JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 23 15 JUNE 2000

Harvesting thermal fluctuations: Activation process inducedby a nonlinear chain in thermal equilibrium

Ramon Reigadaa) and Antonio Sarmientob)

Department of Chemistry and Biochemistry 0340, University of California, San Diego,La Jolla, California 92093-0340

Aldo H. Romeroc)

Department of Chemistry and Biochemistry 0340, and Department of Physics, University of California,San Diego, La Jolla, California 92093-0340

J. M. SanchoDepartament d’Estructura i Constituents de la Mate`ria, Universitat de Barcelona, Avda. Diagonal 647,08028 Barcelona, Spain

Katja Lindenbergd)

Department of Chemistry and Biochemistry 0340, and Institute for Nonlinear Science,University of California San Diego, La Jolla, California 92093-0340

~Received 6 August 1999; accepted 23 March 2000!

We present a model in which the immediate environment of a bistable system is a molecular chainwhich in turn is connected to a thermal environment of the Langevin form. The molecular chainconsists of masses connected by harmonic or by anharmonic springs. The distribution, intensity, andmobility of thermal fluctuations in these chains is strongly dependent on the nature of the springsand leads to different transition dynamics for the activated process. Thus, all else~temperature,damping, coupling parameters between the chain and the bistable system! being the same, the hardchain may provide an environment described as diffusion-limited and more effective in theactivation process, while the soft chain may provide an environment described as energy-limited andless effective. The importance of a detailed understanding of the thermal environment toward theunderstanding of the activation process itself is thus highlighted. ©2000 American Institute ofPhysics.@S0021-9606~00!00723-6#

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I. INTRODUCTION

The search for mechanisms that may induce the sponeous localization of vibrational energy in molecular mateals has surfaced in a variety of contexts where such localenergy may then trigger other events. These may inclswitching and other threshold phenomena, chemical retions, local melting and other deformational effects, and edetonation. In the Kramers problem1,2 a particle moving in abistable potential is used as a model for a chemical procThe trajectory of the particle is associated with the reactcoordinate~RC!. One well of the bistable potential representhe ‘‘reactant’’ state, the other the ‘‘product’’ state, and seprating them is the ‘‘activation barrier.’’ The bistable potentiis connected to a thermal environment, typically throufluctuating and dissipative terms, and every once in a whilarge thermal fluctuation causes the particle to surmountbarrier and move from one well to the other. The averarate of occurrence of these events is associated with theaction rate. This mesoscopic Langevin-type of approach

a!Permanent address: Departament de Quı´mica-Fısica, Universitat de Barcelona, Avda. Diagonal 647, 08028 Barcelona, Spain.

b!Permanent address: Instituto de Astronomı´a, Apdo. Postal 70-264, CiudaUniversitaria, Me´xico D. F. 04510, Me´xico.

c!Present address: Max-Planck Institut fu¨r Festkorperforschung, Heisenbergstr. 1, 70569 Stuttgart, Germany.

d!Electronic mail: klindenberg@ucsd.edu

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mits of an underlying microscopic description of the thermenvironment and its coupling to the bistable system. Forstance, the usual Langevin equation with an instantanedissipation and Gaussiand-correlated fluctuations can be derived from a picture in which the system is harmonicacoupled to an infinite number of harmonic oscillators withuniform spectrum. A generalized Langevin picture involvindissipative memory terms and correlated fluctuations is aciated with a more complex spectrum.3 It is clear, and hasbecome a topic of considerable interest, that the nature oenvironment and its coupling to the bistable system pfoundly influence the transition rate.

