Noise-Activated Escape from a Sloshing Potential Well. Abstract

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1 Noise-Activated Escape from a Sloshing Potential Well Robert S. Maier and D. L. Stein Mathematics and Physics Departments, University of Arizona, Tucson, Arizona Abstract We treat the noise-activated escape from a one-dimensional potential well of an overdamped particle, to which a periodic force of fixed frequency is applied. Near the well top, the relevant length scales and the boundary layer structure are determined. We show how behavior near the well top generalizes the behavior determined by Kramers, in the case without forcing. Our analysis includes the case when the forcing does not die away in the weak noise limit. We discuss the relevance of scaling regimes, defined by the relative strengths of the forcing and the noise, to recent optical trap experiments. PACS numbers: a, Mj, Pm 1 Typeset using REVTEX

2 The phenomenon of weak white noise inducing escape from a one-dimensional potential well in thermal systems), was studied by Kramers [1]. If denotes the noise strength (e.g., and is the well depth, then the escape rate falls off like as "! #. The case when the trapped particle is overdamped is easiest to analyse. If, after each escape, it is reinjected at the bottom of the well, and a steady state has been set up, then in the interior of the well its position will have a Maxwell Boltzmann distribution. Kramers showed that this distribution must be modified near the well top, by being multiplied by a boundary layer function. From the modified distribution, he worked out out the weak-noise limit of the escape rate, including its all-important pre-exponential factor, by computing the outgoing flux. The Kramers formula and its multidimensional generalization have been extended in many ways [2,3]. But a full analysis of escape driven by weak noise, in periodically modulated systems, has not yet been performed. Such an analysis would shed light on the Kramers limit of stochastic resonance. It would also clarify the effects of barrier modulation on phase-transition phenomena. It is now possible to construct a physical system (a mesoscopic dielectric particle that moves, in an overdamped way, within a dual optical trap [4]) that provides a clean experimental test of the three-dimensional Kramers formula. The rate at which thermal noise induces escape agrees well with the predictions of the formula. Adding an external force, of fixed period $&%, would yield a periodically modulated system [5], of the sort that has not yet been fully analysed. A complete treatment of the escape of an overdamped particle from a sloshing potential well of this sort would surely be desirable. Smelyanskiy et al. treated this phenomenon perturbatively, in one dimension [6]. They derived a Kramers prefactor incorporating ', the periodic forcing strength. It applies if the ratio '() is set to a constant as *! #. That is, the forcing is taken to die away in the weak-noise limit. Lehmann et al. [7] treated nonperturbatively the case when ' is independent of, using path integral techniques, and worked out a numerical scheme for computing the ' -dependent prefactor. They also examined the instantaneous escape rate, which in the steady state is a $+% -periodic function of time. In a simulation of a special case (a well with a perfectly harmonic top), they noted that in the weak-noise limit, the instantaneous escape rate maximum cycles slowly around the interval,-#/. $+%0. In this Letter, we go beyond [6] and [7]. By treating the case '1 2 3, where 4 is an arbitrary nonnegative power, we determine the relation between their respective scaling regimes. In the weak-noise, weak-forcing limit, there are three physically important length scales near the oscillating well top, of sizes proportional to 57698, ', and ': Crossover behavior will result if '; < =57698, and the case '> < can itself be viewed as a crossover regime. When ' is -independent, we use facts on noise-induced transport through unstable limit cycles to illuminate the logarithmically slow cycling phenomenon [8]. We compute the instantaneous escape rate as the flux over the oscillating well top, which is an unstable limit cycle. This differs from [6], where the flux through a remote observation point is used. At any? in,-#/. $+%:, our normalized escape rate oscillates periodically in BAC as D! #. We compute the period, and give a physical explanation. More importantly, we place the case of -independent periodic forcing firmly in the Kramers framework, by determining how the Maxwell Boltzmann distribution is modified, in the boundary layer of width EF7 =57698= near the oscillating well top. As 'G! #, it approaches the modified distribution of Kramers [1]. The case when '; < in the weak-noise limit is intermediate between the case of -independent forcing and the case of zero forcing, and its boundary layer structure is 2

