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PHYSICAL REVIEW E 73, 061101 共2006兲

Thermally activated escape rate for a Brownian particle in a tilted periodic potential for all values of the dissipation W. T. Coffey Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland

Yu. P. Kalmykov Laboratoire de Mathématiques et Physique des Systèmes, Université de Perpignan, 52, Avenue de Paul Alduy, 66860 Perpignan Cedex, France

S. V. Titov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Vvedenskii Square 1, Fryazino, Moscow Region, 141190, Russian Federation

B. P. Mulligan Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland 共Received 8 February 2006; published 7 June 2006兲 The translational Brownian motion of a particle in a tilted washboard potential is considered. The dynamic structure factor and longest relaxation time are evaluated from the solution of the governing Langevin equation by using the matrix continued fraction method. The longest relaxation time is compared with the Kramers theory of the escape rate of a Brownian particle from a potential well as extended to the Kramers turnover region by Mel’nikov 关Physics Reports 209, 1 共1991兲兴. It is shown that in the low temperature limit, the universal Mel’nikov expression for the escape rate provides a good estimate of the longest relaxation time for all values of dissipation including the very low damping 共VLD兲, very high damping 共VHD兲, and turnover regimes. For low barriers 共where the Mel’nikov method is not applicable兲 and zero tilt, analytic equations for the relaxation times in the VLD and VHD limits are derived. DOI: 10.1103/PhysRevE.73.061101

PACS number共s兲: 05.40.Jc, 05.10.Gg

I. INTRODUCTION

The translational Brownian motion in the tilted washboard potential V共z兲 = − V0cos共2␲z/a兲 − Fz + const,

共1兲

where z is the coordinate and a is a characteristic length with the constant field driving potential Fz superimposed on the periodic potential −V0cos共2␲z / a兲, arises in a number of important physical applications involving noise and relaxation processes in phase-locked loops. We mention the currentvoltage characteristics of the Josephson junction 关1,2兴, mobility of superionic conductors 关3兴, a laser with injected signal 关4兴, phase-locking techniques in radio engineering 关5兴, dielectric relaxation 共when F = 0兲 of molecular crystals 关6兴, the dynamics of a charged density wave condensate in an electric field 关7兴, ring-laser gyroscopes 关8兴, stochastic resonance 关9,10兴, etc. A comprehensive discussion of the model is given in Refs. 关1,11,12兴. One of the most important characteristics associated with the Brownian motion in any multiwell and single-well potential is the friction and temperature dependence of the greatest 共overbarrier兲 relaxation time which is essentially the inverse of the smallest nonvanishing eigenvalue ␭1 of the characteristic equation or secular determinant of the relevant dynamical system. In other words, ␭−1 1 is the lifetime of the longest lived relaxation mode of the system. The greatest relaxation time may also be obtained by calculating the mean first passage times from each of the wells of the potential 关13兴. As far 1539-3755/2006/73共6兲/061101共11兲

as the calculation of ␭1 is concerned, one may expand the solution of the associated probability density diffusion equation 共usually the specialized form of the Fokker-Planck equation applicable to separable and additive Hamiltonians comprising the sum of the potential and kinetic energies known as the Klein-Kramers equation兲 in Fourier series in the position and velocity variables 关11兴. Alternatively, the secular equation may be generated by averaging the appropriate Langevin equation over its realizations in phase space yielding the hierarchy of differential-recurrence equations governing the decay functions of the system 关12兴. In each of the two methods, the secular determinant results from truncation of the set of differential-recurrence relations at a number ensuring convergence of the resulting set of simultaneous ordinary differential equations. Thus, ␭1 is not in general available in closed form as it is always rendered as the smallest root of a high order polynomial equation. Hence, it is difficult to compare ␭1 so determined with experimental observations of the greatest relaxation time or the relaxation rate. Fortunately 共noting that ␭1 for sufficiently high barriers has exponential dependence on the barrier height兲, a way of overcoming this difficulty is to utilize an ingenious method originally proposed by Kramers 关14兴 in connection with thermally activated escape of particles out of a potential well. Kramers 关14兴共using Einstein’s theory of the Brownian motion as extended both to include the inertia of the Brownian particles and a potential well兲 evaluated the prefactor A 共which is a function of both the dissipative coupling to the bath and the potential shape兲 in an Arrhenius-like equation

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©2006 The American Physical Society


COFFEY et al.

for the escape rate ⌫ over the potential barrier ⌬V 共reaction velocity in the case of chemical reactions兲, viz., ⌫=A

