Thermal decay of a metastable elastic string

Chemical Physics 235 Ž1998. 51–59

Thermal decay of a metastable elastic string C. Cattuto a , M. Borromeo a , F. Marchesoni b

b

a Dipartimento di Fisica, UniÕersita´ di Perugia, I-06100 Perugia, Italy Department of Physics, UniÕersity of Illinois, 1110 West Green St, Urbana, IL 61801, USA and Istituto Nazionale di Fisica della Materia, UniÕersita´ di Camerino, I-62032 Camerino, Italy

Received 27 January 1998

Abstract Thermal nucleation of kink-antikink pairs in an elastic string subjected to a washboard potential is analyzed in the classical limit at low temperature. The nucleation rate is calculated analytically for values of the tilt up to close the instability threshold. Numerical simulation is shown to support our predictions in the diverse parameter regimes. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction Elastic strings provide the simplest solvable model of linear imperfections in solid state physics. The first prominent application of the elastic string paradigm dates back to the heyday of dislocation theory w1–3x. More recently, elastic string models have been shown to provide a full characterization of flux line dynamics in type II superconductors w4–6x. Stationary string currents may be driven by either field gradients Žgeneralized forces. or spatio-temporal asymmetries w7x. The role of disorder in string transport may be relevant, too, depending on the topology and the phase of the string-like objects under consideration. Point-like defects are likely to be as important in dislocation theory, as line or planar defects are to the dynamics of superconducting vortex arrays w5x. One mechanism has been identified as central to string transport by sub-threshold forces: in the presence of a weak bias a line imperfection can jump from a substrate potential trough into an adjacent one

by nucleating kink-antikink pairs, which can be then pulled infinitely apart with almost no effort. Most notably, such a mechanism is thermally assisted, whence its clear-cut experimental signature w8,9x. In order to analyze in detail the nucleation process we focus here on the most tractable string model, namely the damped sine-Gordon ŽSG. string at low temperature. The present work has been originally inspired by two papers co-authored by Prof. Melnikov w12,13x. On occasion of his visit to Perugia eight years ago, he drew our attention to the role of damping in the decay of a metastable Žor driven. string, predicting a wealth of new possible effects which, sadly, he never had time to investigate himself. Over the years he kept encouraging us to improve numerically on his earlier work. The following is a belated report on the results we have been collecting ever since. In Sections 3–5 we determine a piece-wise analytical expression for the kink-antikink pair nucleation rate in a classical, oÕerdamped SG string subjected to a sub-threshold driving force. The formulas for the

0301-0104r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 1 5 6 - 6

C. Cattuto et al.r Chemical Physics 235 (1998) 51–59

52

nucleation rate in the limits of weak w10x and strong tilts w11x are derived in Sections 3 and 4, respectively. In Section 5 the latter formula is improved to account for smaller values of the driving force and is thus proven to match the former one. In Section 6 we estimate the inertial corrections to the nucleation rate of Section 4 in the regime of intermediate to small damping. Our predictions, old and new, are compared in Section 7 with the results of a numerical simulation. Finally, in Section 8 we discuss the possibility of extending the present approach to elastic strings on arbitrary multistable substrates.

appropriate linear superposition of moving kinks fq and antikinks fy with

f " Ž x ,t . s 4arctan  exp Ž "b x y X Ž t . rd . 4

Žmod 2p ., provided that the separation between their centers of mass, X Ž t . ' X 0 q ut, is very large compared with their size d ' c 0rv 0 Ž dilute gas approximation.. In this limit, the equilibrium kink Žantikink. density in a SG theory at finite temperature and with natural boundary conditions f Ž x ™ "`,t . s 0 Žmod 2p ., is w8x n" Ž T . s n0 Ž T . s

2. low temperature SG theory

f t t y c 02 f x x q v 02 sin f s yaf t q F q z Ž x ,t . , Ž 1 . provides an ideal model to study nucleation processes in a variety of periodic physical systems at thermal equilibrium. The coupling of the classical SG field f Ž x,t . to the heat bath at temperature T is described by a viscous term yaf t and a zero-mean Gaussian noise source z Ž x,t .. The damping constant a and the noise intensity are related through the noise autocorrelation function X

X

X

² z Ž x ,t . z Ž x ,t . : s 2 a kTd Ž t y t . d Ž x y x . .

