Thermally activated escape rate for a Brownian

THE JOURNAL OF CHEMICAL PHYSICS 124, 024107 共2006兲

Thermally activated escape rate for a Brownian particle in a double-well potential for all values of the dissipation Yu. P. Kalmykova兲 Groupe de Physique Moléculaire, MEPS, Université de Perpignan, 52, Avenue Paul Alduy, 66860 Perpignan Cedex, France

W. T. Coffey Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland

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

共Received 29 September 2005; accepted 27 October 2005; published online 11 January 2006兲 The translational Brownian motion in a 共2-4兲 double-well potential is considered. The escape rate, the position correlation function and correlation time, and the generalized susceptibility are evaluated from the solution of the underlying Langevin equation by using the matrix-continued fraction method. The escape rate and the correlation time are compared with the Kramers theory of the escape rate of a Brownian particle from a potential well as extended by Mel’nikov and Meshkov 关J. Chem. Phys. 85, 1018 共1986兲兴. It is shown that in the low-temperature limit, the universal Mel’nikov and Meshkov expression for the escape rate provides a good estimate of both escape rate and inverse position correlation time for all values of the dissipation including the very low damping 共VLD兲, very high damping 共VHD兲, and turnover regimes. Moreover, for low barriers, where the Mel’nikov and Meshkov method is not applicable, analytic equations for the correlation time in the VLD and VHD limits are derived. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2140281兴 I. INTRODUCTION

The translational Brownian motion in a 共2-4兲 doublewell potential, V共x兲 = 21 ax2 + 41 bx4,

− ⬁ ⬍ x ⬍ ⬁,

共1兲

where a and b are constants, is almost invariably used to describe the noise-driven motion in bistable physical and chemical systems. Examples are such diverse fields as simple isometrization processes,1–5 chemical reaction-rate 6–14 bistable nonlinear oscillators,15–17 second-order theory, phase transitions,18 and nuclear fission and fusion.19,20 The number of papers devoted to the problem is enormous and many methods of solution have been presented because the stochastic dynamics of barrier crossing transitions is of fundamental significance in physics.21–26 One of the most important characteristics associated with the Brownian motion in any multiwell potential is the friction and temperature dependence of the greatest relaxation time. In the present context we remark that the greatest relaxation time 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.25 As far as the calculation a兲

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of ␭1 is concerned, the secular equation may be generated by averaging the appropriate Langevin equation over its realizations in phase space yielding the hierarchy of differentialrecurrence equations governing the decay functions of the system.27 Alternatively, 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 KleinKramers equation兲 in Fourier series in the position and velocity variables.28 In each of the two methods, the secular determinant results from truncation of the set of differentialrecurrence relations at a number large enough to achieve 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 Kramers6 in connection with thermally activated escape of particles out of a potential well. Kramers6 evaluated the prefactor A in an Arrhenius-type equation for the escape rate ⌫ over the potential barrier ⌬V 共reaction velocity in the case of chemical reactions兲, viz.,

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⌫=A

J. Chem. Phys. 124, 024107 共2006兲

Kalmykov, Coffey, and Titov

␻a −⌬V/kT e , 2␲

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. 25 and 29兲, k is the Boltzmann constant, and T is temperature. Now if the escape rates for mechanical Brownian particles are calculated by the Kramers method, three regimes of damping 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. Kramers6 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, known as the Kramers turnover problem, was solved nearly fifty years later by Mel’nikov and Meshkov.30,31 They postulated from heuristic reasoning that a formula valid for all values of the damping may be given by simply multiplying the general IHD result for ⌫ by their bridging integral. Mel’nikov30 have further extended the bridging integral method to take into account quantum effects in a semiclassical way. Furthermore, Grabert32 and Pollak et al.33 later presented a complete solution of the Kramers turnover problem and have shown that the Mel’nikov and Meshkov universal formula can be obtained without ad hoc interpolation between the weak and strong damping regimes. We remark that the theory of Pollak et al.33 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.34 The universal turnover formula of Mel’nikov and Meshkov has been compared with calculations based on either the solutions of the Klein-Kramers equation or on numerical simulations of the Brownian dynamics for various stochastic systems. In particular, such a comparison with the numerical results for the escape out of a single well was given in Refs. 35 and 36. The study of the one-dimensional translational Brownian motion in a periodic potential has been undertaken by Ferrando et al.37,38 The treatment of the same onedimensional problem and its generalization to diffusion on a surface was undertaken by Pollak and co-workers in Refs. 39–41. Pastor and Szabo42 and Coffey et al.43 tested the Mel’nikov-Meshkov formula in the context of the rotational Brownian motion in double-well potentials. Furthermore, Dejardin et al.44 extended the Mel’nikov-Meshkov calculation to the magnetization relaxation of single-domain ferromagnetic particles possessing nonaxially symmetric potentials of the magnetocrystalline anisotropy. 共We remark that the magnetic relaxation problem differs fundamentally from that of mechanical particles because the undamped equation of motion of the magnetization of a single domain ferromagnetic particle is the gyromagnetic equation. Thus the inertia plays no role; the part played by inertia in the mechanical system is essentially mimicked in the magnetic system for nonaxially symmetric potentials by the gyromagnetic term

which gives rise to the coupling or “entanglement” of the transverse and longitudinal modes.兲 The calculation of the longest relaxation time for various types of magnetocrystalline anisotropy has been accomplished in Refs. 45–47. It is the purpose of this paper to apply Mel’nikov and Meshkov’s universal turnover formula to the analysis of the dynamics of a Brownian particle in the double-well potential given by Eq. 共1兲. The dynamics of this system in the very high damping 共VHD兲 limit, where the inertia of the particle may be neglected, have been extensively studied either by using the Kramers escape rate theory6,25 or by solution of the Smoluchowski equation underlying the problem 共see, e.g., Refs. 11 and 47, 49–5149–51, and references cited therein兲. In the VHD limit, the traditional analysis of the problem has proceeded from the Smoluchowski equation by either converting the solution of that equation to a Sturm-Liouville problem 共e.g., Refs. 11 and 48兲 or by the solution of an infinite hierarchy of linear differential-recurrence relations for statistical moments 共e.g., Refs. 49 and 50兲. Kalmykov et al.50 共see also Ref. 27, Chap. 6兲 have formulated an analytic method of finding exact solutions of differentialrecurrence relations for statistical moments by using continued fractions. The method allows one to calculate in closed form the correlation times, the spectra of the correlation functions, and the generalized susceptibilities for the relevant dynamical variables.27,50 Yet another method of solution was recently proposed by Perico et al.51 They used the first passage time method to derive an integral expression for the correlation time defined, as usual, as the area under the curve of the positional autocorrelation function. By applying the Mori memory function formalism, they were also able to calculate the position correlation function and to compare it with various approximate formulas and computer simulations. If the inertial effects are taken into account, a large number of special solutions exist mostly for particular parameters of the above problem. In the low-temperature limit the transition rate from one well to another is of special interest and can be given analytically. In addition some results for the longest relaxation time have been obtained in the context of the escape theory, e.g., in Refs. 52–54. The position correlation function C共t兲 = 具x共0兲x共t兲典0 / 具x2共0兲典0 共the symbol 具典0 denotes the equilibrium ensemble averages兲 and its spectra for both low and very low dampings were treated in Refs. 16 and 55–57. Fourier transforms of the position correlation functions have also been obtained in Refs. 58 and 59 by a projection operator method. Furthermore, Voigtlaender and Risken60 calculated eigenvalues and eigenfunctions of the Kramers 共Fokker-Planck兲 equation for a Brownian particle in the double-well potential 共1兲 and evaluated the Fourier transforms of the position and velocity correlation functions by applying the matrix-continued fraction method.28 The method is as follows. First the distribution function is expanded in Hermite functions with respect to the velocity as originally done by Brinkman7 and then in Hermite functions with respect to position. Next by inserting this distribution function into the Fokker-Planck equation they obtain a recursion relation for the expansion coefficients. By introducing a suitable vector and matrix notation this recur-