␻a −⌬V/k T B , e 2␲

⌬V kBT,

共2兲

where the attempt frequency ␻a is the angular frequency of a particle executing oscillatory motion at the bottom of a well 共for reviews of applications of Kramers’ method see Refs. 关13,15兴兲, kB is Boltzmann’s constant, and T is temperature. Due to the unique contribution of Kramers to our understanding of the role played by the dissipative coupling in the escape rate, the escape-rate problem is now commonly known as the Kramers problem 关13兴. Now if the escape rates for Brownian particles governed by separable and additive Hamiltonians comprising the sum of the kinetic and potential energy are calculated by the Kramers method, three regimes of damping automatically appear, viz., 共i兲 intermediate-to-high damping 共IHD兲, 共ii兲 very low damping 共VLD兲, and 共iii兲 a turnover region. Kramers 关14兴 obtained so-called IHD and VLD formulas for the escape rate, assuming in both cases that the energy barrier is much greater than the thermal energy so that the concept of an escape rate applies. He mentioned in his paper, however, that he could not find a general method of attack for the purpose of obtaining a formula which would be valid for any damping regime. This problem was solved nearly 50 years later by Mel’nikov and Meshkov 关16,17兴. They postulated from heuristic reasoning that a formula valid for all values of the damping may be given by simply multiplying the general Kramers IHD result for ⌫ by using a certain bridging integral derived by them. Mel’nikov 关17兴 further extended the bridging integral method to take into account quantum effects in a semiclassical way. Later Grabert 关18兴 and Pollak et al. 关19兴 presented an even more complete solution of the Kramers turnover problem showing that the Mel’nikov and Meshkov turnover formula for the escape rate can be obtained without ad hoc interpolation between the weak and strong damping regimes. We remark that the theory of Pollak et al. 关19兴 is also applicable to an arbitrary memory friction and not only in the “white noise” 共memoryless兲 limit. In the semiclassical limit, the latter theory was extended to the quantum regime by Rips and Pollak 关20兴. As far as general accuracy is concerned, the universal turnover formula of Mel’nikov and Meshkov compares favorably with escape rate calculations based on either the solution of the Klein-Kramers equation or on numerical simulations of the Brownian dynamics in various potentials 共not including to date, however, the tilted cosine potential兲. In particular, a comparison with the numerical results for the escape out of a single well was given in Refs. 关21,22兴. For double-well potentials, the Mel’nikov and Meshkov formula has been tested in Refs. 关23–27兴 for both the rotational and translational Brownian motion. Furthermore, Coffey et al. 关15,28兴 have extended the method to the magnetization relaxation of single-domain ferromagnetic particles. 共The magnetic relaxation differs fundamentally from that of inertial Brownian particles because the undamped equation of motion of the magnetization of a single-domain ferromagnetic particle is the gyromagnetic equation and the Hamiltonian is

nonseparable. Thus the inertia plays no role; the part played by inertia in the mechanical system is essentially mimicked in the magnetic system by the gyromagnetic term causing the coupling or “entanglement” of the transverse and longitudinal modes兲. The calculation of the longest relaxation time for various magnetocrystalline anisotropies has been accomplished in Refs. 关29–31兴. It is the purpose of this paper to apply a universal turnover formula for the escape rate to the analysis of the dynamics of a Brownian particle in the tilted periodic potential, Eq. 共1兲. In particular application to a Josephson tunneling junction, Mel’nikov has derived such a universal equation for the longest relaxation time in Ref. 关17兴. In this context, we must remark that the Kramers problem in a tilted periodic potential is qualitatively different from the escape problem from a metastable well because the tilted periodic potential is multistable. The particle having escaped a particular well may again be trapped due to the thermal fluctuations in another well. Moreover, jumps of either a single lattice spacing or of many lattice spacings are possible 关32兴. Thus from a mathematical point of view, one has to take into account the nonperiodic solution of the Fokker-Planck equation. Hence the dynamic structure factor ˜S共k , ␻兲共playing an essential role in neutron and light scattering experiments兲 is used as described by Risken 关11兴 generating an additional 共wave number兲 parameter k in the Fokker-Planck equation. That factor is then averaged over all possible jumps. Yet another difference from the conventional Kramers problem is that the stationary distribution is no longer the Maxwell-Boltzmann distribution. In the VHD limit, the analysis of the problem usually starts from the Smoluchowski equation by either converting the solution of that equation to a Sturm-Liouville problem or to the solution of an infinite hierarchy of linear differential-recurrence relations for statistical moments 关11兴. We have mentioned that a concise method of numerical treatment 共in terms of infinite scalar continued fractions兲 of the model in the VHD limit, where the inertia of the particle may be neglected, has been suggested by Cresser et al. 关33兴 with applications to a ring-laser gyroscope as summarized by Risken 关11兴共see also references cited therein兲. Further development of the continued fraction approach has been given by Coffey et al. 关12,34兴. In the opposite VLD limit, the calculation of the escape rate by Kramer’s method was accomplished in Refs. 关35,36兴. A general method of solution of the problem for all values of dissipation based on a matrix continued fraction representation of the Klein-Kramers equation has been suggested by Risken 关11兴. This method allows one to calculate eigenvalues and eigenfunctions of the KleinKramers 共Fokker-Planck兲 equation for the tilted periodic potential and evaluate the Fourier transforms of various correlation functions for virtually all cases. By applying this method, Ferrando et al. 关32,37兴 have studied the onedimensional translational Brownian motion in a pure periodic potential 共1兲 with F = 0. These authors have evaluated numerically the escape rate ⌫ from the dynamic structure factor ˜S共k , ␻兲 and shown that the friction dependence of ⌫ so obtained is in full agreement with that given by the Mel’nikov universal equation 关17兴. The treatment of the same one-dimensional problem and its generalization to dif-

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THERMALLY ACTIVATED ESCAPE RATE FOR A...

V共x兲 = V共z兲/共kBT兲 + C⬘ = − ␥共cos x + yx兲 + C⬘ ,

共4兲

and the Langevin equation describing the dynamics of the Brownian particle becomes x¨共t⬘兲 + ␤⬘x˙共t⬘兲 + ⳵xV关x共t⬘兲兴/2 = ␭共t⬘兲,

共5兲

where C⬘ = −␥共冑1 − y 2 − 1 + y arcsin y兲共this constant does not affect the equation of motion, however, it is necessary for further calculations兲, ␤⬘ = ␩␨ / m is the dimensionless friction parameter, ␨ is the viscous drag coefficient, and ␭共t⬘兲 is the white noise driving force so that ␭共t⬘兲 = 0,

FIG. 1. Potential V共x兲 = −␥共cos x + yx兲 + C with Vc = 2␥共冑1 − y 2 + y arcsin y兲 and V = 2␲␥y.

