Activated escape is investigated for systems that are driven by noise whose power spectrum peaks at a finite frequency. Analytic theory and analog and digital experiments show that the system dynamics during escape exhibit a symmetry-breaking transition as the width of the fluctuational spectral peak is varied. For double-well potentials, even a small asymmetry may result in a parametrically large difference of the activation energies for escape from different wells

Arrayas, M.

Dykman, Mark

Mannella, R.

McClintock, Peter V. E.

Stein, N. D.

Lancaster E-Prints

VOLUME 84, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 12 JUNE 20005470Symmetry Breaking of Fluctuation Dynamics by Noise ColorM. Arrayás,1 M. I. Dykman,2 R. Mannella,3 P. V. E. McClintock,1 and N. D. Stein11Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom2Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 488243Dipartimento di Fisica, Università di Pisa and INFM UdR Pisa, Piazza Torricelli 2, 56100 Pisa, Italy(Received 10 November 1999)Activated escape is investigated for systems that are driven by noise whose power spectrum peaks ata finite frequency. Analytic theory and analog and digital experiments show that the system dynamicsduring escape exhibit a symmetry-breaking transition as the width of the fluctuational spectral peak isvaried. For double-well potentials, even a small asymmetry may result in a parametrically large differenceof the activation energies for escape from different wells.PACS numbers: 05.40.–a, 05.60.Cd, 73.50.Jt, 73.50.TdThe problem of diffusion and activated escape in sys-tems away from thermal equilibrium is attracting increas-ing attention in diverse contexts, from crystal growth [1]to switching of current in microstructures [2], modes inlasers [3], and biological systems [4]. In many cases thesystem of interest is nonequilibrium because it is drivenby nonthermal noise. Such noise can give rise to new andinteresting phenomena, e.g., the onset of directed diffusionin a spatially periodic potential (ratchet) [5], in which casethe diffusion rate and direction depend on the shape of thenoise power spectrum Fv [6,7].One would expect the escape probabilities to be deter-mined by Fv for v & t21r ,vy where tr and vy are thecharacteristic relaxation time and vibrational frequency ofthe system. However, the situation is more complicated inthe important case of quasimonochromatic noise (QMN)where Fv has a peak at a comparatively high frequencyv0 ¿ t21r ,vy [8–10]. Here, the optimal fluctuation forescape may correspond to “fluctuational preparation of thebarrier,” analogous to phonon-induced barrier preparationin solids [11]. The system is forced by the noise to fluc-tuate at frequency v0. As the amplitude of such randomvibrations increases, the shape of the effective potential fortheir center of motion qc alters, as illustrated for the caseof a bistable potential in Fig. 1. For a large enough am-plitude, the initially occupied metastable potential well forqc may disappear, so that the system then escapes fromit. Such an adiabatic escape scenario, during which thesystem is most likely to move along an oscillating path,was predicted [8] and observed [9] for the case where thehalf-width of the peak in the noise spectrum G ø t21r . Re-cent numerical analysis indicates [10] that, as G increasesto t21r , the most probable escape path (MPEP) becomessmooth.In this Letter we show that the transition with changingnoise spectrum between different types of MPEPs, and thusbetween escape scenarios, is a symmetry-breaking transi-tion. The symmetry is broken in time: it corresponds tothe onset of fast oscillations of the MPEP, with an arbitraryinitial phase, to leading order. We find the location of this0031-90070084(24)5470(4)$15.00transition and show