A different but related set of problems that has attracintense interest in recent years concerns the spontaneoucalization of vibrational energy in periodic nonlinear arrayThe pioneering work of Fermi, Pasta, and Ulam4 demon-strated that a periodic lattice of coupled nonlinear oscillatis not ergodic, and that energy in such a lattice may nevedistributed uniformly. A great deal of work has since folowed in an attempt to understand how energy is distribuin discrete nonlinear systems.5–17 The existence of solitonsand more generally of breathers and other energy-focumechanisms, and the stationarity or periodic recurrenceeven slow relaxation of such spatially localized excitatioare viewed as nonlinear phenomena with important conquences in many physical systems.13,18,19The search for lo-

5 © 2000 American Institute of Physics

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10616 J. Chem. Phys., Vol. 112, No. 23, 15 June 2000 Reigada et al.

calization mechanisms that are robust even when the arare in a thermal environment13,15,16,20has, on the one handnarrowed the problem~because some localization mechnisms are fragile against thermal fluctuations! but on theother hand broadened it~because new entropy-driven locaization mechanisms become possible!. Thermal effects maybe particularly important in biophysical and biochemical aplications at the molecular level.21–23

The interest in the distribution and motion of energyperiodic arrays arises in part because localized energythese systems may bemobile, in contrast with systems wherenergy localization occurs through disorder. Localizedergy that moves with little or no dispersion may appearone location on an array and may then be able to movanother where it can be used in a subsequent process. Ttional harmonic models suffer from the fact that dispersthwarts such a mechanism for energy transfer. Therebeen a surge of recent activity in an attempt to understthe thermal conductivity of nonlinear chains.24–26

The connection between the study of perfect nonlinarrays and the Kramers problem arises because such amay themselves serve as models for a heat bath for osystems connected to them.27–29 Albeit in different contexts,‘‘perfect’’ arrays serving as energy storage and transfersemblies for chemical or photochemical processes areuncommon,30–33 and literature on the subject goes backtwo decades.34–36 We thus consider the following variant othe Kramers problem: a bistable system connected to alinear chain, which is in turn connected to a heat bath inusual Langevin manner~see Fig. 1!. The bistable system isonly connected to the environment through its embeddingthe nonlinear chain, and therefore the ability of the chainspontaneously localize thermal energy and perhaps to trport it to the location of the bistable system can profounaffect the transition rate. We investigate the behavior of tmodel for different types of anharmonic chains and there

FIG. 1. Schematic of a bistable impurity~‘‘reaction coordinate’’ RC! con-nected to a chain that interacts with a heat bath at temperatureT. The chainmasses are indicated by rhombuses. Their interactions~shown schematicallyas springs! may be harmonic or anharmonic. Each mass in the chaisubject to thermal fluctuations denoted byh and the usual accompanyindissipation. The bistable impurity interacts only with the chain, which thprovides its thermal environment. The bistable system is inserted inchain; its detailed interaction with the chain is discussed in the text.

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establish the important role of the nature of the environmon these chemical model systems.

In Sec. II we discuss the energy landscape typicalvarious nonlinear chains in thermal equilibrium. In Sec.the variant of the Kramers problem wherein a bistable stem is connected to each of the different chains is presenIn Sec. IV our results for the transition statistics in thbistable system are detailed. We compare and contrastransition statistics in the different chains and compare thto those found in the standard~Markovian! Kramers and gen-eralized Kramers problems. We conclude with a summand some notes on future directions in Sec. V.

II. NONLINEAR CHAINS

The simplest nonlinear periodic arrays consist of masconnected by springs that may be harmonic or anharmoThe masses may also experience a local harmonic or anmonic potential. In a recent paper we presented a detaview of the thermal landscape of arrays with local hard~the‘‘ f4 model’’!, harmonic, or soft potentials and harmoninteractions.20 Here we present the complementary analy~more interesting, it turns out, in the context of the Krameproblem! of the thermal landscape of masses connectedanharmonic springs~with no local potentials!.

The spontaneous localization of energy in any systemthermal equilibrium is simply a reflection of the thermal flutuations described by statistical mechanics and is unrelto system dynamics. On the other hand, the way in whthese fluctuations dissipate and/or move and disperse, ththe temporal evolution of thermal fluctuations, is dictatedthe system dynamics and, in particular, by the channels cnecting the chain to the thermal environment~dissipation!and the masses to one another~intermolecular interactions!.