fusion on a surface has been given by Pollak and collaborators in Refs. 关38–40兴. Neither Ferrando et al. or Pollak et al. extended their comparisons of the turnover formula with the exact solution to the tilted cosine potential, i.e., F ⫽ 0, which we reiterate is the object of the present paper. This general case differs in many respects from that considered by Ferrando et al. 关32,37兴 as now the dynamic response strongly depends on the tilt parameter F. Here we present the results of a detailed comparison of the Mel’nikov turnover formula with a matrix continued fraction solution for the dynamic structure factor and longest relaxation time for the onedimensional translational Brownian motion in a tilted periodic potential, Eq. 共1兲. In this way, the range of validity of the approximate analytic solutions for dynamic structure factor ˜S共k , w兲 and the escape rate ⌫ may be ascertained. Our matrix continued fraction solution owes much to the method of Risken 关11兴. However, the hierarchy of differentialrecurrence equations for statistical moments 关which is the basis of the evaluation of ˜S共k , ␻兲兴 is derived directly from the underlying Langevin equation without recourse to the Fokker-Planck equation. II. BASIC EQUATIONS

We consider the one-dimensional translational Brownian motion of a particle of mass m in a tilted periodic potential, Eq. 共1兲. On introducing the normalized coordinate x, time t⬘, tilt y, and barrier ␥ parameters as

␭共t1⬘兲␭共t2⬘兲 = ␩␤⬘␦共t1⬘ − t2⬘兲.

共6兲

The overbar means the statistical average over an ensemble of particles which have all started at time t⬘ with the same initial position x共t⬘兲 = x and velocity x˙共t⬘兲 = x˙. Equation 共5兲 is interpreted here as a stochastic differential equation of the Stratonovich type 关11,12兴. Typical examples of physical systems modeled by Brownian motion in a tilted periodic potential and described by the Langevin Eq. 共5兲 are presented in Table I. The corresponding Klein-Kramers 共Fokker-Planck兲 equation for the joint probability density function W共x , x˙ , t⬘兲 of the phase space variables x and x˙ may be written 关11兴

⳵W = LFPW, ⳵ t⬘

共7兲

where the Fokker-Planck operator LFP is given in our dimensionless variables by LFPW = − x˙

冉

冊

⳵W 1 ⳵V ⳵W ⳵ 共x˙W兲 1 ⳵2W + + ␤⬘ + . 共8兲 2 ⳵ x˙2 ⳵ x 2 ⳵ x ⳵ x˙ ⳵ x˙

The first two terms on the right-hand side of Eq. 共8兲 comprise the convective or Liouville term describing in the absence of dissipation the undamped streaming motion along the energy trajectories in phase space corresponding to Hamilton’s equations. The last term 共the diffusion term兲 represents the interchange of energy 共dissipative coupling兲 with the heat bath. The periodic solutions of Eqs. 共5兲 or 共7兲 cannot describe escape of the particle from the well because the potential 共1兲 contains only one well with period 2␲. To investigate the process across the multiwell potential generated by Eq. 共1兲 one has to obtain a nonperiodic solution of Eqs. 共5兲 or 共7兲关11兴. In order to obtain a nonperiodic solution of the KleinKramers Eq. 共7兲, one makes the ansatz 关11兴 W共x,x˙,t⬘兲 =

冕

1/2

˜ 共k,x,x˙,t⬘兲eikxdk, W

共9兲

−1/2

x=

2␲ z, a

y=

aF , 2␲V0

␩=

a 2␲

冑

␥=

V0 , k BT

t⬘ =

m , 2kBT

the potential Eq. 共1兲 takes the form 共see Fig. 1兲

t

␩

,

共3兲

˜ is periodic in x with period 2␲ and it is assumed where W that k is restricted to the first Brillouin zone, −1 / 2 ⱕ k ˜ can then be expanded in a ⱕ 1 / 2. The periodic function W truncated Fourier series in x and in orthogonal Hermite functions in x˙ 关11兴. A similar approach may be used in solution of the Langevin Eq. 共5兲. Following Refs. 关11,32兴, we calculate the dynamic structure factor ˜S共k , ␻兲, which is the time Fourier transform of the

061101-3


COFFEY et al.

TABLE I. Physical systems modeled by Brownian motion in a tilted periodic potential V共x兲 = −␥共cos x + yx兲 and described by the Langevin Eq. 共5兲.

System

Physical Parameters

Translational Brownian motion of a particle in a tilted periodic potential 关3,11兴

x: dimensionless coordinate ␭: random force m: mass of the particle V0: potential magnitude F: constant force ␨: friction coefficient

A point Josephson tunneling junction: the resistively and capacitively shunted junction 共RCSJ兲 model 关1,2,11,12兴

Rotational Brownian motion of a damped pendulum driven by a constant torque 关1,12兴

Dimensionless Magnitude ␥

Tilt Parameter y

Characteristic Time

V0 k BT

aF 2␲V0

m a 2␲ 2kBT

␨␩ m

បIc 2ekBT

Idc Ic

ប C 2e 2kBT

冑

RC

V0 k BT

F V0

I 2kBT

␵␩ I

x: phase difference ␭: noise current Ic: maximum Josephson current Idc: bias dc current e: electron charge ប: Planck constant C: capacity R: resistance x: angular coordinate ␭ random torque I: moment of inertia V0: potential magnitude F: constant torque ␵: friction coefficient

equilibrium translational correlation function S共k , t⬘兲 = 具eik关x共t⬘兲−x共0兲兴典0, where x共0兲 is the initial value of x共t⬘兲, and the angular brackets denotes a stationary averaging. Thus we introduce the set of stationary correlation functions cn,p共k , t⬘兲 defined as cn,p共k,t⬘兲 =

1

冑2 n! n

␩

冑

冑

Dimensionless Damping ␤⬘

␩

for the correlation functions cn,p共k , t⬘兲. This is accomplished as follows. Recalling that the usual rules of analysis apply to stochastic differential equations of the Stratonovich type 关11,12兴 and noting that 关41兴 d Hn共z兲 = 2nHn−1共z兲, dz