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Thermally activated escape rate

rence relation is then cast into a tridiagonal vector recurrence relation. Finally, this vector recurrence relation is solved by matrix-continued fractions. Here we present the results of a detailed comparison of Mel’nikov and Meshkov’s formula with a matrix-continued fraction solution for the position correlation functions and its correlation time, the smallest nonvanishing eigenvalue of the corresponding Fokker-Planck equation, and generalized dynamic susceptibility. Our matrix-continued fraction solution owes much to the work of Voigtlaender and Risken.60 However, it differs from that in two aspects. The hierarchy of differential-recurrence equations for statistical moments is derived directly from the underlying Langevin equation without recourse to the Fokker-Planck equation and the algorithm of the matrix-continued fraction solution is simplified and optimized 共it is about ten times faster and may be applied for higher barriers and smaller damping constants than that used by Voigtlaender and Risken60兲.

FIG. 1. Potential V共y兲 / 共kT兲 = Ay 2 + By 4 for B = 1 and various values of A.

LFPW = − x˙

冉

冊

⳵ W 1 dV ⳵ W ␨ ⳵ kT ⳵2W + + 共x˙W兲 + . ⳵ x m dx ⳵ x˙ m ⳵ x˙ m ⳵ x˙2 共6兲

II. BASIC EQUATIONS

We consider the one-dimensional translational Brownian motion of a particle in the double-well potential 共1兲. The governing nonlinear Langevin equation is60 mx¨共t兲 + ␨x˙共t兲 + ax共t兲 + bx3共t兲 = ␭共t兲.

共3兲

In Eq. 共3兲, x共t兲 specifies the position of the particle at time t , m is the mass of the particle, ␨x˙ is the viscous drag experienced by the particle, and ␭共t兲 is the white noise driving force so that ␭共t兲 = 0,

␭共t兲␭共t⬘兲 = 2kT␨␦共t − t⬘兲.

共4兲

Here 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3兲 is interpreted here as a stochastic differential equation of the Stratonovich type.27,28 Though the nonlinearity changes the motion of x共t兲, many features of the linear equation with a ⬎ 0 and b = 0 are still present for b ⬎ 0 and a ⬎ 0 关now the potential 共1兲 has only one minimum兴. For, b ⬎ 0 and a ⬍ 0, the potential 共1兲 has two minima at x1,2 = ± 冑−a / b separated by a maximum at x0 = 0 and the motion x共t兲 strongly deviates from the linear case. Thus for small energies the particle oscillates either in the left or right well. Even weak noise ensures that the particles do not stay in the same well, they now have a chance to go in the opposite direction, so that particles from the left well may finally reach the right well and vice versa. 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 written60

⳵W = LFPW, ⳵t where the Fokker-Planck operator LFP is given by

共5兲

The first two terms on the right-hand side of Eq. 共6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. Let us introduce the dimensionless variables and parameters as in27,51 y=

x , 2 1/2 具x 典0

A=

a具x2典0 , 2kT

B=

a具x2典20 , 4kT

␨ ␤⬘ = ␩ , m

共7兲

where ␩ = 冑m具x2典0 / 共2kT兲 is a characteristic time. Equation 共3兲 now becomes

␩2y¨ 共t兲 + ␩␤⬘y˙ 共t兲 + Ay共t兲 + 2By 3共t兲 =

␩

冑2mkT ␭共t兲.

共8兲

The normalization condition 具y 2典0 = 1 implies that the constants A and B are not independent,51 B = B共Q兲 =

冋

1 D−3/2共sgn共A兲冑2Q兲 8 D−1/2共sgn共A兲冑2Q兲

册

2

,

共9兲

where Q = A2 / 4B and the Dv共z兲 are Whitaker’s parabolic cylinder functions of order v.61 For A ⬍ 0 共which is the case of greatest interest兲, Q is equal to the barrier height for the potential V共y兲 / 共kT兲 = Ay 2 + By 4 共see Fig. 1兲. For A ⬍ 0 and small Q, B=

⌫2共3/4兲 ⌫共3/4兲关⌫2共1/4兲 + ⌫共− 1/4兲⌫共3/4兲兴冑Q + ⌫2共1/4兲 ⌫3共1/4兲 + O共Q兲 ⬇ 0.1142 + 0.1835冑Q + O共Q兲,

while for A ⬍ 0 and Q Ⰷ 1 , B ⬃ Q. The relevant quantities are the position autocorrelation function


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J. Chem. Phys. 124, 024107 共2006兲


C共t兲 = 具y共0兲y共t兲典0

共10兲

and the correlation time Tc, which is a global characteristic of the relaxation process involved and is defined as the area under the curve of C共t兲, viz.,27,28 Tc =

冕

⬁

共11兲

C共t兲dt

0

because C共0兲 = 1 due to the normalization conditions. According to linear response theory 共see, e.g., Refs. 27 and 28兲, ˜ 共␻兲 having determined the one-sided Fourier transform C ⬁ −i␻t = 兰0 C共t兲e dt 关the spectrum of the equilibrium correlation function C共t兲兴, one can calculate the normalized dynamic susceptibility ␹ˆ 共␻兲 = ␹⬘共␻兲 − i␹⬙共␻兲共Ref. 60兲, ˜ 共␻兲. ␹ˆ 共␻兲 = 1 − i␻C

共12兲

By utilizing general properties of Fourier transforms, we may also obtain simple asymptotic equations for ␹共␻兲 in the low- and high-frequency limits. We have

␹ ⬙共 ␻ 兲 ⬃ ␻

冕

⬁

0

C共t兲dt + ¯ = ␻Tc + ¯

butions from all the eigenvalues ␭k. Thus in order to evaluate C共t兲 and Tc, a knowledge of all the ␭k and ck is required. However, in the high barrier limit 共Q Ⰷ 1兲 , ␭1 ⬃ e−Q Ⰶ ␭k and for symmetric potentials c1 ⬇ 1 Ⰷ ck共k ⫽ 1兲共Refs. 56 and 58兲 so that the approximation Tc ⬇ ␭−1 1 can be used. In other words, the inverse of the smallest nonvanishing eigenvalue, i.e., the greatest relaxation time, closely approximates the correlation time Tc for symmetric potentials in the lowtemperature 共high barrier兲 limit.

III. DIFFERENTIAL RECURRENCE RELATIONS FOR THE CORRELATION FUNCTIONS: LANGEVIN EQUATION APPROACH

By applying a general method of solution of nonlinear Langevin equations developed by Coffey et al.,27 one may recast Eq. 共8兲 as a hierarchy of the differential-recurrence equations for the equilibrium correlation functions 共observables兲 cn,q共t兲 defined as

共13兲

cn,q共t兲 =

for ␻ → 0, and

⫻e关−␬

ត 共0兲/␻3 + ¯ ␹ ⬙共 ␻ 兲 ⬃ C

共14兲

for ␻ → ⬁ 关here we have noted that C˙共0兲 = 0兴. The correlation time Tc may also be defined in terms of the eigenvalues 共␭k兲 of the Fokker-Planck operator LFP from Eq. 共6兲 because the function C共t兲 may be formally written as27,28 C共t兲 = 兺 cke−␭kt .