We pose the following questions:~1! How is the energydistributed in an equilibrium nonlinear chain at any givinstant of time, and how does this distribution depend onanharmonicity? Can one talk aboutspontaneous energy localization in thermal equilibrium, and, if so, what are thmechanisms that lead to it?~2! How do local energy fluctua-tions in such an equilibrium array relax in a given oscillatoAre there circumstances in the equilibrium system wheregiven oscillator remains at a high level of excitation forlong time?~3! Can local high-energy fluctuations movesome nondispersive fashion along the array? Can an arrathermal equilibrium transmit long-lived high-energy fluctutions from one region of the array to another with little dipersion?

In our earlier work20 we showed that in harmonicallycoupled nonlinear chains~‘‘diagonal anharmonicity’’! inthermal equilibrium, high-energy fluctuation mobility doenot occur beyond that which is observed in a harmonic chaHerein we show that the situation might be quite differenthere is ‘‘nondiagonal anharmonicity,’’ that is, if thinteroscillator interactions are anharmonic.

Our model consists of a one-dimensional array ofN unit-mass sites, each connected by a potentialV(xn2xn61) to itsnearest neighbors that may be harmonic or anharmonic:

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10617J. Chem. Phys., Vol. 112, No. 23, 15 June 2000 Harvesting thermal fluctuations

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We assume periodic boundary conditions and consider tprototype potentials:

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At small amplitudes the three potentials are harmonic wthe same force constantk. The independent parametersk andk8 allow control of the harmonic component and the degof anharmonicity of the chain. Elsewhere37,38we have arguedthat the overarching characteristic of anharmonic oscillais the dependence of frequency on energy. For a harmoscillator the frequency isAk independent of energy; forhard oscillator the frequency increases with energy, and fsoft oscillator the frequency decreases with energy.

The set of coupled stochastic equations of motion formasses is that obtained from the Hamiltonian, Eq.~1!, aug-mented by the usual Langevin prescription for couplingsystem to a heat bath at temperatureT:

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]xn@V~xn112xn!1V~xn2xn21!#2g xn1hn~ t !, ~5!

where a dot represents a derivative with respect to time.hn(t) are mutually uncorrelated, zero-centered, Gaussd-correlated fluctuations that satisfy the fluctuationdissipation relation hn(t)h j (t8)&52gkBTdn jd(t2t8). Thenumerical integration of the stochastic equations for allsimulations is performed using the second order Heumethod~which is equivalent to a second order Runge Kuintegration!39,40 with time stepDt50.005. In each simulation the system is initially allowed to relax for enough itertions to ensure thermal equilibrium, after which we take o‘‘measurements.’’

The equilibrium results to be presented here complemour observations, presented elsewhere, on the way in wthese same chains propagate an energy pulse38 as well as asustained signal applied at a particular site.41

A set of energy landscapes is shown in Fig. 2. Alonghorizontal direction in each panel lies a thermalized chainoscillators; the vertical upward progression shows the evtion of this equilibrium system with time. The gray scarepresents the energy, with darker shading reflecting menergetic regions.

Several noteworthy features are evident in the figuThe energy fluctuations are greatest in the soft chain. Tfeature, seen earlier in chains with local anharmopotentials,15,20 is a consequence of the effect that we hacalledentropic localization. In the soft chain not only are ththermal fluctuations greater at a given temperature, a reeasily obtained from a simple virial analysis, but the frenergy is minimized by a nonuniform distribution of enerthat populates regions of phase space where the densistates is high. We have argued that this localization mec

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nism is robust against temperature increases—indeed, itcomes more effective with increasing temperature. A secdistinctive feature of the soft chain is the persistence ofenergy fluctuations: damping is not particularly effective fsoft chains. The only other mechanism for the removallocalized energy from a particular location is along the chaThis is clearly not an effective mechanism, a result that isagreement with our analysis of the propagation of an exnally applied pulse in the soft chain.38 The speed of propagation ~in all chains! of a pulse of a given energy is essetially proportional to the average frequency associated wthat energy, and in the soft chain this average frequedecreases with increasing energy.38 Although we do not seean obvious connection between these excitations and soliat zero temperature~which are not entropic localizationmechanisms!,10,17,31,33–35there may be a closer connectiowith more generalized excitations such as breathers.13,16,21

In the hard chain~Fermi–Pasta–Ulam chain! the totalenergy as well as the energy fluctuations are consider

FIG. 2. Energy~in gray scales! for thermalized chains of 71 oscillators asfunction of time. Thex-axis represents the chain and time advances althey-axis, withtmax5160. The temperature iskBT50.08 and the dissipationparameter isg50.005. Top panel: hard chain withk50.1, k851; middlepanel: harmonic chain withk50.1; lower panel: soft chain withk50.1,k855.