3 I intermediate too. Scaling Regimes. Initially, we work in terms of dimensional quantities. The Langevin equation for a driven Brownian particle in a potential well HJIKHMLONQP is RTS NVUXW RZY N;IZ[\H^]_LONQP:Ua` blocdp0ufe g R Whjikml(LncdP=o (1) Here W is the damping, ` a dimensional measure of the driving, b a dimensionless periodic function of unit amplitude, and l a standard white noise. In the overdamped (large-w ) limit, the inertial term can be dropped, leaving NFIZ[qp Y ] LONQP0UTr(b(LOcdP:Uts uvllocdp=o (2) Here pwikhyxzw R, r>i{`^xzw R, and u I<g}hikCx~W R. The Kramers formula for the r>i{ escape rate is Fƒ LŒ[ŽHyx}hi(kqP gzˆwj Š/ e p ] ] LnN P p ] ] LON P~ gzˆ =Š L[ xu P+ (3) where I e H ] ] LnN Pdx R and I e H ] ] LON P x R are the oscillation frequencies about the bottom N and top N of the well, and I gp. Eq. (3) follows from Kramers s modification of the steady-state Maxwell Boltzmann weighting [\HMLONQP xhji:kc, i.e., [Žg}pšLONQPdxud. If =Š =Š M denotes the inward offset from N, his modifying factor is [^ :x e u xÿ-p +œdožj ] ] LnN P. If rx I{, there are two regimes, depending on the size of r as u. Since u QI " QIJ xzc and r IZ xzc, where denotes length and c denotes time, r will be large in the Kramers limit if it is large compared to a quantity with dimensions xzc, namely e p ] ] LON P~ u. In physical terms, there are two regimes because there are two length scales at the well top. The first is the length scale in Kramers s modification. There is a layer of width e g}hikcxž-h ] ] LON P~, i.e., e u x -p ] ] LON P, within which physics occurs. This FL7u= 7ª P quantity is the diffusion length: the distance to within which the particle must approach, to acquire a substantial chance of leaving the well. If a periodic force is applied, a second length scale becomes important. The well top will oscillate periodically around its unperturbed location N by an amount roughly equal to `^xž-h ] ] LON P~. If this length scale is substantially larger than the first, escape dynamics should be strongly affected. The crossover occurs when ` e gž H ] ] LnN P~hik, i.e., when ` e g R hik. In normalized units, this criterion is r> e p ] ] LnN P~Œu. So if ra«u in the Kramers limit (ü ), K ~x}g and ±² ³< x}g are regimes of weak and strong forcing, respectively. The two regimes should be kept in mind when conducting experiments on escape in periodically driven systems. In the Kramers limit, only when the forcing is much weaker than e g R hi:k is a simple perturbative modification of the Kramers formula likely to apply. An illustration would be the room-temperature dual optical trap experiment of McCann, Dykman, and Golding [4], in which R µ t º¹ _» ¼º½ and I¾L7 ^ÀGgPq ² )Á Â ¹. The corresponding force magnitude e g R hik is approximately ¹ _Ã Newtons. Any repetition of +ž their experiment, with the addition of periodic driving, should take this dividing line into account. 3