共15兲

k

where 兺kck = 1. Now the correlation time Tc contains contri-

d dt

2 y 2共t兲+Ay 2共t兲+By 4共t兲兲/2

2 y 2共t兲+Ay 2共t兲+By 4共t兲兲/2

典0 ,

共17兲

where Hn共z兲 is the Hermite polynomial of order n, y共0兲 is the initial value of y共t兲, ␬ = ␣B1/4, and ␣ is a scaling factor with value chosen so as to ensure convergence of the continued fractions involved as suggested by Voigtlaender and Risken60 共all results for the observables such as ␭1 and Tc are independent of ␣兲. The transformation of Eq. 共8兲 into an equation for the observables is accomplished by a change of variables as follows. Noting that61 and

Hn+1共z兲 = 2zHn共z兲

− 2nHn−1共z兲,

共16兲

k

␩ 兵Hq关␬y共t兲兴Hn关␩y˙ 共t兲兴e共−␬

2 y 2共t兲+Ay 2共t兲+By 4共t兲兴/2

dHn共z兲/dz = 2nHn−1共z兲

Tc = 兺 ck/␭k ,

˙

60

Thus, from Eqs. 共32兲 and 共15兲 we have

= ␩e共−␬

1

冑2n+qn ! q! 具y共0兲Hq关␬y共t兲兴Hn关␩y共t兲兴

共18兲

and that the usual rules of analysis apply to stochastic differential equations of the Stratonovich type,27,28 we have

其

兵2n␩y¨ 共t兲Hq关␬y共t兲兴Hn−1关␩y˙ 共t兲兴 + 2q␬y˙ 共t兲Hq−1关␬y共t兲兴Hn关␩y˙ 共t兲兴

+ 关− ␬2y共t兲 + Ay共t兲 + 2By 3共t兲兴Hq关␬y共t兲兴y˙ 共t兲Hn−1关␩y˙ 共t兲兴其 =

2n␩

冑2mkT

再

Hq关␬y共t兲兴Hn−1关␩y˙ 共t兲兴␭共t兲 − n␤⬘Hq关␬y共t兲兴兵Hn关␩y˙ 共t兲兴 + 2共n − 1兲Hn−2关␩y˙ 共t兲兴其 +

1 Hn+1关␩y˙ 共t兲兴 2␬3

1 ⫻ 2Bq共q − 1兲共q − 2兲Hq−3关␬y共t兲兴 + q关␬4 + A␬2 + 3qB兴Hq−1关␬y共t兲兴 + 关− ␬4 + A␬2 + 3共q + 1兲B兴Hq+1关␬y共t兲兴 2

冎

再

1 n + BHq+3关␬y共t兲兴 − 3 Hn−1关␩y˙ 共t兲兴 4q共q − 1兲共q − 2兲BHq−3关␬y共t兲兴 + 2q关− ␬4 + A␬2 + 3qB兴Hq−1关␬y共t兲兴 4 2␬

冎

1 + 关␬4 + A␬2 + 3共q + 1兲B兴Hq+1关␬y共t兲兴 + BHq+3关␬y共t兲兴 . 2

共19兲


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By averaging Eq. 共19兲 over the realizations of y共t兲 and forming cn,q共t兲共as described in Ref. 27兲, we obtain d dt

␩ cn,q共t兲 = − n␤⬘cn,q共t兲 +

B1/4冑n + 1 关冑共q + 3兲共q + 2兲共q + 1兲cn+1,q+3共t兲 + 冑共q + 1兲共− ␣4 − 2冑Q␣2 + 3共q + 1兲兲cn+1,q+1共t兲 2␣3

+ 冑q共␣4 − 2冑Q␣2 + 3q兲cn+1,q−1共t兲 + 冑q共q − 1兲共q − 2兲cn+1,q−3共t兲兴 −

B1/4冑n 关冑共q + 3兲共q + 2兲共q + 1兲cn−1,q+3共t兲 2␣3

+ 冑共q + 1兲共␣4 − 2冑Q␣2 + 3共q + 1兲兲cn−1,q+1共t兲 + 冑q共− ␣4 − 2冑Q␣2 + 3q兲cn−1,q−1共t兲 + 冑q共q − 1兲共q − 2兲cn−1,q−3共t兲兴. 共20兲

Here we have noted that averaging of the multiplicative noise term in Eq. 共19兲 yields 共see for detail Ref. 27, Chap. 10兲

␩

冑2mkT y 共t兲Hn−1共␩y共t兲兲␭共t兲 = ␤⬘共n − 1兲y Hn−2共␩y兲. q

q

˙

˙

IV. MATRIX-CONTINUED FRACTION SOLUTION

共21兲 The initial conditions for cn,q共t兲 are given by c0,2q−1共0兲 =

冑2

1 2q−1

共2q − 1兲!

具yH2q−1共␬y兲e共−␬

2 y 2+Ay 2+By 4兲/2

典0

Z冑22q−1共2q − 1兲 ! B ⫻

冕

⬁

␰H2q−1共␣␰兲e−共␣

2␰2−2冑Q␰2+␰4兲/2

d␰ ,

共22兲

−⬁

where the partition function Z is given by51 Z=

冕

⬁

e−Ay

2−By 4

dy = 冑␲共2B兲−1/4eQ/2D−1/2共− 冑2Q兲. 共23兲

−⬁

Note that cn,q共0兲 = 0 for n 艌 1 and c0,2q共0兲 = 0 for the equilibrium Maxwell-Boltzmann distribution. The position correlation function C共t兲 = 具y共0兲y共t兲典0, Eq. 共10兲, is given in terms of c0,q共t兲 as ⬁

C共t兲 = 兺 aqc0,q共t兲 q=0 ⬁

= 兺 aq q=0

1

冑2 q! q

具y共0兲Hq关␬y共t兲兴e关−␬

2 y 2共t兲+Ay 2共t兲+By 4共t兲兴/2

典0 ,

共24兲 where, due to the orthogonality properties of the Hermite polynomials, a2q = 0 and a2q−1 =

␬

冑␲22q−1共2q − 1兲!

= c0,2q−1共0兲

␣ZB1/4

冑␲

冕

⬁

yH2q−1关␬y兴e−共␬

2 y 2+Ay 2+By 4兲/2

First we note that the recurrence Eq. 共20兲 may be separated into two independent systems with q + n even and odd. In order to solve Eq. 共20兲 for q + n even, we introduce the column vectors

冢冣冢冣

c2n−2,1共t兲 C2n−1共t兲 = c2n−2,3共t兲 , ⯗

1

=

currence equation for the statistical moments which can be solved by the matrix-continued fraction method.27,28

dy

−⬁

.

Equation 共20兲共originally derived by Voigtlaender and Risken60 from the Fokker-Planck equation兲 is the desired re-

c2n−1,0共t兲 C2n共t兲 = c2n−1,2共t兲 ⯗

共25兲共n 艌 1兲.

Now, Eq. 共20兲 can be rearranged as the set of matrix threeterm recurrence equations d dt

␩ Cn共t兲 = Q−n Cn−1共t兲 − ␤⬘共n − 1兲Cn共t兲 + Q+n Cn+1共t兲, 共26兲 where the matrices Q+n , Q−n , and their elements are given in Appendix A. By one-sided Fourier transformation, we have from Eq. 共20兲 −˜ ˜ 共 ␻ 兲 − Q +C ˜ 关i␩␻ + ␤⬘共n − 1兲兴C n n n+1共␻兲 − Qn Cn−1共␻兲

= ␩␦n,1C1共0兲,

共27兲

where the elements of the column vector C1共0兲 are defined by Eq. 共22兲. By invoking the general method27 for solving the matrix recursion Eq. 共27兲 and noting that Cn共0兲 = 0 for n ⬎ 1, we ˜ 共s兲 in terms of a matrixhave the exact solution for C 1 continued fraction, viz., ˜ 共␻兲 = ␩⌬ 共i␻兲C 共0兲, C 1 1 1

共28兲

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


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J. Chem. Phys. 124, 024107 共2006兲


− ⌬n共i␻兲 = 兵关i␩␻ + ␤⬘共n − 1兲兴I − Q+n ⌬n+1共s兲Qn+1 其−1

共29兲

˜ 共␻兲 and noting and I is the unit matrix. Having determined C 1 ˜ 共␻兲 of the position Eq. 共24兲, we can evaluate the spectrum C correlation function C共t兲 = 具y共0兲y共t兲典0 = 共␣ZB

1/4

⌬ = ␤⬘S/共␩kT兲.