Ž 2. The constant force F represents an external drive, or bias, which breaks the f ™ yf symmetry of the SG equation, thus making the nucleation process possible w8x. Correspondingly, the SG potential V w f x s v 02 Ž1 y cos f . gets tilted by the bias term yFf : the resulting washboard potential retains a multistable structure for < F < - F3 ' v 02 . In the overdamped limit a 4 v 0 , F3 coincides with the static threshold for the locked-unlocked transition w16x. The unperturbed SG equation – obtained from Eq. Ž1. by setting its r.h.s. to zero – has been derived from the covariant Hamiltonian density HSG w f x s

f t2 2

q c 02

f x2 2

qV wf x,

1

2

1r2

E0

ž / ž /

d p

kT

1r2

E0

ž /

exp y

kT

,

Ž 5.

The perturbed SG equation w14,15x

X

Ž 4.

Ž 3.

and bears both extended Žphonons. and localized solutions Žsolitons.. Solitons can be regarded as an

where E0 s HHSG w f " x d x s 8 v 0 c 0 is the rest energy and M0 s E0rc02 is the mass of f ". It follows ŽT . that the dilute gas approximation holds for ny1 0 4 d, that is at low temperature kT < E0 . In the presence of weak perturbations Ži.e. kT < E0 and F < v 02 . the single Žanti.kink is stable, but undergoes a driven Brownian motion with Langevin equation ŽLE. w17x u˙ s ya u . 2p FrM0 q j Ž t . ,

Ž 6a .

where j Ž t . is a zero-mean valued Gaussian noise with autocorrelation function ² j Ž t . j Ž t X .: s 2 a 2 Dd Ž t y tX . and D s kTra M0 . To derive the LE Ž6a. it was assumed that at low temperature kT < E0 the variance of the Žanti.kink speed is much smaller than c 02 , so that the relativistic boost factor b ' Ž1 y u 2rc02 .y1 r2 in Eq. Ž4. may be approximated to unity Žthis is the so-called non-relatiÕistic approximation.. As a matter of fact, one sees immediately from Eq. Ž6a. that the external bias pulls f " in opposite directions with average speed u F s .2p Fra M0 and variance ²Ž u y u F . 2 : s kTrM0 . In the oÕerdamped limit a 4 v 0 the LE Ž6a. can be cast in the Smoluchowski form, that is X˙ s .2p Fra M0 q h Ž t . ,

Ž 6b .

with h Ž t . s j Ž t .ra . Moreover, the assumption of large damping a 4 v 0 affords two major simplifications: Ži. oscillating solutions of Eq. Ž1., like breathers and phonons radiation, are damped out and, therefore, play no role in the nucleation process;

C. Cattuto et al.r Chemical Physics 235 (1998) 51–59

Žii. kink-antikink collisions are always destructiÕe. Indeed, the condition for kinks and antikinks to go through each other in the presence of damping w18x, FrF3 G 2Ž2 arv 0 . 3r2 , is incompatible with the stability requirement F - F3 . Finally, we notice that the Žuncorrelated. drift of single kinks and antikinks determines a net string current j s f t s Ž 2p . 2 n 0 Ž T . u F ,

Ž 7.

whereas their spatial diffusion, with variance ² D X 2 Ž t .: s ²w X Ž t . y X 0 x 2 :, corresponds to the string diffusion 2

D f 2 s Ž 2p . 2 n 0 Ž T . ² D X 2 Ž t . :1r2 .

Ž 8.

In Eqs. Ž7. and Ž8. overbars denote spatial averages, L r2 Ž . i.e. Ž . . . . s lim HyL r2 . . . d x. L™`

3. The kinetic model Let us consider a SG string with natural boundary conditions f Ž x ™ "`,t . s 2p m, m s 0," 1," 2, . . . Žno geometrical Žanti.kinks, n 0 s n " . and subjected to a weak external bias with F ) 0. The string will drift in the F direction by nucleating kink-antikink pairs into the adjacent minimum 2p Ž m q 1. of the V w f x potential. Thermal equilibrium is achieved when independently of the thermalization mechanism, the nucleation and the annihilation rates of the f " pairs coincide. Let G denote the equilibrium nucleation rate per unit of string length; the Žanti.kink lifetime t is thus defined by

G s 2 n 0rt .