that, at the transition, the activation en-ergy of escape displays a nonanalytic dependence on thenoise color parameter.The second effect that we discuss occurs when there isan asymmetry in the potential. Even a small asymmetrymay lead to a parametrically large difference in the activa-tion energies of interwell transitions in opposite directions,giving rise to a dc current in a periodic potential or to strongeffective localization in one of the wells of a double-wellpotential.We will consider such effects for an overdamped systemfluctuating in a double-well potential Uq (cf. Fig. 1),with equation of motionq  2U 0q 1 ft . (1)Here, ft is a quasimonochromatic noise. An example ofQMN is provided by Brownian motion of an underdampedharmonic oscillator, which is described by the equationFIG. 1. The average potential V (7) for motion of the vibrationcenter qc as a function of the vibration amplitude jq2j  jq1j.The initial double-well potential Uq is given by V q, 0, 0.For an asymmetric Uq, only one of the wells disappearswith increasing jq6j. The plot refers to a double-well po-tential Uq  V q,q6  0  2q22 1 q44 2 0.15q. Thebold curves show most probable escape paths for G  0.3.© 2000 The American Physical SocietyVOLUME 84, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 12 JUNE 2000Mˆft  jt, Mˆ d2dt21 2Gddt1 v20 , (2)where v0 and G are the oscillator frequency and damp-ing, respectively, jt is Gaussian white noise, andjtjt0  2D˜dt 2 t0 (for thermal noise, the char-acteristic intensity D˜  2GkBT ). For small Gv0, thepower spectrum Fv of the noise ft has a narrowLorentzian peak at v0, of half-width G. The probabilitydensity of realizations of ft is ~ exp2R0	fD˜ [12],withR0	f  14Z `2`dt 	Mˆft2. (3)The probabilities Wnm of fluctuational transitions be-tween the potential wells n  1, 2 of Uq can be obtained[8,13] (see also [14]) using Feynman’s idea [12] that thetrajectory of the system qt is uniquely determined bythe force ft. Different realizations of ft can lead toa transition, but for small noise intensity D˜, their proba-bility densities are exponentially different. To logarithmicaccuracy, Wnm is determined by the probability density ofthe most probable realization, i.e., the one that minimizesR0	f subject to the constraint (1). In such an optimalfluctuation, the system is driven along an instantonlike op-timal path qt from the bottom qn of the nth well, whichis occupied for t ! 2`, to the top of the potential barrierqb between the wells for t ! `, and the driving optimalforce ft ! 0 for t ! 6` [8]. The optimal trajectoriesqt, ft provide a minimum to the functionalR	f, q  R0	f 1Z `2`dt lt 	 q 1 U 0 2 ft ,(4)where lt is a Lagrange multiplier. The escape rateW ~ exp2RD˜, R  minR	f, q . (5)We will investigate QMN-activated escape in themost interesting case where Gtr can be arbitrary, butthe characteristic noise frequency v0 ¿ G, 1tr [tr maxn	1U 00qn is the relaxation time of the system]. Inthis case the Euler equations for the functional (4) canbe analyzed [8] using a standard averaging method. Themotion of the system is a superposition of fast vibrationsat frequency v0 and slow motion of the vibration centerqc,qt  qct 1Xa6qa expiav0t . (6)The two motions are mixed by the nonlinearity of the po-tential Uq. The time evolution of qc and of q1  q2occurs on the scale 1G, tr and is determined by the time-averaged potential V qc, q1, q2,V 12pZ 2p0dfUqc 1 q1eif 1 q2e2if . (7)To lowest order in Gv0, 1v0tr , the variational prob-lem (4) can be reduced [15] to a much simpler one, R minR¯	qc, q6, withR¯	qc, q6  12 v40Zdt L qc, qc; q6, q6 ,L 12 qc 1 V 0c2 1 4 q1 q2 1 G2q1q2 ,(8)where V 0c  ≠V≠qc. The functional R¯ takes the formof a classical action for a particle with coordinates qc, q6and a Lagrangian L. Escape is determined by the extremetrajectory which starts from qc  qn, q6  0 for t ! 