4 ö! à à Ø æ Õ à É Ù ï É ã Û ó à Æ ä ï É Preliminaries. Our analysis of the ÄXÅÇÆ case, in which È is independent of É, uses optimal trajectories. The ÉDÊ Æ limit is governed by the action functional Ë Ì ÍÏÎ ÊÑÐ ÍdŒÔ ÅÖÕ \ØZÙ ÐMÛÝÜßÞnOÐ Ú à Íd Íä È(á( Ù âqã First, suppose È ÅfÆ. Then the most probable trajectory from ÐŸå to any specified point Ð Þ is the one that minimizes Ë Ì ÍÏÎ Íd Ô ÊÑÐ. The minimum is over all trajectories from ÐŸå to Ð Þ, and all transit times (finite and infinite). There is a single minimizer ÍÏÎ Íd Ê ÐŸæ~ to each side of ÐŸå, which we term an optimal trajectory. The value Ë Ì ÍÏÎ Íd Ô ÊÑÐŸæ~, which depends on Ð Þ and may be denoted çèoð Þ, is the rate at which fluctuations to Ð Þ are exponentially suppressed as É"Ê Æ. In the steady state, the probability density é of the particle will have the limiting form The prefactor ë éÿoð ê<ë OÐ /ìí/î çèoð nð must be computed by other means. Any such ÈÝÅ Æ optimal trajectory must satisfy ÐaÅñÛ"Ü Ú Þ nð, i.e., be a time-reversed relaxational trajectory. This is due to detailed balance [9]. The trajectory from ÐŸå to Ðvò is instanton-like: it emerges from ÐŸå at Í àžó Å and approaches Ðvò as Í Ê. Within the well, çƒnð equals Ì à ŒÔ ÜnÐ ÜšOÐŸå, so ôõwökçƒnðvò equals Ì à ŒÔ ÜšnÐvò ÜšOÐŸå. Also, ë is independent of Ð. If È*ÅZÆ, the model defined by the Langevin equation (2) is invariant under time translations. Íd So the optimal trajectory from ÐŸå to Ðvò is not unique. If Ð>ÅGÐŸæ~ is a reference optimal trajectory, consider the family where the phase shift ü satisfies Íd Æ á( Í Î Ê Ð(øú+û æ dï ð É Ê Íd Í à ü =ð ö{ðÿæ~ ~ýyþ& zý ü, and þ+ is the period of the forcing function á Å. In the Kramers limit of any model with È nonzero but small, the most probable escape trajectory should resemble some trajectory of the form (6). That is, some ü will be singled out as maximizing the chance of a particle being sloshed out. A study of the ÈtÊ Æ limit should yield ü. This was the approach of [6]. Suppose that È ÅZÆ. If ôõ is computed by applying (4) to the unperturbed (È Å Æ ) optimal trajectory Ð ÅÇÐ ø +û Íd, the first-order (i.e., F È ) correction to ôõ, where will be È = ü + ü Ð Ú æøú&û Íd Íd á( It is reasonable to average the Arrhenius factor ì=íî ôõ in the Kramers formula over ü, from Æ to zý. If denotes this averaging, then the escape rate will be modified by the driving, to leading order, by a factor øú&û. If ÄKÅ, i.e., È²ÅÖÈdÉ for some È, then the Kramers formula (3) will be altered to ê ø +û Ü Þ Þ OÐŸå Ù Ü Þ Þ OÐvò zý Clearly, ü should be the phase that minimizes = ü. ì=íî Íä ôõ =ä (4) (5) (6) (7) (8) 4