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10618 J. Chem. Phys., Vol. 112, No. 23, 15 June 2000 Reigada et al.

smaller but quite mobile with little dispersion. In the hachain the average frequency increases with energy and thfore more energetic pulses tend to travel more rapidly.have also shown that the dispersion of energy in a hard cis slow38—this is seen here in the integrity of the spontanously localized pulses over a much longer time than inharmonic chain.

An important question of course concerns the paramregimes where the differences illustrated in Fig. 2 areserved. We have chosen potential parameters that enclear distinctions in the displacement amplitudes associwith the three potentials at the chosen temperature. Therestriction is that the temperature not be ‘‘too low,’’ thatwe avoid the region where all three potentials are essentharmonic. We have chosen very low damping for the illutration. The soft energy landscape is far less sensitivedamping than the hard array. An increase in damping woreadily slow down the motion of the high-energy fluctuatioand would shorten their lifetime. Further, while the speedthe energy fluctuation pulses is sensitive to the potentialrameters, their lifetime and dispersion properties are les~as long as one is in the highly anharmonic regime!. On theother hand, the persistence of the fluctuations in thearray is quite sensitive to the harmonic contribution topotential. All else remaining fixed, the landscapes remqualitatively similar as temperature increases: the flucttions in the soft array become even stronger relative toothers, and the pulse speeds in the hard array becomehigher.

Suppose now that a bistable ‘‘impurity’’ is embeddedeach of these chains, as illustrated in Fig. 1. When sufficenergy reaches the impurity, a transition may occur fromwell to the other. The statistical and dynamical propertiesthese transitions are not obvious, and are explored in thesection.

III. KRAMERS PROBLEM AND STATEMENT OF OURVARIANT

A. Traditional Kramers problem

The original Markovian Kramers problem1 describes thereaction coordinatey evolving in the bistable potential

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where gb is the dissipation parameter~the subscript for‘‘bistable’’ distinguishes this from the other dissipation prameters! and hb(t) represents Gaussian, zero-centerd-correlated fluctuations that satisfy the fluctuationdissipation relation appropriate for temperatureT,^hb(t)hb(t8)&52gbkBTd(t2t8). The rate coefficientkr fortransitions from one metastable well to the other is expresas

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This is the highest possible rate because it assumes nocrossings of the barrier when the particle moves from owell to the other. The ‘‘transmission coefficient’’k,1 cap-tures the effects of recrossings. The dependence ofk on thevarious parameters of the problem has been the subjecintense study over many years.2,42–44 Its dependence ongb

and temperature is exemplified in the simulations shownFig. 3. In particular, we note the occurrence of a maximuas predicted by Kramers, the transmission coefficient at hfriction ~diffusion-limited regime! decays asgb

21 ~and is in-dependent of temperature!; at low friction ~energy-limitedregime! Kramers predicted thatk is proportional togb /kBT.

An important generalization of the original Krameproblem, the so-called Grote–Hynes problem,45 reformulatesthe model in terms of the generalized Langevin equation

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where G(t2t8) is a dissipative memory kernel and thfluctuation–dissipation relation is now generalized^hb(t)hb(t8)&5kBTG(t2t8). The dissipative memory kernel reflects the dynamics of the thermal environment andcharacterized by its own time scales. A frequent choice isexponential, but other forms that have been used includGaussian and a decaying oscillatory memory kernel. Thetroduction of additional parameters of course changesbehavior of the transmission coefficient.

The main point here is to call attention to the fact ththe transmission coefficient has the same value for two

FIG. 3. Transmission coefficientk versus dissipation parametergb for twotemperatures obtained from direct simulation of Eq.~7!. Solid circles:kBT50.025; triangles:kBT50.05.