共37兲

S = 养well冑−2mV共x兲dx is the action in the well which can be calculated as

冕冑 0

/冑␲兲CT1 共0兲C1共t兲,

共30兲

S=2

x2⬘

− 2mV共x兲dx = 2

共31兲

Furthermore, according to Eqs. 共11兲, 共28兲, and 共31兲, the correlation time Tc is now given by ˜ 共0兲 = 共␩␣ZB1/4/冑␲兲CT共0兲⌬ 共0兲C 共0兲. Tc = C 1 1 1

共32兲

ᠮ 共0兲 Noting Eq. 共26兲, one can show that C 1 −3 + − = −␤⬘␩ Q1 Q2 C1共0兲 so that the third derivative of the correlation function C共t兲 is

ត 共0兲 = ␣ZB CT共0兲C ᠮ 共0兲 C 1 冑␲ 1 =−

␤⬘␣ZB1/4

␩ 冑␲ 3

= kT␩

CT1 共0兲Q+1 Q−2 C1共0兲.

det关␭1␩I + Q1 + Q+1 ⌬2共− ␭1兲Q−2 兴 = 0.

共34兲

The relaxation times Tc and 1 / ␭1 yielded by the matrixcontinued fraction method will now be compared with those obtained by the Mel’nikov-Meshkov method.30,31

冋冑冋冑

1 2␲␩ ⫻

共33兲

Equation 共33兲 allows one to estimate the high frequency behavior of the susceptibility ␹ˆ 共␻兲 from Eq. 共14兲. The smallest nonvanishing eigenvalue ␭1 of the FokkerPlanck operator 关that is, ␭1 of the hierarchy of Eq. 共20兲兴 can also be estimated by using matrix-continued fractions from the secular equation27,28,60

冑− 2mV共x兲dx

8冑2Q3/4 . 3B1/4

Here x1,2 ⬘ = ± 共4具x2典20Q / B兲1/4 are the solutions of the equation V共x兲 = 0. For Q Ⰷ 1, S ⬃ 共8 / 3兲kT␩冑2Q. The time ␶IHD is the longest relaxation time for the IHD damping 共␤⬘ 艌 1兲 defined −1 IHD IHD IHD IHD IHD = ⌫12 + ⌫21 = 2⌫12 , where ⌫12 and ⌫21 are the as ␶IHD Kramers escape rates from the wells 1 and 2, respectively, IHD IHD = ⌫21 兲 so 共due to the equivalence of the wells 1 and 2, ⌫12 that −1 ␶IHD =

1/4

x1⬘

0

where the sign “T” designates transformation of the column vector C1共0兲 to a row vector. Thus one has ˜ 共␻兲 = 共␣ZB1/4/冑␲兲CT共0兲C ˜ 共␻兲. C 1 1

冕

=

␤ ⬘2 ␩ 2 ␤⬘ + 兩V⬙共0兲兩 − 4 2 m

V⬙共x1兲 V共x 兲/共kT兲 e 1 + 兩V⬙共0兲兩

冑

冑 2 冑冑2␲␩ 共 ␤⬘ + 8 QB − ␤⬘兲, e−Q

册

V⬙共x2兲 V共x 兲/共kT兲 e 2 兩V⬙共0兲兩

册共38兲

where x1,2 = ± 冑−a / b are coordinates of the minima and we have noted that V共xi兲 / 共kT兲 = −Q, 共␩2 / m兲兩V⬙共0兲兩 = 2冑QB, and 共␩2 / m兲V⬙共xi兲 = 4冑QB. The leading factor on the right-hand side of Eq. 共35兲 is the correction to the IHD result due to Mel’nikov and Meshkov 共the depopulation factor兲. As shown by Mel’nikov and Meshkov,31 A⬘共⌬兲 → 1 as ⌬ → ⬁ and A⬘共⌬兲 / ⌬ → 1 as ⌬ → 0. Thus if Q Ⰷ 1 and ␤⬘ → ⬁, we have from Eq. 共35兲 the VHD formula

␶VHD =

␲␩␤⬘

2冑2Q

共39兲

eQ ,

V. MEL’NIKOV-MESHKOV UNIVERSAL EQUATION

A general theoretical treatment of an escape rate from an arbitrary double-well well potential has been given by Mel’nikov and Meshkov.30,31 By solving the Fokker-Planck equation converted to an energy-action diffusion equation by the Wiener-Hopf method, Melnikov and Meshkov30,31 evaluated the longest relaxation time ␶ in the high barrier limit for all values of the dissipation. That is, they solved the Kramers turnover problem. Their results for the potential 共1兲 yield the universal formula

␶= where

A⬘共2⌬兲 ␶IHD , A⬘2共⌬兲

共35兲

冋冕

A⬘共⌬兲 = exp

1 ␲

⬁

0

which is the result of Larsson and Kostin 共in our notation兲.21 In like manner, in the VLD limit 共␤⬘ → 0兲, we have

␶VLD =

册

共36兲

Q

.

共40兲

Equation 共40兲 can also be obtained from the Kramers formula for the escape rates in the VLD limit. In the present −1 VLD VLD VLD VLD = ⌫12 + ⌫21 = 2⌫12 , where ⌫12 problem that yields ␶VLD 共again due to the equivalence of the wells the escape rate VLD VLD is equal to ⌫21 兲 is given by6 ⌫12 VLD ⌫12 ⬃

ln关1 − exp兵− ⌬共␭2 + 1/4兲其兴 d␭ , ␭2 + 1/4

3␲␩

e 8 冑2 ␤ ⬘ Q

⌬ 2␲

冑

V⬙共x1兲 V共x 兲/共kT兲 e 1 . m

The longest relaxation time ␶ yielded by the asymptotic Eqs. 共35兲, 共39兲, and 共40兲 will be compared with that evaluated from the matrix-continued fraction solution in Sec. VII.