5 _ š a ] _ I M R 6 a P 6 M _ f M R f X Eq. (8) is essentially the formula of Smelyanskiy et al. [6]. But our derivation makes it clear that their perturbative approach requires that "$# % rapidly as &'# %, i.e., that ( be sufficiently large. Estimating the minimum of )+*-,/. by applying it to unperturbed optimal trajectories yields a correction to 021 which is valid only to 354"7698. If " is independent of &, then by Laplace s method as &\# :;/<=>?BADCEFHG AJI %. This would seemingly suggest that L ^ R R ^ R R 4`_bac8ed MON 4`_ f/8gd L MONQPSRK R 6 4UT VW89" K L MONQPSR R 4UT VW89" EXY;/<=>?ZAD[QCEF & 6 EXbhYij & 6 (9) 4kl021nmo&p8 (10) is the (rqs% Kramers formula, with 021 shifted by " 4T VW8 to leading order. But the prefactor in (10) is correct only in the small-" limit. If "utv&9w, the 354"76c8 correction to 0x1 will be of size 354U& w 8. If ( q K, it will alter the prefactor, as in (8). But when (zy K m, 354" 8 terms will also affect the prefactor. The most difficult case is ({qz%, where terms of all orders in " will affect the qz% and }D~ are fixed, by replac- prefactor. A nonperturbative treatment is called for. Analysis. We first remove explicit time-dependence, when "$ ing (2) by ^ R _ qvk 4`_8Q " ƒ 4 ˆ8Q & Š 4` c8œ 2q KoŽ (11) Here %ry }D~, and is periodic: rq} ~ is identified with rq %. The state space with coordinates 4 _7Œ 8 is effectively a cylinder. On this cylinder, the oscillating well bottom =C =C _ q a 4` c8 is a stable limit cycle, and the oscillating well top _ q š 4` c8 is an unstable limit =C B=YC cycle. To stress " -dependence, we B=C denote them and f. To study escape through f as &\# %, we can use results of Graham and B=C Tél [10,11]. The &\# % limit B=C is governed B=C by a helical, instanton-like optimal trajectory œqz 4 c8, which spirals out B=YC of and into f. It is the most probable escape B=C path in the steady state. 0x1 equals )+*ž Ÿ# 4` c8., which must be computed numerically. would first be computed nonperturbatively, by integrating B=C Euler Lagrange equations. As nears the oscillating well top, it increasingly B=C resembles a time-reversed relaxational =C trajectory. So at any specified, the th winding < of, as it spirals into, has an inward offset that hyij shrinks ^ R R B=YC geometrically, like, as # ª. Here «q+ b4 ˆ8 and are " -dependent, and q *- d 4Dš f 4` c8c8 dh ±.. hij The form (5) for the steady-state probability density generalizes to ²³4 8 4Hk µs4` 89mo&p8. To compute µ and ² =C at any specified, an optimal trajectory ending at is needed; in general, one different from. An asymptotic analysis of the Smoluchowski equation for the probability density [9,12] shows that µ satisfies the Hamilton Jacobi equation $4 Œ¹2º»8¼q+%, where the satisfies Hamiltonian ½4` Œ¾ 8 equals ¾À Á+ g¾ m ÃÂ\4` Ä8 g¾. Also, along any trajectory, ² ²»qÅkn4Æ Â5ÈÇÊÉÌËDÍ±É`Í ËDµ$m 8c² Ž (12) 5