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10619J. Chem. Phys., Vol. 112, No. 23, 15 June 2000 Harvesting thermal fluctuations

ferent values of the dissipation parametergb , and that there-fore one cannot conclude whether the system is in onegime ~diffusion-limited! or another~energy-limited! simplyfrom the value of the transition rate. One needs to knowtrend with a changing dissipation parameter, and onequires further information about the dynamics underlyinggiven transition rate. Not surprisingly, these dynamics tout to be entirely different in different regimes.46 The timedependence of the transmission coefficient is a direct refltion of the explicit trajectories of the particle as it transfrom one well to the other. A number of investigators halooked at the time dependence of the transmission coefficin the diffusion-limited47–51 and energy-limited47,49–51 re-gimes, and also at the effect of different types of memkernels.50–52

Of interest to us here are the different dynamical behiors in the diffusion-limited regime and the energy-limiteregime. In Fig. 4 we show two views of each of two typictrajectories of the reaction coordinate for the MarkoviKramers problem. The transmission coefficients associa

FIG. 4. Trajectory of a bistable impurity described by the Langevin eqtion, Eq. ~7!. The temperature iskBT50.08. First panel:gb55.0. Secondpanel: gb50.02. The third~fourth! panel zooms in on a portion of thehigh-damping~low-damping! trajectory.

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with these two trajectories are not too different~see below!,but they correspond to different damping, placing themopposite sides of the turnover in thek vs gb curve. Thetrajectory in the first panel is in the diffusion-limited regimthe third panel shows an expanded view of a portion of ttrajectory. The particle performs rather erratic motion withone well and once in a while it surmounts the barrier. Whthe particle surmounts the barrier it does not spend mtime in the barrier region before being trapped again in oor the other well. The crossing trajectories thus tend tovolve only one or a very small number of crossingrecrossings. The trajectory in the second panel, a portiowhich is expanded in the fourth panel, is energy-limited. Tparticle performs a fairly periodic motion within one welBarrier crossing events tend to retain the particle in the brier region for several recrossings; a phase space anashows that the associated trajectories are rather smoothcillations from one side to the other of the potential wabove the barrier.49 Correlation functions associated witthese trajectories are presented and discussed in Sec. IV

B. Variant of the Kramers problem

We would like to understand the way in which the vedifferent thermal landscapes described in Sec. II affectdynamics of a reaction coordinate evolving in a bista‘‘impurity’’ embedded in these environments. The connetion of the bistable impurity to the thermal environment ocurs only through its connection to the chain, that is, wegb50.

We need to specify how the bistable system interawith the chain. We insert the impurity along the chain btween sitesi and i 11 and connect it to each of these twsites~see Fig. 1!. It is customary to choose a simple interation potential with a harmonic dependenceVint(x,y)}(x2y)2 for each chain site connected to the impurity. Herey isthe reaction coordinate andx stands for the coordinate of thchain site connected to the impurity. However, this intertion tends to destabilize the bistability in that it causesneighbors to pull the bistable particletoward the barrierrather than toward its natural metastable states. The intetion thus lowers the barrier of the bistable impurity. Sincedo not want to ‘‘bias’’ the problem in this way, we havchosen an interaction that instead tends to favor the alremetastable states:

Vint~x,y!5kint

2 S y221

22xD 2

. ~11!

Near the bistable minima~which are shifted by the interaction! the total potential for the reaction coordinate is sharmonic, and near the maximum aty50 it is still parabolic.The barrier height is modulated by the motion of the neigbors ~somewhat reminiscent of the barrier fluctuationsresonant activation problems!. At large values ofy the inter-action hardens the bistable potential.

The equations of evolution then have the following cotributions. For a site in the chain not connected to tbistable impurity we have, as before, Eq.~5!. For the bistableimpurity,

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10620 J. Chem. Phys., Vol. 112, No. 23, 15 June 2000 Reigada et al.

y52dVb~y!

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and similarly forxi 11 . Comparisons are made for the samtemperature, damping coefficients, and interaction param(kint50.1 throughout this work! varying only the nature ofthe chain.