具eik关x共t⬘兲−x共0兲兴eipx共t⬘兲+V关x共t⬘兲兴/2Hn关x˙共t⬘兲兴典0 ,

Hn+1共z兲 = 2zHn共z兲 − 2nHn−1共z兲, 共11兲

共10兲 where Hn共z兲 is the Hermite polynomial of order n 关41兴. Equation 共5兲 now may be recast as a hierarchy of equations

one has

d i共p+k兲x共t⬘兲+V关x共t⬘兲兴/2 兵e Hn关x˙共t⬘兲兴其 = 兵2nx¨共t⬘兲Hn−1关x˙共t⬘兲兴 + 关i共p + k兲 + ⳵xV关x共t⬘兲兴/2兴x˙共t⬘兲Hn关x˙共t⬘兲兴其ei共p+k兲x共t⬘兲+V关x共t⬘兲兴/2 dt⬘ = 兵n共2␭共t⬘兲Hn−1关x˙共t⬘兲兴 − 2␤⬘共n − 1兲Hn−2关x˙共t⬘兲兴 − ␤⬘Hn关x˙共t⬘兲兴 − Hn−1关x˙共t⬘兲兴⳵xV关x共t⬘兲兴兲 + 共i共p + k兲 + ⳵xV关x共t⬘兲兴/2兲共nHn−1关x˙共t⬘兲兴 + Hn+1关x˙共t⬘兲兴/2兲其ei共p+k兲x共t⬘兲+V关x共t⬘兲兴/2 . By averaging Eq. 共12兲 over the realizations of x共t⬘兲, using the identity 关12兴 061101-4

共12兲



F关x共t⬘兲兴Hn−1关x˙共t⬘兲兴␭共t⬘兲 = ␤⬘共n − 1兲F共x兲Hn−2共x˙兲

共13兲

共which is valid for an arbitrary function F关x共t⬘兲兴兲, and forming the stationary averages cn,q共t兲共as described in Ref. 关12兴兲, we have d cn,p共k,t⬘兲 = − n␤⬘cn,p共k,t⬘兲 + dt⬘ +

冑再冋 n 2

冑

n+1 2

册

再冋

i共p + k兲 −

册

␥y i␥ cn+1,p共k,t⬘兲 − 关cn+1,p+1共k,t⬘兲 − cn+1,p−1共k,t⬘兲兴 2 4

␥y i␥ cn−1,p共k,t⬘兲 + 关cn−1,p+1共k,t⬘兲 − cn−1,p−1共k,t⬘兲兴 . 2 4

i共p + k兲 +

Having determined the c0,p共k , t兲 from Eq. 共14兲, the function S共k , t⬘兲 can be evaluated as ⬁

⬁

p=0

p=0

共15兲 where the initial conditions c0,p共k , 0兲 for c0,p共k , t⬘兲 are given by 关11兴

冕

2␲

eipx−V共x兲/2dx,

Z=

0

冕

2␲

e−V共x兲dx

0

共16兲

␶−1共k兲 = lim i␻ ␻→0

eikx共t⬘兲 = eV关x共t⬘兲兴/2 兺 a pei共p+k兲x共t⬘兲 , p=0

冕

␶M = ␶IHD/A共␤⬘␦, ␥y兲,

0

Equation 共14兲共equivalent to that derived by Risken 关11兴 from the Fokker-Planck equation兲 is the desired recurrence equation for the statistical moments. This equation can be solved by the matrix continued fraction method to yield the one-sided Fourier transform of the dynamic structure factor ˜S共k , ␻兲 = 兰⬁S共k , t⬘兲e−i␻␩t⬘dt⬘ 共see Appendix A兲. Having deter0 mined ˜S共k , ␻兲, the longest relaxation time ␶ and decay rate ⌫ ⬇ ␶−1 can be evaluated as follows. Defining the decay rate ⌫ ⬇ ␶−1 as

冕

共19兲

共20兲

where ␶IHD is the longest relaxation time in the IHD limit which is given by the Kramers IHD formula 关17兴

␶IHD = 4␲␩共冑␤⬘2 + 2␥ − ␤⬘兲−1e⌬V ,

共21兲

⌬V is the height of the lowest barrier 共see Fig. 1兲, viz., ⌬V = 2␥共冑1 − y 2 + y arcsin y − ␲y/2兲.

共22兲

The function A in Eq. 共20兲 is defined as 关17兴

1/2

␶−1共k兲dk,

.

III. MEL’NIKOV’S UNIVERSAL EQUATION

* dxe−ipx−V共x兲/2 = 共2␲兲−1Zc0,p 共k,0兲.

␶−1 =

册

−1

An analytical approximation to the decay rate ␶−1 in a tilted periodic potential has been obtained by Mel’nikov 关17兴 by reducing the Klein-Kramers equation to an integral equation of Wiener-Hopf type. The Mel’nikov expression for the longest relaxation time ␶ is 关17兴

⬁

2␲

冋

˜S共k,0兲 −1 ˜S共k, ␻兲

共18兲

The dynamic structure factor ˜S共k , ␻兲 can be calculated exactly by solving the differential recurrence Eq. 共14兲 using the matrix continued fractions 关11兴共as described in Appendix A兲 and may be compared with the characteristic time ␶共k兲 and the decay rate ⌫ ⬇ ␶−1 evaluated from Eqs. 共19兲 and 共17兲.

共the asterisk denotes the complex conjugate兲. Here we have noted that

a p = 共2␲兲−1

共14兲

h共k兲 , i␻ + ␶−1共k兲

˜S共k, ␻兲 =

where ␶−1共k兲 can be extracted as 关32兴

* 共k,0兲c0,p共k,t兲, S共k,t⬘兲 = 兺 a pc0,p共k,t兲 = 共2␲兲−1Z 兺 c0,p

c0,p共k,0兲 = Z−1

冎

冎

共17兲

A共d,g兲 =

−1/2

冕

1/2

w共k,d,g兲dk,

共23兲

−1/2

where the time ␶共k兲 is a characteristic time associated with the long time behavior of the function S共k , t⬘兲 which can be approximated as t → ⬁ by an exponential S共k , t⬘兲 ⬃ h共k兲e−t/␶共k兲关32兴. In the frequency domain as ␻ → 0, this approximation corresponds to