2`and approaches qc  qb , q6  0 as t ! `.From (8), the equation for q6 is of the form2q¨a 114 qc 1 V 0cV00c,2a 1 G2qa  0, a  6 ,(9)where V 00c,a  ≠2V≠qc≠qa . The potential V (7) dependson q1,q2 only in terms of the product q1q2. Thereforethe phase argq1  2 argq2 is a cyclic variable. From (8),R¯ is minimal for ddt argq1  0, and so by choosingthe time origin in (6) one can make q1t  q2t. An-other important consequence is that Eq. (9) has a trivialsolution q6t  0. It corresponds to a smooth (nonoscil-lating) optimal path qt, withqct  qt  V 0cqc, 0, 0  U0q . (10)The path q  U 0q is the well-known solution for theMPEP in an overdamped system in thermal equilibrium.Such a system is described by (1) with ft being whitenoise with a flat power spectrum of height 2kBT . In thepresent case the solution (10) arises because the noise spec-trum Fv has a broad low-frequency plateau, of widthv0 which is parametrically larger than the relaxation rateof the system, and of height 2D˜v40 . Correspondingly,R  v40DU [DU  Uqb 2 Uqn], and the escape rateW ~ exp2v40DUD˜.The smooth optimal path (10) provides a minimum tothe functional R¯ if the eigenvalue problemZdt0Xj	d2R¯dqitdqjt0cnjt0  lncnit (11)has only non-negative eigenvalues ln [here, i, j  c,6,and the derivatives are calculated for the path (10),cni6`  0]. The matrix d2R¯dqidqj  0 for i  c,j  6 or i  j  6. Using standard arguments [16]one can show that all eigenvalues of d2R¯dqcdqc arepositive except for the trivial zero eigenvalue with theeigenfunction ~ qc.Of special interest in (11) is the equation for cn1 cn2,2c¨n6 112V 0cV000c,2,1 1 2G2cn6  lncn6 . (12)5471VOLUME 84, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 12 JUNE 2000Since jV 0cj, jV 000c,2,1j  1tr , the eigenvalues are alwayspositive for large Gtr . The smooth path (10) is indeedthe optimal escape path in this case.As Gtr decreases, the lowest eigenvalue of (12) l0 maybecome equal to zero. This signals instability of the solu-tion (10) and defines the critical Gcr where it occurs. Forstill smaller G the minimum of R¯ is provided by the solu-tion with finite q6. From (6), such a solution oscillates intime, with an arbitrary phase in the approximation (8), i.e.,the time symmetry of the MPEP is spontaneously broken.The activation energy R is a nonanalytic function of Gfor G  Gcr . The situation is similar to a second orderphase transition, with R¯ and c06t playing the roles ofthe free energy functional and the soft mode. Indeed, forsmall jG 2 Gcr j, one can expand the functional R¯	qc, q6(8) in deviations dqi Pcncni of the actual trajectoryfrom the extreme path (10), with cni being the normal-ized eigenfunctions of (11) for G  Gcr . The coefficientof jc0j2 in dR¯v40 is 2G2 2 G2cr . Therefore c0 playsthe role of an order parameter. For G . Gcr the mini-mum of R¯ is reached for c0  0. For G , Gcr , it isreached for finite jc0j determined by higher-order termsin the expansion of R¯. The nontrivial part of dR¯v40 ,after appropriate renormalization, takes the familiar formajc0j4 1 2G2 2 G2cr  jc0j2, with a being a G-independentconstant. Therefore Rv40  DU 2 G2 2 G2cr 2a forG , Gcr , and d2RdG2 is discontinuous for G  Gcr .Away from Gcr , the prefactor in the escape rate is deter-mined by the eigenvalues ln (cf. [16]). It blows up as Gapproaches Gcr .We note an interesting similarity between the abovetime-symmetry-breaking transition and breaking ofthe spatial symmetry of the MPEP in two-variablewhite-noise-driven systems that was first found and inves-tigated by Maier and Stein [17]. As in (8), the Lagrangianfor the optimal fluctuational paths analyzed in Ref. [17]was even in one of the variables. In a way, a one-variableQMN-driven system provides a natural realization of themodel [17].A solution