The same situation exists in other simulation studies: Tucker et al. [ 91 and Frishman and Pollak [ 161. In ref. [ 91, the memory friction time was not too large and the rate at the critical coupling was not calculated. Nevertheless, in the case closest to this limit, the ratio of the VTST rate to the GH rate turned out to be 0.6. As seen in Table 1, this ratio has approached the value 0.5 for critical coupling. Numerical studies in the scaling region on this model should prove to be extremely interesting.

Another task that needs to be accomplished is the reformulation of the rate problem as a renormalization group problem [20]. This approach was initially

S. Singh. G. W. Robinson /Chemical Physics 183 (1994) 365-373 373

designed for the scaling region, but is known to give excellent numerical results even far away from this region. This will help provide rates for realistic poten- tials to a reasonable numerical accuracy. In any case, the casting of the ABC problem as a problem in critical theory allows the systematic mathematical dissection of the rate problem, as illustrated here and in ref. [ 51, and should lead to many new approaches and new results.

Acknowledgement

Financial support at SPQR laboratory is provided by the Welch Foundation and the National Science Foun- dation.

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