where

061101-5

w共k,d,g兲 = 4 sin共␲k兲sin关␲共k + ig兲兴e−␲g+⌶共k,d,g兲−⌶共0,2d,0兲/2 , 共24兲


COFFEY et al. ⬁

⌶共k,d,g兲 = 兺 n−1cos关␲n共2k + ig兲兴

再冋冉冊册

n=1

冑nd

⫻ erfc

冋冑冉

+ erfc

nd

1 ␲g + 2 d

1 ␲g − 2 d

冊册

e n␲g

冎

e−n␲g ,

共25兲

and erfc共x兲 is the complementary error function defined as 2 erfc共x兲 = 冑2␲ 兰⬁x e−t dt 关41兴. 共Other forms of the function A are given in Ref. 关17兴兲. For zero tilt 共y = 0兲, Eq. 共24兲 can be simplified to yield 关32兴 w共k,d,0兲 = 4 sin2共␲k兲e⌶共k,d,0兲−⌶共0,2d,0兲/2 ,

共26兲

冑nd / 2兲cos共2␲nk兲. The pawhere rameter ␦ is the dimensionless action associated with the path along the top of lowest barrier given by ⬁ n−1erfc共 ⌶共k , d , 0兲 = 2兺n=1

␦=2

冕

−arcsin y

冑− V共x兲dx

x0

= 2 冑␥

冕

−arcsin y

冑冑1 − y2 − cos x + y共x + arcsin y兲dx,

x0

共27兲 where x0 is an appropriate solution of the equation V共x0兲 = 0. On this path, a particle starts with zero velocity at the top of the barrier and, having descended into the well, returns again to the top of the barrier 关36兴. For y 1, ␦ has the following behavior: 1 ␦ = 1 − 兵␲关9 − 4 log共␲y/4兲兴y + 冑␲y 3/2其 + O共y 2兲, 32 ␦0

FIG. 2. The normalized action ␦ / ␦0 vs the tilt parameter y. Solid line: Eq. 共27兲. Dashed line: Eq. 共28兲. Dashed-dotted line: Eq. 共29兲.

equations in terms of the equilibrium 共stationary兲 distribution function W0 and diffusion coefficient D共2兲 only. The advantage of such a method is that it allows us to obtain VHD and VLD solutions, valid for all barrier heights including very low barriers, where asymptotic methods 共like that of Mel’nikov兲 are not applicable. In Sec. IV, we apply the theory of MFPT for evaluation of the decay rate ␶−1 in the VHD and VLD limits at zero tilt y = 0. Here the results can be given in a closed form. IV. THE MFPT ASYMPTOTES FOR THE DECAY RATE AT ZERO TILT

In the VHD limit 共␤⬘ 1兲, the appropriate single variable Fokker-Planck 共Smoluchowski兲 equation for the probability density function W共x , t⬘兲 is 关11兴

再

where ␦0 = 8冑2␥ is the action for zero tilt. If y approaches 1, ␦ can be approximated as 关36兴

␦ ⬇ 3␦0关2共1 − y兲兴5/4/10.

共29兲

The normalized action ␦ / ␦0 evaluated from Eqs. 共27兲–共29兲 is shown in Fig. 2. Now A共d兲 → 1 at d → ⬁ and A共d兲 / d → 2 at d → 0. Thus for very high damping, ␤⬘ → ⬁, Eq. 共20兲 yields the VHD asymptote

␶VHD = 4␲␩␤⬘␥−1e⌬V .

␶VLD =

␲␩

␤⬘␦冑␥/2

e⌬V ,

Thus, noting that the diffusion coefficient D共2兲 = 共2␤⬘兲−1, the longest relaxation time is given by 关42兴 MFPT ␶ ⬃ ␶VHD = 2 ␤ ⬘␩

共30兲

In the VLD limit 共␤⬘ → 0兲, one has from Eq. 共20兲共31兲

which is the result of Büttiker and Landauer 关36兴共in our notation兲. The Mel’nikov method 共as other asymptotic methods兲 is valid in the high barrier 共low temperature兲 limit only. In order to accurately estimate ␶ for the low barrier, one should use other methods such as mean first passage time 共MFPT兲关13兴. The MFPT may be easily calculated for all systems with dynamics governed by single variable Fokker-Planck

冎

1 ⳵ ⳵ ⳵2 关␥ sin xW共x,t⬘兲兴 + 2 W共x,t⬘兲 . W共x,t⬘兲 = ⳵ t⬘ 2␤⬘ ⳵ x ⳵x 共32兲

共28兲

= 2 ␤ ⬘␩

冕冕 ␲

x

−␲

−␲

冕

␲

−␲

1 W0共x兲

冕

x

−␲

W0共␾兲d␾dx

e␥共cos ␾−cos x兲d␾dx,

共33兲

where the equilibrium Boltzmann distribution function W0共x兲 is given by W0共x兲 =

e␥ cos x 2 ␲ I 0共 ␥ 兲

共34兲

and I0共z兲 is the modified Bessel function 关41兴. The time MFPT ␶VHD is the time needed for the Brownian particle starting at the top of the barrier x = −␲ to reach, having attended a well bottom, the neighboring top at x = ␲. In the opposite low damping limit 共␤⬘ 1兲, in order to obtain a single variable Fokker-Planck equation, one may

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introduce as variables the energy of the particle ␧ = x˙2 − ␥ cos x

共35兲

and the time w 共phase兲 measured along a closed trajectory in phase space. These comprise the action-angle variables 关13兴 of the problem. The energy ␧ varies very slowly with time. Consequently, it is a slow variable in comparison to the phase w. By using the method of Praestgaard and van Kampen 关43兴, i.e., averaging the Fokker-Planck equation 共7兲 over the fast phase variable w, we have a single variable Fokker-Planck equation for the probability density function W共␧ , t兲 in energy space

冋冉

册

冊

1 ⳵ ⳵ 2 ⳵2 W共␧,t⬘兲 = 2␤⬘ x˙ 共␧兲 − + 2 x˙2共␧兲 W共␧,t⬘兲, 2 ⳵ t⬘ ⳵␧ ⳵␧ 共36兲 where the double overbar denotes averaging over the fast phase variable. Noting that the diffusion coefficient is

关

兴

D共2兲 = 2␤⬘x˙2共␧兲 = 2␤⬘ ␧ + ␥cos x共␧兲 , the time ␶ is then given by

MFPT ␶ ⬃ ␶VLD =

␩ 2␤⬘

冕

␥

−␥

W−1 0 共␧兲

冕

␧

−␥

W0共x兲dx

␧ + ␥ cos x共␧兲

d␧.

