Mean exit times over fluctuating barriers

Volume 136, number 7,8

PHYSICS LETTERS A

17 April 1989

MEAN EXIT TIMES OVER FLUCTUATING BARRIERS D.L. STEIN Department ofPhisles, University ofAri:ona, Tucson, AZ 85721, USA

R.G. PALMER Department of Phi’sics, Duke University, Durham, NC 27706, USA

LL. VAN HEMMEN Universität Heidelberg, Sonderforschungbereich 123, 6900 Heidelberg 1, ERG

and Charles R. DOERING Department ofPhvsics, Clarkson University, Potsdam, NY 13676, USA Received 22 January 1989; accepted for publication 1 February 1989 Communicated by A.R. Bishop

We investigate the problem of thermal activation over a fluctuating barrier. Three regimes are considered: the fluctuations slow compared to the mean crossing time tA of the average barrier height, fluctuations on roughly the same timescale as ~. and fluctuations extremely fast compared to ~A• In the latter two cases, the mean barrier crossing time is reduced. The relevance ofthese results to a variety of problems in complex systems is discussed.

The problem of the dynamical evolution of highly constrained, strongly interacting many-body systems with multiple locally stable states occurs in a wide variety of contexts, and has often served as a unifying framework for studying the behavior of physically diverse systems. Some examples include slow relaxation in glasses [1—4],chemical kinetics in large biomolecules [5—7], and biological evolution [8— 121. All involve barrier crossing in some form, and many display complex behavior such as nonexponential relaxation in time or non-Arrhenius temperature dependence ofcrossing rates. In some sense this is not surprising in view of the multiple barrier crossings needed to achieve relaxation of some perturbation; however, theories based on multiple hops over static barriers have not satisfactorily explained may of the observed phenomena. There is an additional complication, which in fact is crucial in determining the complex behavior of

these systems their dynamics is governed by many relevant timescales, from very fast to very slow, and often no obvious timescale gaps exist so that one can separate the “fast” from the “slow” degrees of freedom. It therefore makes sense to expect that during a typical barrier crossing event the barrier itself does not remain static; it will likely be modulated by some strongly coupled degrees of freedom which are changing on a timescale comparable to the crossing time itself. In such cases one must consider the problem of activation over a fluctuating barrier. Some simple examples may clarify the picture we have in mind. In many cases, the difficulty arises from constraints [131 e.g., in a glassy liquid, a given atom A may be blocked from relaxing or diffusing because of the presence of other atoms B, C, “in the way”. When enough of these atoms happen to move away, opening up a channel or lowering a barrier, atom A is able to move. This can actually be

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seen explicitly in a short movie based on molecular dynamics simulations of the disrupted-tetrahedral silicate Na~O3SiO2over the timescale range lO—~ to 10’’ s produced by Angell and Cheeseman [14]. One notices from the simulation that the relatively small sodium ion is normally unable to diffuse because it is trapped in a small region controlled by a “gate” of larger oxygen ions. Occasionally, thermal fluctuations result in the oxygen ions moving relalively far apart for a short period of time, enabling the sodium ion to diffuse through. It is interesting to note that a similar gating process controls the flow of small ligand molecules (0.. CO) into large biomolecules such as myoglobin [15. 161. and has applications to interstitial diffusion in solids [171. On can picture this process in another way. The larger “gate” ions present a barrier to the smaller diffusing ion. Because of the physical origin of the barncr, its height is fluctuating randomly due to thermal noise. Jumps over the barrier are more likely when the barrier is relatively low, but the nature of the barrier crossing depends on relative barrier height, size of fluctuations, correlations between fluctuations, etc. Hence, the effect of constraints can be incorporated in a natural and quantitative way by considering a model of barrier crossing where the Itarrier is fluctuating randomly due to thermal noise ci In this Letter we will examine the mean exit time in the simplest possible case of a single fluctuating barrier. Clearly, this will not be sufficient to describe the complex dynamical behavior of glasses, for cxample, in which one needs to self-consistently incorporate many barrier crossings such that the suecessful crossing of one barrier may strongly influence the fluctuations of another (since the barriers themselves are controlled by the local atomic arrangements). We will discuss here only some simple cases which illustrate the general features of the problem, and leave a more detailed study to a report to be published elsewhere, We consider an ensemble of particles which at time zero are trapped in a potential well. The average •

.

.

Anderson and UlIman [181 ha~e ln\estigated a problem along similar lines, in which molecular relaxation is affected by a fluctuating ‘free volume within the context of the free volume thcor\ of glasses. We are grateful to Dr. J.L. Skinner for bringing this work to our attention.

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height of the barrier to be crossed is A, but the barncr is also fluctuating with an amplitude that in general will be temperature-dependent. The correlation time t~of the barrier fluctuations will be an important parameter in the problem, and may also be ternperature-dependent. perhaps even activated, though for now we simple treat it as an independent pararneter. Therefore the noise due to barrier fluctuations is colored. It is natural to model the process as stationary, Markovian. and Gaussian: by Doob’s theorem [191, the fluctuations therefore correspond to an Ornstein—Uhlenbeck process. The barrier fluetuations are controlled by the process ~(t) defined by the stochastic differential equation 2/T. ‘7 (1) (I) f= ~(i) + \. where t~ 1(t) is 6-correlated white noise, and all times will be dimensionless. Hence, the stationary distribution of ~ is .