Ž 9.

Following Ref. w10x, we calculate t by having recourse to the LE formalism Ž6. – Ž8.. In the overdamped limit Ž6b. the mean-square displacement of f " is ² D X 2 Ž t . : s u 2F t 2 q 2 Dt ,

53

equals the relevant mean-square free path n 0 ŽT .y2 . A simple calculation yields

(

G s 2 Dn30 Ž T . 1 q 1 q Ž FrFc .

2

Ž 11 .

with Fc s kTn 0 ŽT .r2p . The physical meaning of Fc is discussed in Section 5. Two limits are of particular interest: for F < Fc

G 0 s 4 Dn30 Ž T .

Ž 12 .

Ž zero bias limit. and for F 4 Fc

G 1 s 2 u F n20 Ž T .

Ž 13 .

Ž weak bias limit.. We make now a few important remarks: Ž1. In view of Eq. Ž5., G 0 and G 1 are Arrhenius rates with activation energies 3 E0 and 2 E0 , respectively. While G 1 points to an underlying two-body nucleation mechanism Žsee Section 4., G 0 hints at a gas kinetics. In the absence of external bias the f ™ yf symmetry of the SG theory may be broken locally, only: the presence of at least one Žanti.kink spectator is required to make the decay of a sub-critical nucleus possible. Ž2. Buttiker and Christen w19,20x criticized the ¨ zero-bias limit of Eq. Ž12. and in particular the predicted 3 E0 activation energy, on the basis of phenomenological arguments. Although indirect numerical evidence Žsee Section 7 and Ref. w21x. supports our viewpoint, a systematic simulation work is required to assess the validity of the kinetic model. Ž3. The drift current corresponding to G 1 can be easily computed: since the nucleated kink-antikink ŽT . under the partners travel a relative distance ny1 0 action of the bias F, the resulting net current is j s Ž2p . G 1rn 0 , whence the result in Eq. Ž7.. More notably, in the weak bias limit the string current turns out to be proportional to the driving force F, as expected in linear response theory. Analogously, on making use of expression Ž12. for G 0 , one recovers the string diffusion law Ž8.. This proves the internal consistency of the kinetic model.

Ž 10 .

with D s Ž kTrE0 .Ž c 02ra .; moreover, we know that the collision between a kink and an antikink is always destructive, whereas two Žanti.kinks bounce off one another almost elastically. The f " lifetime t is then determined by the condition that ² D X 2 Žt .:

4. The two-body model Let us address, now, the question as how a kinkantikink pair may be nucleated starting from a vacuum configuration, e.g. f Ž x,t . s 0. Thermal fluctua-

C. Cattuto et al.r Chemical Physics 235 (1998) 51–59

54

tions are expected to trigger the process by activating a critical nucleus w8,22,23x, the size of which is known to increase with decreasing F – see Eq. Ž18., below. Provided that the critical nucleus size is small enough to ignore many-body effects on the length ŽT . – see Section 4 – we can describe the scale ny1 0 nucleation process as a two-body mechanism. The ensuing nucleation model can be treated as an escape process in a multidimensional system with one neutral equilibrium Žor zero. mode w24x. Thermal fluctuations may activate, with finite probability, a nucleus f N Ž x, X . of length 2 X which encroaches upon the adjacent V w f x minimum 2p . For X 4 d, f N Ž x, X . is well described by the linear superposition of a kink and an antikink centered at .X, respectively, f N Ž x , X . s fq Ž x q X ,0 . q fy Ž x y X ,0 . s 4arctan sinh Ž Xrd . rcosh Ž xrd . ,

Ž 14 . where f " Ž x,t . is defined in Eq. Ž4. with b s 1. w f N Ž x, X . has been centered at the origin for convenience.x The energy of the nucleus D EN is a function of its size 2 X, namely D EN s HSG f N Ž x , X . d x

H

s 2 E0 1 y

ž

= 1q

Ž 16 .

where for R 4 0

cosh Ž 2 Xrd . q 1

sinh Ž 2 Xrd .

overdamped limit the decay of the critical nucleus is fully described by the reduced Žor nucleus. coordinate R s 2 X ŽFig. 1.. On adding the kink-antikink interaction term to the LE Ž6b. for f ", we obtain R˙ s yVNX Ž R . q hR Ž t . ,

1 2 Xrd

Fig. 1. Ža. The critical nucleus potential VN Ž R . for arbitrary values of a MR . The straight line represents the bias potential y2p FR r a MR . For the kink-antikink potential we plotted the function D EN Ž R .y2 E0 with Rs 2 X and D EN Ž R . given in Eq. Ž15.. Žb. Sketch of a critical nucleus; the attractive kink-antikink forces Žinwards arrows. and the bias pulling forces Ždouble arrows. are marked for reader convenience; the vertical arrow points in the direction of the string drift.