共37兲

The calculation of W0共␧兲 and the integrals in Eq. 共37兲 is described in Appendix B. The areas of applicability of the VHD and VLD asymptotes 关Eqs. 共33兲 and 共37兲兴 are the same as those of the corresponding Fokker-Planck equations 共32兲 and 共36兲, viz., the VHD and VLD limits, respectively. In practice, Eqs. 共33兲 and 共37兲 may be used at ␤⬘ ⬎ 5 and ␤⬘ ⬍ 0.01. V. RESULTS AND DISCUSSION

The greatest relaxation time predicted by the turnover formula Eq. 共20兲 and the inverse decay rate calculated numerically by matrix continued fraction methods are shown in Figs. 3 and 4 as functions of ␤⬘ for different values of the barrier height and tilt parameters. Here, the IHD 关Eq. 共21兲兴, VHD 关Eq. 共30兲兴, and VLD 关Eq. 共31兲兴 asymptotes for ␶ are also shown for comparison. Apparently in the high barrier limit, Eq. 共20兲 provides a good approximation to the decay rate for all values of the friction parameter ␤⬘ including the VHD, VLD, and the Kramers turnover regions. In spite of very good overall agreement between numerical results and the universal Eq. 共20兲, a marked difference of order of 20% between numerical and analytical results exists in the VLD region at moderate barriers 共this difference decreases with increasing ␥, see Fig. 3兲. Such a difference has already been noted for other systems 共see, e.g., Refs. 关22,37兴兲. In order to improve the accuracy of the universal turnover formula in this region, Mel’nikov 关44兴 suggested a systematic way of accounting for finite-barrier corrections. Analysis of the translational Brownian motion in a periodic cosine potential has demonstrated 关45兴 that if such a correction is included,

FIG. 3. The longest relaxation time ␶ / ␩ vs the friction parameter ␤⬘ for the tilt parameter y = 0.3 and different values of the barrier parameter ␥ = 5, 10, and 15. Solid line: the universal Mel’nikov equation 共20兲; dashed line: the VHD equation 共30兲; dashed dotted line: the IHD equation 共21兲; dotted line: the VLD equation 共31兲; filled circles: exact numerical solution, Eq. 共17兲.

the accuracy of the universal formula is considerably improved for Brownian motion in a periodic potential with tilt y = 0. One would expect a similar improvement for nonzero tilt. For zero tilt, y = 0, the greatest relaxation time ␶ predicted by the Mel’nikov universal equation 共20兲 and the inverse decay rate calculated numerically by matrix continued fraction methods are shown in Fig. 5 as functions of ␤⬘ for different values of the barrier height including the very low barrier ␥ = 0.1. Here, the VHD 关Eq. 共33兲兴 and VLD 关Eq. 共37兲兴 asymptotes for ␶ calculated via the theory of MFPT are also shown for comparison. In the VHD and VLD limits, these asymptotes may be used to estimate ␶ for all barrier heights. For small barriers 共e.g., ␥ = 0.1兲, the Mel’nikov universal for-

FIG. 4. ␶ / ␩ vs the friction parameter ␤⬘ for the barrier parameter ␥ = 10 and different values of the tilt parameter y = 0.0, 02, 04, and 0.6. Solid line: the universal Mel’nikov equation 共20兲; dashed line: the VHD equation 共30兲; the IHD equation 共21兲; dotted line: the VLD equation 共31兲; filled circles: exact solution, Eq. 共17兲.

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COFFEY et al.

FIG. 5. ␶ / ␩ vs ␤⬘ for the tilt parameter y = 0 and ␥ = 0.1, 5, and 10. Solid line: the universal Mel’nikov equation 共20兲; dashed line: the VHD MFPT equation 共33兲; dotted line: the VLD MFPT equation 共B7兲; dashed dotted line: the ad hoc extrapolating equation 共38兲; filled circles: exact solution, Eq. 共17兲.

mula is obviously not valid. However, here the simple ad hoc extrapolating equation 关13兴 MFPT MFPT ␶ ⬃ ␶VHD + ␶VLD

共38兲

provides a satisfactory estimate of the longest relaxation time ␶ for all damping 共see Fig. 5兲. The real and imaginary parts of the normalized dynamic structure factor ˜S共k , ␻兲 / ˜S共k , 0兲, for various values of the tilt parameter y are shown in Fig. 6 with barrier parameter ␥ = 10, the friction coefficient is ␤⬘ = 10, and k = 0.2. For comparison, we also show in this figure the pure Lorentzian spectra ˜S 共k, ␻兲 1 L , = ˜S共k,0兲 1 + i␻␶k