024107-7

J. Chem. Phys. 124, 024107 共2006兲


VI. CORRELATION TIME IN THE VHD AND VLD LIMITS

The calculation of the correlation time Tc is a much more complicated problem than the evaluation of the smallest nonvanishing eigenvalue alone since all the other eigenvalues contribute to Tc 共see Sec. II兲. Fortunately, in this problem an accurate method of estimating the estimation Tc in the VHD and VLD limits exists. The method, based on the mean first passage time, was first suggested by Szabo62 in the context of the theory of polarized fluorescent emission in uniaxial liquid crystals. However, it may be used for all systems with dynamics governed by a single variable 共x兲 Fokker-Planck equation. Namely, one may calculate in integral form the correlation time Tc defined as the area under the curve of the normalized autocorrelation function Cy共t兲 = 具Y关x共0兲兴Y关x共t兲兴典0 for an arbitrary function Y共x兲. Here 具典0 designates the statistical average over the stationary 共equilibrium兲 distribution function Wst关x共0兲兴 with x共0兲 defined in the range x1 艋 x共0兲艋 x2 and it is assumed that 具Y典0 = 0. The pertinent feature of these is that an exact formula for the correlation time Tc may be given in terms of the diffusion coefficient D共2兲共x兲 and Wst共x兲 only 共see, e.g., Refs. 27 and 28兲, viz., 1 Tc = 2 具Y 典0

冕

x2

x1

1 Wst共x兲D共2兲共x兲

冋冕

x

Y共z兲Wst共z兲dz

x1

册

2

Since in the VHD and VLD limits, the dynamics of the present system are governed by a single variable, we can obtain accurate VHD and VLD asymptotes by applying Eq. 共41兲. In the VHD limit 共␤⬘ Ⰷ 1兲, the appropriate single variable Smoluchowski equation for the probability density function W共y , t兲 is51

冊

共42兲

Thus noting that D共2兲共y兲 = 1 / 共2␤⬘␩兲, the correlation time Tc of the position autocorrelation function C共t兲 is given by the general Eq. 共41兲, which reads for the present problem as51 Tc = TVHD = 2␩␤⬘

冕

⬁

−⬁

1 Wst共y兲

冋冕

y

y ⬘Wst共y ⬘兲dy ⬘

−⬁

册

2

␩

= ␤⬘

21/4冑␲eQ/2D−1/2共− 冑2Q兲共D−3/2共− 冑2Q兲兲2

−冑Q

61

where erf共x兲 is the error function

冕

y

−⬁

y ⬘e−Ay⬘

冕

⬁

冑x + 冑Q

4

共46兲

where the double overbar denotes averaging over the fast phase variable. Now the correlation time Tc is given by Tc ⬇ TVLD

冕

⬁

1 共2兲 D 共␧兲Wst共␧兲

冋冕

␧

¯¯y 共␧⬘兲W 共␧⬘兲d␧⬘ st

−Q

册

2

d␧, 共47兲

where the diffusion coefficient D共2兲共␧兲 is given by D共2兲共␧兲 = 2␤⬘␩y˙ 2共␧兲 =

2␤⬘

␩

关␧ − Ay 2共␧兲 − By 4共␧兲兴.

By calculating the integrals in Eq. 共47兲 as described in Appendix B, we obtain 3␲27/4eQ/2Q1/4

TVLD

␩

=

␤⬘D−3/2共− 冑2Q兲 ⫻

冕

x sinh2共Qx2/2兲

1

0

冑1 + x

再冋册冋册冎 2x

E

1+x

− 共1 − x兲K

dx,

2x

1+x

共48兲

VII. RESULTS AND DISCUSSION

and we have noted that

冑 ⬘ dy ⬘ = ␲/B eA2/4B关erf共冑By 2 + A/2冑B兲 − 1兴.

2−By 4

册

⳵2 2 y˙ 共␧兲 W共␧,t兲, ⳵ ␧2

共44兲

2

ex 关1 − erf共x兲兴2

+ ␩2

冊

dx,

4

Here Wst共y兲 = e−Ay −By / Z is the equilibrium Boltzmann distribution function 关which is a stationary solution of Eq. 共42兲兴. For A ⬍ 0, we have by changing the variables in Eq. 共43兲 Tc

冋冉

2␤⬘ ⳵ 1 ⳵ W共␧,t兲 = ␩2y˙ 2共␧兲 − 2 ⳵t ␩ ⳵␧

where K共m兲 and E共m兲 are complete elliptic integrals of the first and second kinds, respectively.61 For high barriers, Q Ⰷ 1, the behavior of TVHD and TVLD given by Eqs. 共44兲 and 共48兲 is very similar to that of ␶VHD, Eq. 共39兲, and ␶VLD, Eq. 共40兲, respectively. The advantage of Eqs. 共44兲 and 共48兲 is that they allow us to evaluate the correlation time Tc in the VHD and VLD limits at all barrier heights including low barriers 共Q 艋 1兲, where asymptotic methods 共such as the Melnikov-Meshkov one兲 are not applicable.

dy. 共43兲

2

共45兲

and the time w 共phase兲 measured along a closed trajectory in phase space as action-angle variables.6,25 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,63 i.e., averaging the Fokker-Planck equation 共5兲 over the fast phase variable w, one can derive a single variable Fokker-Planck equation for the probability density function W共␧ , t兲 in energy space

−Q

共41兲

冉

␧ = ␩2y˙ 2 + Ay 2 + By 4

=

dx.

1 ⳵ ⳵ ⳵ W共y,t兲 = + 2Ay + 4By 3 W共y,t兲. ⳵t 2 ␤ ⬘␩ ⳵ y ⳵ y

In the opposite low damping limit 共␤⬘ Ⰶ 1兲, one may, in order to obtain a single variable Fokker-Planck equation, introduce the energy of the dipole

The exact matrix continued fraction solution 关Eq. 共28兲兴 we have obtained is easily computed 共algorithms for calculating matrix-continued fractions are discussed in Refs. 27 and 28兲. As far as practical calculations of the infinite matrix-


024107-8


FIG. 2. The numerically exact results for the correlation time Tc / ␩ 关Eq. 共32兲, solid lines兴 and the inverse of the smallest nonvanishing eigenvalue 共␩␭1兲−1 关Eq. 共34兲, stars兴 vs ␤⬘ are compared with the universal Mel’nikovMeshkov result 关Eq. 共35兲, dashed lines兴 for high barriers Q = 5 and 10. The relaxation times ␶IHD / ␩ 关Eq. 共38兲, dotted lines 1兴, ␶VHD / ␩ 关Eq. 共39兲, dotted lines 2兴, and ␶VLD / ␩ 关Eq. 共40兲, dashed-dotted lines 3兴 are also shown for comparison.

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 M ⫻ M. N and M were determined in such way that a further increase of N and M did not change the results. Both N and M depend on the dimensionless barrier 共Q兲 and damping 共␤⬘兲 parameters and must be chosen taking into account the desired degree of accuracy of the calculation. The final results are independent of the scaling factor ␣. The advantage of choosing an optimal value of ␣ is, however, that the dimensions N and M can be minimized.60 Both N and M increase with decreasing ␤⬘ and increasing Q. In practical calculations, very low values of ␤⬘共⬃0.001兲 and very high values of Q共⬃100兲 can be handled. The greatest relaxation time ␶ predicted by the Mel’nikov and Meshkov universal Eq. 共35兲 and the inverse of the smallest nonvanishing eigenvalue ␭1 calculated numerically by matrix-continued fraction methods are shown in Fig. 2 as functions of ␤⬘ for relatively high values of the barrier height parameter Q = 5 and 10. Here, the VHD 关Eq. 共39兲兴, IHD 关Eq. 共38兲兴, and VLD 关Eq. 共40兲兴 asymptotes for ␶ are also shown for comparison. Apparently in the high barrier limit, Eq. 共35兲 provides a good approximation to 1 / ␭1 for all ␤⬘ including the VHD, VLD, and turnover regions. The quantitative agreement in damping behavior may be explained as follows. The behavior of the escape rate as a function of the barrier height parameter Q for large Q is approximately Arrhenius-type and arises from an equilibrium property of the system 共namely, the Boltzmann distribution at the bottom of the well兲. On the other hand, the damping dependence of the escape rate is due to nonequilibrium 共dynamical兲 properties of the system and so is contained in the prefactor A only, the detailed nature of which depends on the