—

~—

~

P(~)=

(2)

2it .

,.

its time average (ç(t)> =0. and its autocorrelation .

.

function is (~~= 1 (~(I )~(I ) > = e —

.

(3)

We shall therefore investigate the mean exit time for crossing of a single barrier given by AF=A + ~( 1) (4) ~

.

where A and B have dimensions of energy and the behavior of~/t)is governed by eqs. ( l)—(3). While other prefactors (which set the amplitude scale of the fluctuations) of ~(t) can be considered, the \/ 7’/i~ term is both simple and plausible: the ternperature dependence can be gotten from a simple model of “gate” controlled by atoms resting in parabolic wells and undergoing thermal fluctuations, and the t~ dependence is necessary to insure a sensible white noise limit. We will examine three cases of interest: the correlation time t. extremely fast compared to all other intrinsic timescales in the problem (the white noise limit), r~extremely slow compared to all other timescales, and a regime intermediate between these two. Consider first the case where r~is extremely large

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compared to all other timescales. There are three of these in the case where the fluctuations vanish Ta’ Tr cor(B= 0); we shall denote them by rr, tg’ and responds to the attempt time for particles near the bottom of the potential well and Tg to the transit time for particles near the top of the well; in the high-friction limit these times are governed by the curvatures near the top and bottom of the potential well. Ta corresponds to the mean exit time; ifa particle typically Tg’

stays tin the for times much longer than Tr or then 131,well where /3=l/T. If r~ is long compared 1—e to all of these times, then the problem reverts to that ofan ensemble of static barriers with a Gaussian distribution of barrier heights (assuming that the barrier evolves on a timescale long compared to i~ before the crossing process begins.) One then recovers a static distribution of relaxation times, leading to non-exponential decay, and an increased effective barrier height, due to very large contributions to the average exit time from upward fluctuations of the barrier potential near the maximum. We therefore turn to the more interesting cases where r~is shorter than the average exit time when B=o. The first of these is the intermediate regime where tr’ Tg<> ~r, Tg;(B=0) here N(t) is theN(t)=N(0)e’~, number of particles maining in the well after time t. Under these circumstances we may write, for B~0(we will always assume that B<

/N(t)\ =jr ~

=5 0

dt

17 April 1989

the barrier has large downward fluctuations, we need to examine that situation more closely. Eq. (5) assumes that the rate variable k(t) —~e~1’~. When ~(t) becomes large and negative, k(t) becomes large and positive. This is unphysical; the peak rate occurs at ~ where escape over the barrier is diffusive rather than activated. What is the contribution to the integral from the regime çc<~ 0?The fraction of particles lost during these unphysically fast transitions is of order —~——

5

d~e~2{I —exp[ —exp(~/’~7~. r~~) ]}.

Since this is close to ~ for /1 large, we are losing one half of our particles during unphysically fast transitions. However, the temperature dependence of this expression is small; d/d/3 of the above expression vanishes extremely rapidly asfl—*cc. Hence, eq. (5) is useless for calculating the prefactor of the escape rate, but can be used to find the effective barrier /Jlog as long as flBr~>1. Averages such as that occurring in eq. (5) can be computed by re-expressing the integral in the harmonic oscillator formalism [20,21] <~> = , —

(6)

where 1

~ •

=

ye

—

~

d dx-

1 2r~

(7)

~ and (8)

Since /JA>> 1, we can use standard first-order perturbation theory to find =y’e1~4~ (9) or an effective barrier of

(exp(_Jeut~’it )dt))dt,

(5)

0

where txF is given by eq. (4). The derivation of eq. (5) relies on breakingup the time into intervals small compared to r~so that ~.Fis piecewise constant, and assuming that the chance of escape in an interval is small. Since this is clearly false for intervals where

~

(10)

The effectivebarrier is reduced, which perhaps is not surprising in view of the opportunities for barrier crossing when the barrier is relatively low. Finally, we study the white noise limit (z~—*0), where the barrier is fluctuating rapidly compared to all other timescales in the problem. Diffusion across 355

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the barrier in the high friction limit is governed by the Langevin equation ~(t) W’ (x) I [V (x)+~/~7~ —

+\/2Tt/~(t), where

(11)

2(t) is 6-correlated white noise, and primes denote derivatives with and respect to x.theV(x) represents theofstatic potential well W(x) fluctuating part the barrier. In the white noise limit, adiabatic elimination of the fast variable ~ leads to the Fokker—Planck equation for the probability density function p(x, I): I)

=

(v’ (x) +

-~-

+TW’(x) ~

W’(x))P(x, I)

.