共39兲

where the relaxation time ␶k = 1 / Re关␶−1 M 共k兲兴 is related to the ␶M from the universal equation 共20兲 via ␶−1 M −1 = 2兰1/2 0 Re关␶ M 共k兲兴dk. Apparently the simple equation 共39兲 describes perfectly the low frequency part of the dynamic structure factor ˜S共k , ␻兲 / ˜S共k , 0兲. Thus we have demonstrated how the matrix continued fraction solution of nonlinear Langevin equations may be successfully applied to a Brownian particle moving in the tilted periodic potential, Eq. 共1兲, for wide ranges of the barrier parameter ␥, tilt parameter y, and the damping parameter ␤⬘. We have shown that in the low temperature limit, the Mel’nikov formula for the longest relaxation time, Eq. 共20兲, yields satisfactory agreement with the numerical results for all values of damping. Moreover, the Mel’nikov equation 共20兲 allows one to accurately estimate the damping dependence of the low-frequency parts of the dynamic structure factor ˜S共k , ␻兲 via the simple approximate analytic formula, Eq. 共39兲. In practical calculations, Eq. 共20兲 may be used for ␥ ⱖ 1.5 and 0 ⱕ y ⱕ 0.9. For 1.0ⱖ y ⬎ 0.9 共where a parabolic

FIG. 6. The real and imaginary parts of the normalized dynamic structure factor ˜S共k , ␻兲 / ˜S共k , 0兲 vs ␻␩ for various values of the tilt parameter y and for the barrier parameter ␥ = 10, the friction coefficient ␤⬘ = 10 and k = 0.2. Solid lines: numerical calculation. Asterisks: Eq. 共39兲.

approximation of the barrier top is no longer valid兲, the matrix continued fraction solution must be used. For small ␥ 共where asymptotic methods like Mel’nikov one are not applicable兲 and y = 0, Eqs. 共33兲, 共37兲, and 共38兲 yield a good estimate for the longest relaxation time. To conclude, we have shown that the universal turnover formula for evaluating the longest relaxation time ␶ as a function of the dissipation parameter for Brownian particles in a tilted periodic potential provides in the low-temperature limit excellent agreement with the exact continued fraction solution for all values of the dissipation parameter including the VLD,VHD, and turnover regions. A similar conclusion may be drawn 关22–29兴 for various stochastic systems modeled by Brownian motion in multiwell potentials with equivalent and nonequivalent wells, where the validity of the universal equation for ␶ ⬃ ⌫−1 共⌫ is the escape rate兲 has been verified by comparison with numerical solutions of the underlying Langevin or Fokker-Planck equations. Thus the universal turnover equation for the escape rate appears to yield an effective and powerful tool for evaluating the damping dependence of the prefactor A in Eq. 共2兲 for a wide class of nonlinear stochastic systems even as in the present problem where the stationary solution differs from the MaxwellBoltzmann distribution. It is obvious that the description of the relaxation processes in the context of Eq. 共20兲 neglects quantum effects. These effects are important at very low temperatures and necessitate an appropriate quantum mechanical treatment. As mentioned in Sec. I, Mel’nikov 关17兴 and Rips and Pollak 关20兴 have extended the turnover formula for mechanical particles to account for quantum tunneling in a semiclassical way. We have seen that classical turnover

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formula for the escape rate may be confirmed as an accurate approximation to the exact escape rate of a mechanical Brownian particle because one may exactly solve the corresponding Fokker-Planck equation describing the evolution of the distribution function in phase space using matrix continued fractions. In order to verify formulae for the escape rate which incorporates quantum effects, it is necessary to identify the appropriate quantum mechanical master equation underlying the relaxation process, which becomes the FokkerPlanck equation in the classical limit 关46,47兴. An appropriate candidate seems to be the Caldeira-Leggett 关48兴 quantum Fokker-Planck equation for the time evolution of the Wigner transform of the reduced density operator 共here the relationship between the quantum density operator and the semiclassical distribution function is given by the Wigner transformation 关49兴兲. The Caldeira-Leggett approach may be used for all values of damping. In the VHD limit, one can use the quantum Smoluchowski equation, which to leading order coincides with the classical Smoluchowski equation, but contains essential quantum corrections 关50–52兴. Such an approach to the quantum mechanical problem also lends itself to solution by continued fraction methods 关46,47兴 so that the universal quantum escape rate equations can be tested in a similar manner to that which we have described here.

冢冑冢

Q+n 共k兲 = i

冑

n 2

=i

n−1 2

¯

By introducing the column vectors Cn共k , t⬘兲 defined as 关11兴

¯

cn−1,−1共k,t⬘兲

Cn共k,t⬘兲 =

0

⯗

共A2兲

˜ 共k , s兲 is The exact solution of Eq. 共A2兲 for the spectrum C 1 given by in terms of a matrix continued fraction 共A3兲

where the matrix continued fraction ⌬n共k , s兲 is defined by the recurrence equation

n = 1,2,3, . . . ,

Eq. 共14兲 can be written in the vector tridiagonal form d Cn共k,t⬘兲 = Q−n 共k兲Cn−1共k,t⬘兲 − ␤⬘共n − 1兲Cn共k,t⬘兲 dt⬘ + Q+n 共k兲Cn+1共k,t⬘兲, where the matrices

Q+n 共k兲

and

⯗

⯗

− ␥/4

0

¯

k + i␥y/2

− ␥/4

¯

⯗

¯ − 1 + k − i␥y/2 − ␥/4

,

⯗

⯗

¯

cn−1,0共k,t⬘兲

cn−1,1共k,t⬘兲

␥/4 ⯗

¯

冢冣 ⯗

␥/4 0 ⯗

˜ 共k,s兲 − Q+共k兲C ˜ 共k,s兲 − Q−共k兲C ˜ 共k,s兲关␩s + ␤⬘共n − 1兲兴C n n+1 n−1 n n

˜ 共k,s兲 = ⌬ 共k,s兲C 共k,s兲, C 1 1 1

APPENDIX A: MATRIX CONTINUED FRACTION SOLUTION

⯗

By Laplacian transformation, Eq. 共A1兲 can be rearranged as the set of matrix three-term recurrence equations

= ␦n,1C1共k,0兲.

This publication has emanated from research conducted with the financial support of Science Foundation Ireland 共Project No. 05/RFP/PHY/0070兲.