6 ø ö Ï õ ö Ï H ì õ ð S E õ ð ø ø ö Ï õ Here ÎWÏ Ð7ÑˆnÔÕÏÖ 'Ø Ï ÐÚÙzÛ Ü Ï` ÑÞÝ is the drift on the cylinder, and Ï ßnàžá ãâåä æèçoé ÏÝoÑpê is the diffusion tensor. It follows from the Hamilton Jacobi equation that the Hessian Ï ë±à ë á ìå obeys a Riccati equation along any optimal trajectory [12,13]. This gives a scheme for computing í³ï î Ø, starting with an ï5ïý value for í on î«ðbñyò ó. In principle, the steady-state escape rate ô can be computed by the Kramers method [1]: evaluating the probability flux through î«ðbñò. But this is intricate, due to a subtle problem discovered by Graham and Tél [10,11]. Optimal trajectories that are perturbations of the escape trajectory ö\ ø î ðbñyò intersect one another wildly near î«ðbñò õ. This is because öw ø î ð ñò is a delicate object: a saddle connection in the Hamiltonian dynamics sense. In consequence, any î Ø near î«ð ñò is reached by an infinite discrete set of optimal trajectories, indexed by ú, the number of times a trajectory winds around the cylinder before reaching î Ø. The density asymptotics are [12] û Ï î Úüþý/ í ð ò Ï î Ï Öì ò Ï`î pñ êñ (13) since ì and í are infinite-valued, not single-valued. It is known [10 12] that at any fixed, any ìvð ò is not quadratic but linear in the offset from î õ ðbñò : ò Ï þöìì Ö Ï! #"$ &%('*) (14) ì+,.- ê is what, in the absence of multivaluedness, the Hessian matrix element ë " ì/ oë0 " would equal at â ê. Along î«ð ñò, it obeys the scalar Riccati equation ëbì+,& oë xâ Öì " Ù1%± Ø Ø Ï2 Ð õð ñò Ï ˆ9ì ) (15) ì+, â ì+,ˆï` is the 3,4 -periodic solution of this equation, which is easy to solve numerically. At any, ì+, equals %± Ø Ø Ï`Ð õ to leading order in Û. Deviations from this value are due to anharmonicity of at the well top. It is also known [12] that the second term on the right-hand side of (12) tends rapidly to zero along î ðbñò, as it spirals into î«ðbñyò õ. So with each turn, í is multiplied by 65 ö?> Ö87Ï 9;:/Î =<, i.e., ö?> for some A»âDAÊÏ`. Since by 65 7l Ø Ø Ï$2 Ð ð ñò c=<. This factor equals 0. So í ð ò übac ü, it follows that along î ðbñyò, í ü E as ê. Here E ÔFAG (, like ì+,, is a 34 -periodic function of, which quantifies the linear falloff of í. On 5 êˆñ 34Q, E turns out to be proportional to ì+, [14]. It can be found by integrating (12) along î ðbñyò õ. It is the ölø H limit of the quotient íi (. Non-constancy, if, as it spirals into î«ðbñò any, is due to anharmonicity of. Substituting (14) and í ð as the Summing from Ö L KMONPRQTS VU ý òqüje K into (13) yields $ W YX Ìì Z\[] 5Ö Ï! #" &%^O _ (16) ê steady-state probability density û, at an inward offset from the oscillating well top. to H is acceptable since the relative errors it introduces are exponentially, and. small, and ignorable. The dependence here on ö, i.e., on, is due to ì, E Discussion. The cycling phenomenon, and much else, follow from the infinite sum (16). To determine its behavior on the ï5ï PR" diffusive length scale near the oscillating well top, set 6

7 `badced f grh with c fixed, and also multiply by d$i0f grh. (As in the case of no periodic driving, a steady-state density kj that is normalized to total probability l within the well must include an d$i0f grh factor.) The resulting expression is invariant under dnmo pikhqd. So ksr`tactd j f grhu vxwyjz{} ~ r c uxvxw $W r#ƒ Yˆ d w$u dno u (17) where the quantity z {} ~ rc u vxw, for any c and any v in Š u,œw, is periodic in Ž d with period Ž p. In the steady state, the instantaneous escape rate r vxw through the oscillating well top, which equals rdqˆ w r sˆ 0` w k j! =, satisfies r vxwy r l ˆ w d f grh z {š ~] r u vxw W r#ƒ Yˆ d w u dno œ (18) So at any v in Š u Œw, the instantaneous escape rate, divided by d f grh W r#ƒ ˆ d w, ultimately oscillates in Ž d with period žž p, i.e., with period =Š Ÿ r j {š ~ r vxw w # v?. Lehmann et al. [7] noted that on Š u Œw, the peak of the function r w may shift when d is decreased. The preceding results indicate this phenomenon is widespread. Its cause is physical. As d o, the most probable trajectory taken by an escaping particle is the helix v mo {š $~ r vxw, along which it moves in a ballistic, noise-driven way. However, once it gets within an ª r d f grh w distance of the oscillating well top, it moves diffusively, instead. It is easily checked that such a changeover must take place at a location that cycles slowly around Š u Œw, as d«o. If d mo p!ikhqd, the changeover returns to its original location. If the well top is perfectly harmonic, so that and f do not depend on v, and the bottom is too, it is straightforward to integrate r vxw over Š u Œ w. We find y f$± r² s³ w d f grh W r ƒc ˆ d w œ µ Œ r² w (19) It is useful to compare (19) with the perturbative formula (10). They are consistent if f diverges like i0f grh as o. An i0f grh divergence was seen in this special case by Lehmann et al. [7], and it occurs widely [14]. It has major consequences. f is the normal derivative of the prefactor. But is ª r l w on ³ {} ~, and is well-behaved in the well interior as o. So there must be a layer near the top, of width ª r f grh w, in which declines linearly to zero. This has been seen numerically [14]. We can now compare the steady-state probability density (17), which is valid on the ª r d f grh w length scale near the oscillating well top, to the density when a¹. The analog of z {} ~ rc u vxw, when a, is (up to a constant),ºx»²¼ ½ ƒ ± r² w #ci¾ 8d i0f grhs $W ÁÀ r² w c hqâ*u (20) The ºx»²¼ is the Kramers boundary function [1], and the exponential is from the Maxwell Boltzmann distribution. How can z {š ~ r c u vxw, which is defined by an infinite sum, degenerate into such a classical (and v -independent) form when o? The origin of this weak-driving discontinuity is clear: the o limit passes through an intermediate scaling regime, namely Ã a l, where the sum (16) is not valid. Light is thrown on this by an integral representation forkr` j u vxw of Smelyanskiy et al. [6], which is valid if Ã a l. In our notation, it is (up to a constant) 7