Figure 5 shows trajectories of the bistable impurity ebedded in each of the three chains. In the hard chaintrajectory is rather similar to that of a Markovian Krameparticle in the diffusion-limited regime, while in the sochain it is closer to that of the energy-limited regime. Thisa direct reflection of the behavior seen in Fig. 2, that is,the fact that in the hard chain independent thermal flucttions created elsewhere along the chain have a good chof reaching the bistable impurity, causing erratic motion.occasional large fluctuation causes a transition over therier, usually unaccompanied by recrossings: the same enmobility that brings independent fluctuations to the impuralso makes it easy for the impurity to then lose a particuenergy fluctuation back to the chain. In the soft chain, onother hand, the particle performs fairly periodic motiowithin one well. Only fluctuations in the sites immediateadjacent to the impurity can excite the impurity; fluctuatiooriginating elsewhere do not travel to the impurity. Strofluctuations are therefore rarer but more energetic and mpersistent, so transition events occur less often. Howeonce such a fluctuation occurs it tends to remain in thatgion for a long time; the reaction coordinate thereforecrosses the barrier a large number of times until it eventuloses the excess energy and is trapped again in one owells.

FIG. 5. Trajectory of a bistable impurity embedded in a chain of 30 oslators withkBT50.08 andg50.005. Top panel: hard chain withk50.1 andk851. Middle panel: harmonic chain withk50.1. Bottom panel: soft chainwith k50.1 andk855.

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A second set of trajectories associated with the sabistable impurity in the same three chains at the same tperature but with a~10-fold! higher dissipation parameter ishown in Fig. 6. Not surprisingly, the trajectories are nomore similar to one another, but nevertheless there areimportant and revealing differences that will be made edent in our discussion in the next section. Furthermorecomparison of the two sets will allow important observatioconcerning the trends associated with increased dampin

In the next section we provide a quantitative charactization of the differences in the trajectories and a compariof these results with those of the traditional Kramers prolem.

IV. RESULTS FOR TRANSITION RATES

A useful description of the bistable system in differeregimes is provided by the normalized correlation functio

C~t![^y~ t1t!y~ t !&

^y2~ t !&, ~14!

where the brackets indicate an average overt. Since^y(t)&50, this correlation function decays to zero. When the thmal environment strongly and rapidly changes the partmomentum, the trajectory is erratic and the correlation fution decays monotonically and exponentially. The dectime is a measure of the mean time between crossing evfrom one well to the other, and its inverse can be identifiwith the transition ratekr . If on the other hand the effects othe thermal environment are weak, then the trajectory istermined mainly by the deterministic potential and remacorrelated over much longer periods of time.

The correlation functions for the Markovian Krametrajectories of Fig. 4 are shown in Fig. 7. In the higdissipation regime the correlation function is monotonic adecays exponentially over essentially all times. This is aflection of the essentially random motion within each wand between wells~the correlation functions for portions o

-FIG. 6. Trajectory of a bistable impurity embedded in a chain of 30 oslators. All parameters are the same as in Fig. 5 except that the dissipparameter has been increased tog50.05.

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10621J. Chem. Phys., Vol. 112, No. 23, 15 June 2000 Harvesting thermal fluctuations

the trajectory entirely within one well are also monotonicadecreasing, albeit not to zero!. The slope of the high-gb

curve in the right panel leads to a mean time between crings of tc'250.

The oscillations in the low-gb correlation function reflectmainly the systematic periodic motion of the particle witheach well, i.e., of the portions of the trajectory that evolvea long time neary51 or neary521. The period of theseoscillations for the parameters used here istbottom5A2p, andthis is very nearly the period of the oscillations in the figuCrossing events from one well to the other are mostly serated by long times and are essentially independent~how-ever, see a further discussion below!. Hence the logarithmicrendition in the right panel gives a straight line. Its sloleads to a mean time between crossing events oftc'450.

The Kramers correlation functions serve as a pointreference for an interpretation of the correlation functioassociated with our variant of the Kramers problem. Thare shown in a number of figures starting with Fig. 8, whshows the correlations functions associated with the trajeries in Figs. 5 and 6. We stress that in each panel theg and

FIG. 7. Correlation functions for the Markovian Kramers problem assoated with the trajectories of Fig. 4. Dashed curves:gb55.0, dotted curves:gb50.02. First panel: short-time behavior. Second panel: correlation futions on a logarithmic time scale.