J. Chem. Phys. 124, 024107 共2006兲

FIG. 3. The numerically exact results for the correlation time Tc / ␩ 关Eq. 共32兲, solid lines兴 and the inverse of the smallest nonvanishing eigenvalue 共␩␭1兲−1 关Eq. 共34兲, stars兴 vs Q for ␤⬘ = 10−3 共low damping兲 and ␤⬘ = 10 共high damping兲. The universal Mel’nikov-Meshkov formula 关Eq. 共35兲, dashed lines兴 and the relaxation times ␶VHD / ␩ 关Eq. 共39兲, triangles兴 and ␶VLD / ␩ 关Eq. 共40兲, filled circles兴 are also shown for comparison.

behavior of the energy distribution function at the barrier points.29 The Mel’nikov-Meshkov approach30,31 yields the distribution function at the barrier point for all values of the damping allowing one to evaluate the damping dependence of the prefactor A in Eq. 共2兲. We remark that as emphasized by Kramers, it is hardly ever of any practical importance to improve on the accuracy of the IHD or VLD formulas themselves because in experimental situations where relaxation is studied, one has only estimates of the prefactor within a certain degree of accuracy which is difficult to evaluate. For example, little detailed information about the value of ␤⬘ exists. Nevertheless, it is important to predict the behavior of the relaxation times as a function of ␤⬘ using analytical methods such as the one used here because of the detailed information such methods yield about the mechanisms underlying the relaxation process. In Fig. 3, we compare the VLD and VHD asymptotes of Tc, Eqs. 共44兲 and 共48兲, with the Mel’nikov-Meshkov Eq. 共35兲 and numerical solutions for the correlation time Tc and the inverse of the smallest nonvanishing eigenvalue ␭1 for low 共␤⬘ = 0.001兲 and high 共␤⬘ = 10.0兲 values of damping. As one can see, the Mel’nikov-Meshkov formula provides a good approximation both for Tc and 1 / ␭1 at Q ⬎ 2. In Fig. 4, we compare the VHD and VLD correlation times, Eqs. 共44兲 and 共48兲, with the exact numerical solution for Tc and 1 / ␭1 at small barrier, Q = 0.5, where the Melnikov-Meshkov universal formula is not applicable. Equations 共44兲 and 共48兲 may be used for the estimation of Tc for ␤⬘ Ⰷ 1 and ␤⬘ Ⰶ 1, respectively, and for all barrier heights Q including very low barriers. We also remark that for intermediate and high damping, ␤⬘ 艌 1, numerical values of 1 / ␭1 and Tc are very close to each other while for low damping, ␤⬘ Ⰶ 1 , 1 / ␭1 differs considerably from Tc. The imaginary ␹⬙共␻兲 part of the dynamic susceptibility for various values of the barrier height Q and the friction


024107-9

J. Chem. Phys. 124, 024107 共2006兲


FIG. 4. The numerically exact results for the correlation time Tc / ␩ 关Eq. 共32兲, solid line 1兴 and the inverse of the smallest nonvanishing eigenvalue 共␩␭1兲−1 关Eq. 共34兲, stars兴 vs ␤⬘ for a small barrier Q = 0.5. They are compared with the universal Mel’nikov-Meshkov result 关Eq. 共35兲; dashed line 2兴 and the correlation times TVLD / ␩ 关Eq. 共48兲; dashed-dotted line 3兴, and TVHD / ␩ 关Eq. 共44兲, dotted line 4兴.

coefficient ␤⬘ are shown in Fig. 5. The low- and highfrequency asymptotes 关Eqs. 共13兲 and 共14兲兴 are also shown in Fig. 5 for comparison. One relaxation band dominates the low-frequency part of the spectra and is due to the slow overbarrier relaxation of the particles in the double-well potential. As seen in Fig. 5, the low-frequency part of the spectrum may by approximated by the Debye equation

␹ˆ 共␻兲 =

1−␦ + ␦, 1 + i␻/␭1

共49兲

where ␦ is a parameter accounting for the contribution of the high-frequency modes. The characteristic frequency ␻R ⬇ ␭1 of the low-frequency band strongly depends on Q as well as on the friction parameter ␤⬘. Regarding the barrier height Q, the frequency ␻R decreases exponentially as Q is raised. This behavior occurs because the probability of escape of a dipole from one well to another over the potential barrier exponentially decreases with increasing Q. As far as the dependence of the low-frequency part of the spectrum for large and mod-

erate friction 共small inertial effects兲 ␤⬘ 艌 1 is concerned, the frequency ␻R decreases as ␤⬘ increases as is apparent by inspection of curves in Fig. 5共a兲. For small friction 共large inertial effects兲 ␤⬘ ⬍ 0.1 the frequency ␻R decreases with decreasing ␤⬘ for given values of Q. A very high-frequency band is visible in all the figures due to the fast inertial oscillations of the particles in the potential wells. For Q Ⰷ 1, the characteristic frequency of oscillation ␻L increases as ⬃2Q3/4␩−1 with increasing Q. As far as the behavior as a function of ␤⬘ is concerned, the amplitude of the highfrequency band decreases progressively with increasing ␤⬘, as one would intuitively expect. On the other hand, for small friction 共large inertial effects兲 ␤⬘ Ⰶ 1, a fine structure appears in the high-frequency part of the spectra 关due to resonances at high harmonic frequencies28 of the almost free motion in the 共anharmonic兲 potential兴. Thus we have demonstrated how the matrix-continued fraction approach of the solution of nonlinear Langevin equations27 may be successfully applied to the nonlinear Brownian oscillator in a double well potential, Eq. 共1兲 for wide ranges of the barrier height parameter Q, and the damping parameter ␤⬘. We have shown that in the lowtemperature limit, the Mel’nikov-Meshkov formula, Eq. 共35兲, for the longest relaxation time bridging the VLD and IHD escape rates as a function of ␤⬘ yields satisfactory agreement with the numerical results for all values of damping. Moreover, the Mel’nikov-Meshkov Eq. 共35兲 allows one to estimate accurately the damping dependence of the lowfrequency parts of the spectra of the equilibrium correlation function C共t兲 and the complex susceptibility ␹共␻兲 and to evaluate the contribution of the overbarrier relaxation mode to the correlation time Tc of the position correlation function C共t兲. We have given an exact as well as a simple approximate analytic formulas for the correlation time Tc and the dynamic susceptibility ␹共␻兲.

ACKNOWLEDGMENT

The TCD Trust is gratefully acknowledged for financial support for one of the authors 共S.V.T.兲.

APPENDIX A: EQUATIONS FOR C1„0…, Q+n, AND Q−n

The column vector C1共0兲 and the matrices Q+n and Q−n are given by

冢冣 c0,1共0兲

C1共0兲 = c0,3共0兲 , ⯗

FIG. 5. The imaginary part of the normalized dynamic susceptibility ␹⬙ vs ␩␻ for various values of ␤⬘ and Q. Solid lines are the continued fraction solution 关Eqs. 共12兲 and 共31兲兴. The Debye spectra 关Eq. 共49兲兴 are shown by dotted lines with asterisks. Dotted and dashed lines: the low- and highfrequency asymptotes 关Eqs. 共13兲 and 共14兲兴.