(12)

The average exit time from x=0 to x=~ is then easy to compute [22]: 1 ~

=

To get a quantitative feeling for the amount of the barrier reduction, we studied the simple case where V(x)=~x2— ~x3, W(x)=De~”~ :, (15) We confined our study to a> 3 so that W(x) had

is governed by eq. (1), p

~(t)

p (x,

17 April 1989

I

~

significant curvature over the range 0
fluctuating barrier in the fast, intermediate, and slow regimes. We find that if the barrier fluctuates on a timescale short compared to the crossing time of the corresponding static problem, the mean exit time is reduced; if the fluctuations are sufficiently slow, the mean exit time is increased. The amount of decrease depends on the correlation time of the fluctuations: if these are temperature dependent, as one might reasonably expect, the crossing behavior will be nonArrhenius. This possibility is currently under study, as is the case where r~is itself affected by the crossing

xexp(

x

5

1j dy ~ V’ (y)(~,)2)‘\

this will enable us to examine both the time- and dz \/Ti~~2

temperature-dependence of the barrier crossing 0.02

-

xexp(

/

event. We eluding theare distribution also investigating of escape the times, full problem, for all inr~

~Jd

——

V’(u) u13/()2).

~

(13)

0

The exit time given by eq. (13) looks very much like that of the standard Kramers problem, but with an “effective” potential

001

Jdy(

d2F

1 V~(~_+iTlog[1+w~(y)2]) +W’(y) The second term is negligible at low temperatures, and the usual steepest descents analysis yields an Arrhenius temperature dependence of the mean exit time with the effective potential ~(x)

5

V

(~

2

(14)

dy ~ (y) Once again, the effective barrier is always reduced.

0,00

0 1

0

0.3

2F=~Fs

~ from eq. (15) and ~ = 10.d The effective barrier reducFig. I. =Effective barrier reduction 1a1,~AF1o1ai. where ~~Iat,c

= 0

356

0.2

tion is proportional to to ~/i~in eq. (4).

D2.

Note that D in eq. (15) corresponds

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problem. The results from these studies will be reorted elsewhere p This work was initiated when three of us (RGP, JLvH, and DLS) were visitors at the Institute for Theoretical Physics at Santa Barbara. DLS acknowledges the hospitality of the Physics Department of Clarkson University and CRD that of the Center for the Study of Complex Systems at the University of Arizona, where the present work was completed. One of us (DLS) is an Alfred P. Sloan Fellow.

References [I] CA. Angel! and W. Sichina. Ann. N.Y. Acad. Sci. 279 (1976)53. [2] 1. Wong and C.A. Angel!. Glass: structure by spectroscopy (Dekker, New York, 1976). [3}R.G. Palmer and D.L. Stein, in: Relaxations in complex systems. eds. K.L. Ngai and G.B. Wright (U.S. Government Printing Office, Washington DC 1985-461-700/20001, 1985) p. 253. [41C.A. Angell, in: Relaxations in complex sysiems, eds. K.L. Ngai and G.B. Wright (U.S. Governmeni Printing Office, Washington DC 1985-461-700/20001, 1985) p.3. [51I-I. Frauenfelder and P.G. Wolynes, Science 229 (1985) 337. [6] DL. Stein, Proc. Nail. Acad. Sci. USA 82 (1985) 3670.

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[7] A. Ansan, J. Berendzen. S.F. Bowne, H. Frauenfelder, lET. Iben, T.B. Sauke. E. Shyamsunder and R.D. Young. Proc. NatI. Acad. Sci. USA 82 (1985) 5000. [81 P.W. Anderson, Proc. Nail. Acad. Sci. US.A 80 (1983) 3386. [9] D.L. Stein and P.W. Anderson. Proc. Nat!. Acad. Sci. USA 81(1984)1751. [10] S.A. Kauffman. J. Theor. Biol. 22 (1969) 437.

[111G.Weisbuch. CR. [12]C.M.

Acad. Sci. 298 (1984) 375. Newman, i.E. Cohen and C. Kipnis, Nature 315

(1985) 400.

113] R.G. Palmer, D.L. Stein, E.

Abrahams and P.W. Anderson, Phys.. Rev. Lett. 53 (1984) 958. [141CA. Angel! and P.A. Cheeseman. movie shown ai ITP. 1987: C.A. Angell, CA. Scamehorn. C.C. Phifer, R.R. Kadiyala and PA. Cheeseman. Phys. Chem. Miner. 15 (1988) 221. [151 D. Beece, L. Eisenstein, H. Frauenfelder, D. Good. M.C. Marden. L. Reinisch, A.H. Reynolds. L.B. Sorensen andK.T. Yue. Biochemistry 19 ( l980) 5147. [161 M. Karplus and J..A. McCammon. CRC Crit. Rev. Biochem. 9(1981)293. [171 A.M. Stoneham, Ads’. Phys. 28 (1979) 457. [l8]J.E. Anderson and R. UlIman, 1. Chem. Phys. 47 (1967) 2178. [191 N.G. van Kampen, Stochastic processes in physics and chemistry (North-Holland, Amsterdam, 1981). [201 B. Simon, Functional integration and quantum physics (Academic Press. New York. 1979) theorem 4.7. [21 ] J.L. van Hemmen and K. Rzi~ewski, Phys. Rev. A 28 (1983) 474. [22] C.W. Gardiner, Handbook of stochastic methods. 2nd Ed. (Springer, Berlin, 1983) section 5.4.

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