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kBT are the same in all cases, as is the coupling of the chto the bistable system; only the nature of the chainchanged. The first panel shows a correlation function forharmonic chain that is oscillatory at early times, and qusimilar to the energy-limited Markovian Kramers case~seealso the corresponding trajectories in Figs. 4 and 5!. Weconjecture that the harmonic chain provides a thermal eronment comparable to the low-damping Markovian Kraers environment. The correlation function associated withhard chain is similar to the behavior at higher damping inKramers case. The correlation function associated withsoft chain also decays in an oscillatory fashion, but in a mcomplex way than in the energy-limited Markovian Kramecase.The alternation in the amplitudes is a consequencethe presence of sustained bursts of energy that cause a fifraction of the trajectory to occur above the barrier, leadinto many correlated recrossing events. The particle oscillatesabove the barrier for intervals much longer than in the Mkovian Kramers trajectory. The typical oscillation perioabove the barrier is about twice as long astbottom in ourexample~detailed discussions of these times can be found

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c-FIG. 8. Correlation functions associated with the trajectories of Figs. 5~firstpanel! and 6~second panel!. Dashed curves: hard chain; solid curves: hamonic chain; dotted curves: soft chain.

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-

10622 J. Chem. Phys., Vol. 112, No. 23, 15 June 2000 Reigada et al.

our earlier work49–51!. This effect is already slightly visiblein the low-gb Kramers correlation function in Fig. 7, but it imuch stronger in the soft chain. To reproduce this behain the Kramers model it is necessary to consider thegener-alized Kramers model with a memory friction: thereclearly an additional memory effect in the soft chain thallows the energy to remain trapped in the region ofbistable impurity for a long time. This is in accord with thnotion that transitions in the soft chain are caused by lothermal fluctuations in the nearest neighbors of the impurThe impurity may periodically exchange energy with theneighbors before the fluctuation eventually dissipates awand this causes repeated recrossings. This in turn leads tconclusion that the memory friction in the Kramers modneeded to reproduce the soft chain environment would mlikely be oscillatory.51

The periods of oscillations in the harmonic and schains are somewhat different from those of the Kramcurve in Fig. 7 and from each other. This is due to diffeences in the effective potentials.

The eventual decay of the correlation functions forthree chains is exponential. We find for the times betwetransition events~single or bursts as appropriate! tc'101 forthe hard chain,tc'453 for the harmonic chain, andtc

'1540 for the soft chain.A similar set of correlation functions associated with t

higher-damping-parameter trajectories of Fig. 6 is shownthe second panel of Fig. 8. The dynamics of the bistasystem in the hard chain with increasing damping dochange in character, whereas the oscillations in the harmand soft chains become less pronounced as these sysmove toward the diffusion-limited regime. First we note thall the curves become steeper, which translates to ashortermean time between crossing events and therefore ahighertransition rate for all three chains. The specific valuesobtain aretc'72 for the hard chain,tc'170 for the har-monic chain, andtc'215 for the soft chain. The decreasetime between crossing events is most pronounced for thechain. This is consistent with the notion that the soft chainin the energy-limited regime where small increases in efftive damping cause the greatest increases in the transrate~see Fig. 3!. The harmonic chain lies closer in this sento the turnover region, and the hard chain even closerThe second point is that this apparent trend for the hard cindicates that it, too, lies on the low-damping side of tturnover in spite of the diffusion-limited aspects of its dnamics. This is the reason for the very small oscillatiovisible at the earliest times in the hard chain correlation futions. It is apparent that neither the trajectory itself nor evthe shape of the correlation function at one value ofdamping provides unequivocal information to determwhich side of the turnover regime one is on; it is necessarinvestigate the trend.

It is interesting to investigate whether our variants of tthermal environment can actually be ‘‘pushed’’ acrossturnover point by increasing the damping on the chain.this purpose we present a series of correlation functionseach of the chains for different values of the damping para

r

te

al.

ey,thelst

ts

-

ln

nleticms

t

e

ofts-on

t.in

s-ne

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er

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eter. The other parameters, including the temperature, remfixed and equal to the values given earlier.