+ Q2n−1

=B

1/4

冑2n − 1 2␣3

冢

d+0 d−1 e1 e0

d+2

0

e2

0

¯

d−3 d+4

d−5 ¯ ¯

e3 ¯

0

0

e4

d+6

⯗

⯗

⯗

⯗

冣

,


024107-10

+ Q2n

J. Chem. Phys. 124, 024107 共2006兲


=B

1/4

冑2n 2␣3

− =−B Q2n−1

− =−B Q2n

冢

d−0 e0

0

0

¯

d−1

e2

0

¯

e4 ¯ d−6

¯

⯗

d−2 d−3

0

e3

d−4 d−5

⯗

⯗

⯗

e1

冢冢

冑 1/4 2n − 2 2␣

3

冑 1/4 2n − 1 2␣3

冣

d−0 d+1 e1

0

m = m共␧兲 = ,

By noting that sn2共u 兩 m兲 + cn2共u 兩 m兲 = 1 and m sn2共u 兩 m兲 + dn2共u 兩 m兲 = 1,61 one has y 2共t兲

冣冣

¯

e0

d−2

0

e2

d+3 d−4

0

0

e4 d−6 ¯

⯗

⯗

⯗

e3 ¯

d+5 ¯ ⯗

0

0

¯

d−1

e2

0

¯

e1

d+2 d−3

0

e3 d−5 d+6 ¯

⯗

⯗

d+4 e4 ¯ ⯗

=

,

d+0 e0

⯗

e2 − e1 2 = . e2 1 + 共1 + ␧/Q兲−1/2

再

e2关1 − m sn2共冑Be2t/␩ + w兩m兲兴, − Q 艋 ␧ 艋 0,

e2关1 − sn2共冑B共e2 − e1兲t/␩ + 冑mw兩m−1兲兴, 0 ⬍ ␧ ⬍ ⬁.

共B3兲

Next we recall the Fourier series for the Jacobi functions64,65 qn ␲ 2␲ n␲u + , 兺 2n cos 2K K n=1 1 + q K

cn共u兩m兲 =

2␲ 共2n + 1兲␲u qn+1/2 cos , 1/2 兺 m K n=0 1 + q2n+1 2K

共B5兲

冉冊

共B6兲

,

⬁

nqn n␲u E 2␲2 , m sn 共u兩m兲 = 1 − − 2 兺 2n cos K K n=1 1 − q K

冋

APPENDIX B: EVALUATION OF AVERAGES IN THE UNDAMPED LIMIT

In the very low damping limit 共␤⬘ Ⰶ 1兲, the energy of the particle is not conserved but will vary very slowly with time 共quasistationarity兲. Thus the dynamics of the system are described by a one-dimensional Fokker-Planck equation 共46兲 and differ but little from those of the undamped limit 关␤⬘ = 0, when the Langevin force in Eq. 共3兲 vanishes兴. The undamped limit has been treated in Refs. 16, 55, and 60. We apply these results for evaluation of integrals in Eq. 共48兲. In the undamped limit, the energy ␧, Eq. 共45兲, is a constant of the motion, viz., ␧˙ = 0. Equation 共45兲 can be rearranged as the deterministic nonlinear differential equation describing the undamped dynamics of the particle 共B1兲

where z共t兲 = y共t兲 / 冑e2 and e1,2 = 冑Q / B共1 ⫿ 冑1 + ␧ / Q兲 are the roots of the quadratic equation ␧ + 2冑QBx − Bx2 = 0. Equation 共B1兲 has a solution55,60 in terms of the Jacobian doubly periodic elliptic function cn共u 兩 m兲 and dn共u 兩 m兲共Ref. 61兲, y共t兲 =

再

w=

± 冑e2dn共冑Be2t/␩ + w兩m兲, − Q 艋 ␧ 艋 0,

± 冑e2cn共冑B共e2 − e1兲t/␩ + w冑m兩m−1兲, 0 ⬍ ␧ ⬍ ⬁,

冎

共B2兲

冕

1

y共0兲/冑e2

1

冑共x2 − e1/e2兲共1 − x2兲 dx,

册

1 4␲2 E 2 + m − 2共1 + m兲 + 3 K 3K2

m2 sn4共u兩m兲 =

冑e2B dt

共B4兲

册

2

eq = 冑共q + 3兲共q + 2兲共q + 1兲.

z共t兲 =

冋

⬁

d±q = 冑q + 1关3共q + 1兲 − 2␣2冑Q ± ␣4兴,

± 冑关z2共t兲 − e1/e2兴关1 − z2共t兲兴,

冉冊

⬁

dn共u兩m兲 =

where

␩ d

冎

⬁

⫻兺

n=1

冋

册

冉冊

n 2␲ 2 nqn n␲u − m − 1 , 2 2n cos 4K 1−q K 共B7兲

where q = exp关−␲K共1 − m兲 / K共m兲兴 and K = K共m兲 and E = E共m兲 are complete elliptic integrals of the first and second kinds, respectively.61 Thus, from Eqs. 共B3兲–共B7兲, we can obtain averages ¯¯y , y 2, and y 4 over the phase w. In particular, in the domain −Q 艋 ␧ 艋 0, we have ¯¯y 共␧兲 = 1 K

冕

K

y共␧,w兲dw = ± ␲冑e2/共2K兲,

y 2共␧兲 = e2E/K, y 4共␧兲 =

共B8兲

0

共B9兲

e22 关m − 1 + 共4 − 2m兲E/K兴. 3

共B10兲

Accordingly, on noting that Wst is the equilibrium MaxwellBoltzmann distribution W0, viz., W0关y共0兲,y˙ 共0兲兴dy共0兲dy˙ 共0兲 =

␩

e Z冑␲

−␩2y˙ 2共0兲+2冑QBy 2共0兲−By 4共0兲

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


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W0共␧兲d␧ =

25/4e−Q/2

␲Q1/4D−1/2共− 冑2Q兲

Re兵K关m共␧兲兴其e−␧

冑1 + 冑1 + ␧/Q

具y 2典0 =

d␧. 共B11兲

211/4Q1/4e−Q/2

␲D−3/2共− 冑2Q兲 ⫻

冕

⬁

冑1 + 冑1 + ␧/QRe关E共m兲兴e−␧d␧ = 1.

−Q

¯ The average of a dynamical quantity ¯X共␧兲 over ␧ is defined as ¯¯ 典0 = 具X

冕

⬁

¯¯ X共␧兲W0共␧兲d␧,

−Q

冕

⬁

W0共␧兲d␧ = 1.

冕 ⬘ 冑

By using Eqs. 共B7兲–共B11兲 in Eq. 共47兲 and noting that ¯¯y = 0 at 0 ⬍ ␧ ⬍ ⬁, we obtain

冑e2e␧共eQ − e−␧兲2 d␧, 2 2 −Q 关3␧ + 共1 − m兲Be2兴K + 2关3冑QBe2 − 共2 − m兲Be2兴E

3␩␲3/2

4␤ Z B

具y共0兲y共t兲典0 =

0

25/4␲e−Q/2

D−3/2共− 冑2Q兲

−Q

共Q + ␧兲1/4

冑mK共m兲

冋

冉

1

册冊

n=1

冕

共Q + ␧兲1/4 q2n−1 兺 K共m−1兲 n=1 共1 + q2n−1兲2

+ 8兺 +8

再冕

0

n␲冑Be2 q2n t 2n 2 cos 共1 + q 兲 ␩K共m兲

⬁

⬁

0

冋

⫻cos

e−␧d␧

⬁

册

冎

共2n − 1兲␲冑B共e2 − e1兲 −␧ t e d␧ . 2␩K共m−1兲共B13兲

D. Chandler, J. Chem. Phys. 68, 2959 共1978兲. B. J. Berne, J. L. Skinner, and P. G. Wolynes, J. Chem. Phys. 73, 4314 共1980兲. 3 D. L. Hasha, T. Eguchi, and J. Jonas, J. Chem. Phys. 73, 1571 共1981兲; J. Am. Chem. Soc. 104, 2290 共1982兲. 4 D. K. Garrity and J. L. Skinner, Chem. Phys. Lett. 95, 46 共1983兲. 5 B. Carmeli and A. Nitzan, J. Chem. Phys. 80, 3596 共1984兲. 6 H. A. Kramers, Physica 共Utrecht兲 7, 284 共1940兲. 7 H. C. Brinkman, Physica 共Utrecht兲 22, 29 共1956兲; 22, 149 共1956兲. 8 C. Blomberg, Physica A 86, 49 共1977兲. 9 P. B. Visscher, Phys. Rev. B 14, 347 共1976兲. 10 J. L. Skinner and P. G. Wolynes, J. Chem. Phys. 69, 2143 共1978兲; ibid. 72, 4913 共1980兲. 11 R. S. Larson and M. D. Kostin, J. Chem. Phys. 69, 4821 共1978兲. 12 S. C. Northrup and J. T. Hynes, J. Chem. Phys. 69, 5246 共1978兲; ibid. 69, 5261 共1978兲; 73, 2700 共1980兲; R. F. Grote and J. T. Hynes, ibid. 73, 2715 共1980兲. 13 M. Mangel, J. Chem. Phys. 72, 6606 共1980兲. 14 K. Schulten, Z. Schulten, and A. Szabo, J. Chem. Phys. 74, 4426 共1981兲. 15 M. Bixon and R. Zwanzig, J. Stat. Phys. 3, 245 共1971兲. 2