¯ − 1 + k + i␥y/2

Q−n 共k兲

ACKNOWLEDGMENTS

1 + k + i␥y/2 ¯ ⯗

⯗

0 ␥/4 k − i␥y/2 ␥/4 − ␥/4 1 + k − i␥y/2 ⯗ ⯗

Q−n 共k兲

冣冣

共A1兲

are

,

¯

¯

.

¯

− ⌬n共k,s兲 = 兵关␩s + ␤⬘共n − 1兲兴I − Q+n 共k兲⌬n+1共k,s兲Qn+1 共k兲其−1

˜ 共k , s兲 and and I is the unit matrix. Having determined C 1 noting Eq. 共15兲, one can calculate the dynamic structure factor ˜S共k , ␻兲 as ˜S共k, ␻兲 = 共2␲兲−1ZC†共k,0兲⌬ 共k,i␻兲C 共k,0兲, 1 1 1

共A4兲

where the symbol “†” designates transformation of the column vector C1共k , 0兲 to a row vector and its conjugation. The exact matrix continued fraction solution, Eq. 共A4兲, we have obtained is easily computed 共algorithms for calculating matrix continued fractions are discussed in Refs.

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COFFEY et al.

关11,12兴兲. As far as practical calculations of the infinite matrix continued fraction are concerned, we approximate it by a matrix continued fraction of finite order 共by putting ⌬n+1 = 0 at some n = N兲; simultaneously, we confine the dimensions of the infinite matrices Q−n , Q+n , and I to a finite value 共2Q + 1兲 ⫻ 共2Q + 1兲. The N and Q were determined in such way that a further increase of N and Q did not change the results. Both N and Q depend on the dimensionless magnitude 共␥兲 and damping 共␤⬘兲 parameters and must be chosen taking into account the desired degree of accuracy of the calculation. Both N and Q increase with decreasing ␤⬘ and increasing ␥. APPENDIX B: UNDAMPED LIMIT FOR ZERO TILT

For ␤⬘ 1, the dynamics of the system differ but little from those of the undamped limit 共␤⬘ = 0兲 when the Langevin force vanishes. For ␤⬘ = 0, the energy of the particle 关see Eq. 共35兲兴 is a constant of the motion and so the dynamics of the particle in the potential well are described by the following deterministic nonlinear differential equation:

m sn2共u兩m兲 = 1 −

共B5兲 where K共m兲 and E共m兲 are the complete elliptic integrals of the first and second kind, respectively 关41,53兴, and q = exp关−␲K共1 − m兲 / K共m兲兴. Thus we have from Eqs. 共B2兲, 共B3兲, and 共B5兲 cos x共␧兲 = 2

= d cos x共t⬘兲 = ± 冑关␧ + ␥ cos x共t⬘兲兴关1 − cos2 x共t⬘兲兴. dt⬘ 共B1兲

For −␥ ⬍ ␧ ⬍ ␥, Eq. 共B1兲 has a solution cos x共t⬘兲 = 1 − 2m共␧兲sn2共t⬘冑␥/2 ± w兩m共␧兲兲,

共B2兲

关1 − m共␧兲sin2 x⬘兴−1/2dx⬘ ,

0

共B3兲 m共␧兲 =

1 2 e−x˙ 共0兲+␥ cos x共0兲dx共0兲dx˙共0兲 2 ␲ I 0共 ␥ 兲

␥+␧ . 2␥

3/2

by making the transformation of the variables 兵x共0兲 , x˙共0兲其 → 兵w , ␧其, and by integrating the distribution function W0共␧兲 over the phase w, we have

where sn共u 兩 m兲 is the Jacobian doubly periodic elliptic function 关41兴, arcsin兵冑m共␧兲 sin关x共0兲/2兴其

共B6兲

W0关x共0兲,x˙共0兲兴dx共0兲dx˙共0兲

or

冕

E共m兲 − 1, K共m兲

where the double overbar denotes averaging over the fast phase variable, the dependence of the modulus m on ␧ is given by Eq. 共B4兲. In order to evaluate equilibrium averages, one needs an equation for the stationary distribution function Wst. On noting that Wst is the equilibrium Maxwell-Boltzmann distribution W0, viz.,

x˙共t⬘兲2 = ␧ + ␥ cos x共t⬘兲

w=

冋册

2␲2 nqn n␲u E共m兲 ⬁ − 2 , 兺 2n cos n=1 K共m兲 K 共m兲 1−q K共m兲

W0共␧兲d␧ =

冦

冑2K关m共␧兲兴e−␧ d␧ 共− ␥ ⬍ ␧ ⬍ ␥兲 ␲3/2I0共␥兲冑␥ 冑2K关m−1共␧兲兴e−␧ d␧ 共␧ ⬎ ␥兲 ␲3/2I0共␥兲冑␥m共␧兲

,

with 兰−⬁␥W0共␧兲d␧ = 1. Thus according to Eqs. 共37兲 and 共B6兲, in the context of the MFPT approach, ␶ in the VLD limit is given by

共B4兲 1 ␶ ⬃ ␩ 4 ␤ ⬘␥

In order to proceed, we recall the Fourier series for sn 共u 兩 m兲关53兴 2

关1兴 G. Barone and A. Paterno, Physics and Applications of the Josephson Effect 共Wiley, New York, 1982兲. 关2兴 K. K. Likharev, Dynamics of Josephson Junctions and Circuits 共Gordon and Breach, New York, 1986兲. 关3兴 W. Dieterich, P. Fulde, and I. Peschel, Adv. Phys. 29, 527 共1980兲. 关4兴 W. W. Chow, M. O. Scully, and E. W. Van Stryland, Opt. Commun. 15, 6 共1975兲. 关5兴 A. J. Viterbi, Principles of Coherent Communication 共McGraw-Hill, New York, 1966兲.

冧

冕

␥

−␥

冕

␧

e␧−xK关m共x兲兴dx

−␥

m共␧兲K关m共␧兲兴 + E关m共␧兲兴

d␧.

共B7兲

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