of the variational problem (8) that is quali-tatively different from (10) can be obtained in the limitof small Gtr [8]. It has the form q6t  q01 exp2Gjtjfor jtjtr ¿ Gtr 212, with V 0cqc, q6  0. This solu-tion corresponds to the vibration center qc following theincrease of q6 adiabatically and staying at the minimumof the average potential V for given q6 until this minimummerges with the barrier top (see Fig. 1). This occurs forsome q1  q2  q01 such that V0c  V00cc  0. From (8)R  4v40Gjq01 j2. (13)The above solution corresponds to the “barrier prepa-ration” for the vibration center qc by the slowly varyingvibration amplitude q6 [8] and gives an activation energyproportional to the small parameter G. However, it appliesonly if the potential minimum of V from which the system5472escapes merges with the barrier top with increasing q6.It is clear from Fig. 1 that, in the case of an asymmetricdouble-well potential, this only happens for one of thewells. For the other well, the barrier for qc remains fi-nite for all q6. Therefore the value of R, although lessthan v40DU, is not proportional to G. This means that,even for a weakly asymmetric potential, activation en-ergies of escape from different wells are parametricallydifferent. Similarly, for a periodic potential of the ratchettype (see [5]), the activation energies for diffusion in oppo-site directions differ, leading to strongly directed diffusionfor low noise intensities [15].To test these predictions, we have performed analogand digital simulations of a QMN-driven system. Thetechniques are described in [18]. The activation energieswere extracted from measured escape rates as functions ofnoise intensity. We chose the simplest system: a biaseddouble-well Duffing oscillator, with potentialUq  212q2 114q4 2 Aq . (14)For A  0 the two wells are symmetrical. The relaxationtime tr  12. For large Gtr , when (10) applies, theescape activation energy Rv40  14 whereas, from(13), Rv40  2G3 for Gtr ø 1. The critical value Gcr ,where time symmetry of the MPEP (10) breaks down, canbe obtained from Eq. (12) using the explicit form of theoptimal path (10) qct  7 exp2t2 2 cosht212 [thesigns 7 correspond to escape from the left and rightwells of Uq, respectively]. The soft mode c06t Ccosht212, and Gcr  12.A sharp change of R as a function of G for G  12 isobserved in the experimental data (plusses) in Fig. 2. Asexpected for a symmetry-breaking transition, R  constfor G . 12, whereas dR ~ G 2 Gcr 2 for small posi-tive Gcr 2 G. The data are in excellent agreement withthe results (crosses) obtained by solving the Euler equa-tions for the reduced functional R¯ (8). They are also closeto results (squares) obtained from the Euler equations forthe functional R (4) [8] using the numerical method ofEinchcomb and McKane [10(b)]. At criticality the ex-treme paths of R¯, which emanate from jqcj  1, q6 0, are focused at the barrier top q6  qc  0 (see insetof Fig. 2), which gives rise to blowing up of the prefac-tor in the escape rate. Similar focusing was observed andcarefully analyzed at a spatial symmetry-breaking transi-tion in a white-noise-driven system [17].For A fi 0 the potential (14) is asymmetric. Break-ing of the time symmetry of the MPEP occurs in bothwells, but for different G. For the well at positive qin Fig. 1 (well 1), Gcr 2 12  2Ap , to first orderin A. This well is deeper for A . 0 [clearly, the es-cape activation energy R1A  R22A]. This meansthat the fluctuational barrier preparation in the deeperwell starts for larger G. Therefore, unexpectedly, theratio r  R1R2 is a nonmonotonic function of G.VOLUME 84, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 12 JUNE 2000FIG. 2. The activation energy of escape R for a symmet-ric Duffing oscillator as a function of the noise bandwidth G,showing discontinuity of d2RdG2 at the symmetry-breakingtransition for Gcr  0.5. Analog experimental data (1) forv0  10 are compared to numerical results (3) for the re-duced Lagrangian (8) and appropriately extended numerical re-sults (squares) [10(b)] for the full variational problem. Linesshow the asymptotics. Inset: Trajectories for the Lagrangian(8) that start from qc  21,q1  q2  0 for t ! 