8 Ä Å0ÆxÇ(ÅKÈ?É Ê&ËRÌÍÎ Ï where åçæèå«é ã ä êxëíìïîzð1ñjò óõô ö&ø0ãsú òsû ügý ý éþþ0 ëû é êxë úžôö&ø Ç(ÐÑÔ!Å $ÕRË Ö Ø Ê0Ù Ù]ÐVÚ Û$Ü²Ø ÜËRÌ?Ç(Å0ÐšÝ ËRÌÞÜ}ßZà?ÐšáÜ&â!ã ä (21). The expression (21) is a Maslov WKB form [15], which we study further elsewhere [14]. It is easy to see that (21) provides an interpolation, across the æ regime, from the small- portion of the æ regime to the æ case treated by Kramers. When is nonzero and small, but fixed as Ä, applying Laplace s method to (21) yields an infinite sum resembling our sum (16). And when æ, evaluating (21) yields Kramers s factor. This research was supported in part by NSF grant PHY

9 REFERENCES [1] H. A. Kramers, Physica 7, 284 (1940). [2] Theory of Continuous Fokker Planck Systems, edited by F. Moss and P. V. E. McClintock (Cambridge University Press, Cambridge, 1989). [3] New Trends in Kramers Reaction Rate Theory, edited by P. Talkner and P. Hänggi (Kluwer, Dordrecht, 1995). [4] L. I. McCann, M. I. Dykman, and B. Golding, Nature 402, 785 (1999). [5] A. Simon and A. Libchaber, Phys. Rev. Lett. 68, 3375 (1992). [6] V. N. Smelyanskiy, M. I. Dykman, and B. Golding, Phys. Rev. Lett. 82, 3193 (1999). [7] J. Lehmann, P. Reimann, and P. Hänggi, Phys. Rev. Lett. 84, 1639 (2000). [8] M. V. Day, Stochastics 48, 227 (1994). [9] R. S. Maier and D. L. Stein, Phys. Rev. E 48, 931 (1993). [10] R. Graham and T. Tél, Phys. Rev. Lett. 52, 9 (1984). [11] R. Graham and T. Tél, Phys. Rev. A 31, 1109 (1985). [12] R. S. Maier and D. L. Stein, Phys. Rev. Lett. 77, 4860 (1996). [13] D. Ryter and P. Jordan, Phys. Lett. A 104, 193 (1984). [14] R. S. Maier and D. L. Stein, in preparation. [15] R. S. Maier and D. L. Stein, J. Stat. Phys. 83, 291 (1996). 9

Scaling and crossovers in activated escape near a bifurcation point

PHYSICAL REVIEW E 69, 061102 (2004) Scaling and crossovers in activated escape near a bifurcation point D. Ryvkine, M. I. Dykman, and B. Golding Department of Physics and Astronomy, Michigan State University,