1 2

共B12兲

which after some simplifications leads to Eq. 共48兲. The correlation function 具y共0兲y共t兲典0 for the undamped motion can be derived from Eqs. 共B2兲, 共B4兲, 共B5兲, and 共B9兲 and is given by60 共in our notation兲

1

␩2具y˙ 2典0 = 具␧ + 2冑QBy 2 − By 4典0 = .

−Q

Thus we have from Eqs. 共B10兲 and 共B11兲

TVLD ⬇

One can also verify that the equipartition theorem holds, viz.,

16

M. I. Dykman, S. M. Soskin, and M. A. Krivoglaz, Physica A 133, 53 共1985兲. 17 P. Hänggi, Phys. Lett. A 78, 304 共1980兲. 18 J. A. Krumhansl and J. R. Schriefier, Phys. Rev. B 11, 3535 共1975兲. 19 J.-D. Bao and Y.-Z. Zhuo, Phys. Rev. C 67, 064606 共2003兲. 20 V. M. Kolomietz, S. V. Radionov, and S. Shlomo, Phys. Rev. C 64, 054302 共2001兲. 21 R. S. Larson and M. D. Kostin, J. Chem. Phys. 72, 1392 共1980兲. 22 W. T. Coffey, M. W. Evans, and P. Grigolini, Molecular Diffusion and Spectra 共Wiley, New York, 1984兲. 23 C. Blomberg, Physica A 86, 67 共1977兲. 24 L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 共1998兲. 25 P. Hänggi, P. Talkner, and M. Borcovec, Rev. Mod. Phys. 62, 251 共1990兲. 26 M. I. Dykman, G. P. Golubev, D. G. Luchinsky, P. V. E. McClintock, N. D. Stein, and N. G. Stocks, Phys. Rev. E 49, 1935 共1994兲. 27 W. T. Coffey, Yu. P. Kalmykov, and J. T. Waldron, The Langevin Equation, 2nd ed. 共World Scientific, Singapore, 2004兲. 28 H. Risken, The Fokker-Planck Equation, second ed. 共Springer, Berlin, 1984兲. 29 W. T. Coffey, D. A. Garanin, and D. J. McCarthy, Adv. Chem. Phys. 117, 528 共2001兲. 30 V. I. Mel’nikov, Physica A 130, 606 共1985兲; Phys. Rep. 209, 1 共1991兲. 31 V. I. Mel’nikov and S. V. Meshkov, J. Chem. Phys. 85, 1018 共1986兲. 32 H. Grabert, Phys. Rev. Lett. 61, 1683 共1988兲. 33 E. Pollak, H. Grabert, and P. Hänggi, J. Chem. Phys. 91, 4073 共1989兲. 34 I. Rips and E. Pollak, Phys. Rev. A 41, 5366 共1990兲. 35 M. Topaler and N. Makri, J. Chem. Phys. 101, 7500 共1994兲. 36 A. N. Drozdov and P. Talkner, J. Chem. Phys. 109, 2080 共1998兲. 37 R. Ferrando, R. Spadacini, and G. E. Tommei, Phys. Rev. A 46, R699 共1992兲. 38 R. Ferrando, R. Spadacini, and G. E. Tommei, Phys. Rev. E 48, 2437 共1993兲. 39 E. Pollak, J. Bader, B. J. Berne, and P. Talkner, Phys. Rev. Lett. 70, 3299 共1993兲. 40 Yu. Georgievskii and E. Pollak, Phys. Rev. E 49, 5098 共1994兲. 41 E. Hershkovitz, P. Talkner, E. Pollak, and Yu. Georgievskii, Surf. Sci. 421, 73 共1999兲. 42 R. W. Pastor and A. Szabo, J. Chem. Phys. 97, 5098 共1992兲. 43 W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov, J. Chem. Phys. 120, 9199 共2004兲; Yu. P. Kalmykov, S. V. Titov, and W. T. Coffey, J. Chem. Phys. 123, 094503 共2005兲. 44 P. M. Déjardin, D. S. F. Crothers, W. T. Coffey, and D. J. McCarthy, Phys. Rev. E 63, 021102 共2001兲.


024107-12

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Yu .P. Kalmykov, J. Appl. Phys. 96, 1138 共2004兲. Yu. P. Kalmykov, W. T. Coffey, and S. V. Titov, Fiz. Tverd. Tela 共S.Peterburg兲 47, 260 共2005兲关Phys. Solid State 47, 272 共2005兲兴. 47 Y. P. Kalmykov, W. T. Coffey, B. Ouari, and S. V. Titov, J. Magn. Magn. Mater. 292, 372 共2005兲. 48 A. Schenzel and H. Brand, Phys. Rev. A 20, 1628 共1979兲. 49 I. I. Fedchenia, J. Phys. A 25, 6733 共1992兲. 50 Yu. P. Kalmykov, W. T. Coffey, and J. T. Waldron, J. Chem. Phys. 105, 2112 共1996兲. 51 A. Perico, R. Pratolongo, K. F. Freed, R. W. Pastor, and A. Szabo, J. Chem. Phys. 98, 564 共1993兲. 52 B. J. Matkowsky, Z. Schuss, and E. Ben-Jacob, J. Appl. Math. 42, 835 共1982兲. 53 C. W. Gardiner, J. Stat. Phys. 30, 157 共1983兲. 54 H. X. Zhou, Chem. Phys. Lett. 164, 285 共1989兲. 55 Y. Onodera, Prog. Theor. Phys. 44, 1477 共1970兲.

K. Voigtlaender and H. Risken, J. Stat. Phys. 41, 825 共1985兲. M. I. Dykman, R. Mannella, P. V. E. McClintock, F. Moss, and S. M. Soskin, Phys. Rev. A 37, 1303 共1988兲. 58 J. B. Morton and S. Corrsin, J. Stat. Phys. 2, 153 共1970兲. 59 K. Matsuo, J. Stat. Phys. 18, 535 共1978兲. 60 K. Voigtlaender and H. Risken, J. Stat. Phys. 40, 397 共1985兲; Chem. Phys. Lett. 105, 506 共1984兲. 61 Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun 共Dover, New York, 1964兲. 62 A. Szabo, J. Chem. Phys. 72, 4620 共1980兲. 63 E. Praestgaard and N. G. van Kampen, Mol. Phys. 43, 33 共1981兲. 64 E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, fourth ed. 共Cambridge University Press, Cambridge, England, 1927兲. 65 S. V. Titov, P. M. Déjardin, and Yu. P. Kalmykov, J. Chem. Phys. 120, 4852 共2004兲.

45

56

46

57