2` and arefocused into the barrier top qc  q6  0 for G  Gcr .For large G both R1 and R2 are independent of G,and r  const . 1 for A . 0. As G decreases, R1starts decreasing first, and r decreases. However, withfurther decrease in G, r starts sharply increasing, as seenin Fig. 3. This happens because, for small G, it is only forthe shallow well that the barrier of the average potentialV can disappear, as seen in Fig. 1. Therefore R2 ~ Gfor small G, and r becomes large; see Fig. 3. It isseen from Fig. 1 that the MPEPs in different wells havealready become very different even when G has fallenonly slightly below Gcr .FIG. 3. The ratio of the activation energies of escape r R1R2 from the wells 1 and 2 of the biased Duffing oscilla-tor (14).In conclusion, for bistable systems driven by a quasi-monochromatic noise the activation energies of escapefrom different wells are parametrically different, as arealso the activation energies for diffusion in opposite di-rections in a periodic potential. This allows the use ofQMN for highly selective species separation. The tran-sition from most probable escape along a smooth path, tothat along an oscillating path, is a symmetry-breaking tran-sition. The second derivative of the activation energy withrespect to the bandwidth of the noise is discontinuous at thetransition.It is a pleasure to acknowledge helpful discussions andcorrespondence with A. J. McKane and M. M. Millonas.The research was supported by the NSF through GrantNo. PHY-0071059, by the Engineering and Physical Sci-ences Research Council (U.K.) under GR/L99562, and bythe EC under ERBCHBGCT940738.[1] J. R. Smith and A. Zangwill, Phys. Rev. Lett. 76, 2097(1996).[2] S. Kogan, Phys. Rev. Lett. 81, 2986 (1998).[3] R. Roy, R. Short, J. Durnin, and L. Mandel, Phys. Rev.Lett. 45, 1486 (1980); M. B. Willemsen, M. U. F. Khalid,M. P. van Exter, and J. P. Woerdman, Phys. Rev. Lett. 82,4815 (1999).[4] A. Fersht, Structure and Mechanism in Protein Science(W. H. Freeman, New York, 1999).[5] F. Jülicher, A. Ajdari, and J. Prost, Rev. Mod. Phys. 69,1269 (1997).[6] M. M. Millonas and M. I. Dykman, Phys. Lett. A 183, 65(1994).[7] M. M. Millonas, Phys. Rev. Lett. 74, 10 (1995).[8] M. I. Dykman, Phys. Rev. A 42, 2020 (1990).[9] M. I. Dykman, P. V. E. McClintock, N. D. Stein, and N. G.Stocks, Phys. Rev. Lett. 67, 933 (1991).[10] (a) S. J. B. Einchcomb and A. J. McKane, Phys. Rev. E 49,259 (1994); (b) Phys. Rev. E 51, 2974 (1995).[11] Quantum Tunnelling in Condensed Media, edited by Yu.Kagan and A. J. Leggett (Elsevier, Amsterdam, 1992).[12] R. P. Feynman and A. R. Hibbs, Quantum Mechanics andPath Integrals (McGraw-Hill, New York, 1965).[13] M. I. Dykman and M. A. Krivoglaz, Sov. Phys. JETP 55,30 (1979).[14] A. J. Bray and A. J. McKane, Phys. Rev. Lett. 62, 493(1989); A. J. McKane, Phys. Rev. A 40, 4050 (1989).[15] M. I. Dykman (unpublished).[16] J. S. Langer, Ann. Phys. (N.Y.) 41, 108 (1967).[17] R. S. Maier and D. L. Stein, Phys. Rev. Lett. 71, 1783(1993); Phys. Rev. E 48, 931 (1993); J. Stat. Phys. 83,291 (1996).[18] D. G. Luchinsky, P. V. E. McClintock, and M. I. Dykman,Rep. Prog. Phys. 61, 889 (1998); R. 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Symmetry breaking of fluctuation dynamics by noise color.

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Symmetry breaking of fluctuation dynamics by noise color.

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