Figure 9 shows results for the soft and harmonic chaWith increasing damping the early-time oscillations in tcorrelation function in the soft chain first lose some of t‘‘alternation’’ features typical of a long oscillatory dissipative memory kernel and eventually the correlation functiloses its oscillatory character altogether. The crossingcontinues to increase as damping increases, so througthis series one is still on the low damping side of the tuover. The second panel of Fig. 9 shows the correlation futions for the impurity in the harmonic chain. The trendsimilar to that of the soft chain but, in all respects, indicatiof the fact that the harmonic environment is closer toturnover region than the soft environment. Thus the oscitions disappear sooner, and the increase in the transitionwith increasing damping is smaller.

Perhaps the most interesting features are seen in FigHere we clearly see the very small short-time oscillatiowhich disappear as damping increases. The transition raquite insensitive to damping in the rangeg50.005–0.1shown in the figure~the line forg50.2, not explicitly shown,

FIG. 9. Correlation functions for the bistable impurity in the soft chain~firstpanel! and harmonic chain~second panel! for various values of the dissipation parameter.

twe

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ized

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10623J. Chem. Phys., Vol. 112, No. 23, 15 June 2000 Harvesting thermal fluctuations

also falls in the same regime!. The turnover value mustherefore be within this range. To ascertain if this is so,also exhibit the correlation function for a considerably largvalue of the dissipation parameter,g51.0. The slower decayfor g51 is clear in both panels.

V. CONCLUSIONS

There has been a dearth of information on the effectsthe activation process ofnonlinearitiesin the environment.We have taken an approach here that goes part way, muthe tradition of modeling efforts for a variety of systeminteracting with a complex environment: the ‘‘immediasurroundings’’ of the reaction coordinate are describedcroscopically, while the interaction of this immediate enronment with other degrees of freedom is handled phenenologically.

We find that the dynamics of the activation processsome parameter regimes are profoundly affected by theture of the chain. If the damping parameter connectingchain to the heat bath is sufficiently low, a soft chain pvides an environment very similar to that of the GroteHynes model with an oscillatory memory kernel in th

FIG. 10. Correlation functions for the bistable impurity in the hard chainvarious values of the dissipation parameter. First panel: short-time behaSecond panel: correlation function on a logarithmic scale.

er

n

in

i-

-

a-e-

energy-limited regime,51 while a hard chain provides an environment akin to that of the Kramers model in the diffusiolimited regime.49 This in turn means that in such parametregimesthe hard chain is a more effective mediator of tactivation process than is the soft chain.

A number of interesting questions concerning these stems are currently under investigation. One concerns thefluence of boundary conditions on the behavior that we hdescribed.37 A second problem concerns the effect on treaction coordinate of a pulse or a sustained signal appsomewhere else along the chain. We have showed thapropagation of such a pulse or signal is strongly affectedthe nature of the chain,38,41 and we expect these differencein turn to affect the response of a bistable impurity to theexcitations. Such models are interesting in the contextphysical or biophysical situations wherein energy is releaat some location~provided perhaps by a chemical reactionan absorption process at that location!, which must thenmove to another location~that of the bistable impurity! toeffect some further chemical process~represented by the activation process!. The usual linear chain models are plaguby the excessive dispersion that would make such a tramission inefficient. Nonlinearities in the environment mprovide the necessary mobility with little attendant dispsion, thus greatly increasing the efficiency of such a proc

ACKNOWLEDGMENTS

R.R. gratefully acknowledges the support of this rsearch by the Ministerio de Educacio´n y Cultura throughPostdoctoral Grant No. PF-98-46573147. A.S. acknowledsabbatical support from DGAPA-UNAM. This work wasupported in part by the Engineering Research Programthe Office of Basic Energy Sciences at the U. S. Departmof Energy under Grant No. DE-FG03-86ER13606, andpart by the Comisio´n Interministerial de Ciencia y Tecnologıa ~Spain! Project No. DGICYT PB96-0241.

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