/

.

VN Ž R . s y

Ž 15 .

The components of a large nucleus experience two contrasting forces ŽFig. 1.: an attractive one with potential function "4E0 expŽy2 Xrd ., see Eq. Ž15., due to the vicinity of the nucleating partner, and a repulsive one with effective potential "2p FX, due to the external bias which pulls the nucleus partners f " apart. The critical nucleus configuration f N Ž x, R N r2. is attained for a distance R N Ž F . between f " such that the two competing forces balance each other. The critical nucleus f N Ž x, R N r2. is thus the field saddle-point configuration in the escape process associated with the nucleation. In the SG theory w14x f N Ž x, R N r2. admits of one unstable mode, only, with negative eigenvalue l0N, and one neutrally stable translation mode with null eigenvalue Žthe so-called Goldstone mode.. In the

2p F

a MR

Ry

4 E0

a MR

eyR r d .

Ž 17 .

Here, MR s M0r2 and hR Ž t . is the same as h Ž t ., but for the substitution of D with DR s kTra MR in its autocorrelation function. The size of the critical nucleus is set by the condition that VNX Ž R .< R N s 0, whence R N Ž F . s ydln

ž

p F 16 v 02

/

Ž 18 .

and the negative eigenvalue

l0N s VNXX Ž R N . s y

p F 2 a

.

Ž 19 .

The two-body nucleation rate in Gaussian approximation w24x reads

G2 s

< l 0N < ZNX 2p L Z0

eyD E N r kT .

Ž 20 .

C. Cattuto et al.r Chemical Physics 235 (1998) 51–59

Here, the activation energy D EN is given by Eq. Ž15. for 2 X s R N ; Z0 and ZNX denote the Žeffective w24x. partition functions of a SG string with length L ™ ` in its vacuum and saddle-point configuration, respectively. The entropy factor ZNX rZ0 can be factorized as ZNX Z0

s L w 4p kTM0 x

1r2

2p kT

P

Ž a < l0N < .

1r2

P

ZN

ž / Z0

, ph

Ž 21 . where the contributions from the Goldstone mode, the unstable mode and the phonon modes are clearly identifiable. On neglecting corrections of the second order in Frv 02 to the phonon spectrum, the factor Ž ZN rZ0 . ph boils down to 4v 02rŽ2p kT . 2 w14x. On making use of Eqs. Ž18., Ž19. and Ž21. we finally obtain for the nucleation rate Ž20. the well-known result w8,22,10x

G2 s

2 v0

pd

< l0N <

1r2

2 E0

ž / ž / 2pa

kT

1r2

eyD E N r kT .

Ž 22 .

The validity of this formula is restricted to kT < Fd < E0 . The upper bound corresponds to the non-relativistic approximation F < v 02 and guarantees large critical nucleus sizes R N Ž F . 4 d and activation energies D EN close to 2 E0 . The lower bound is required for the Gaussian approximation Ž20. at the saddle-point of VN Ž R . to hold true. An estimate of G 2 for F smaller than, but close to v 02 is reported in Ref. w22x: those authors, however, neglect relativistic effects altogether with no justification.

noted in Section 3 that the rate G 1 is consistent with the prescriptions of the linear response theory; this led us to figure that the discrepancy in Eq. Ž23. should be caused by the assumptions introduced to derive G 2 . Eventually, we concluded that the Gaussian approximation implied by Langer’s formula Ž20. is inadequate to describe the decay process of the critical nucleus represented by the LE Ž16. for the nucleus coordinate R. The approach of Section 4 allows a simple estimate of the non-Gaussian corrections to G 2 . Without entering the intricacies of Langer’s formalism w24x, we remind the reader that the pre-factor in Eq. Ž20. is inÕerse proportional to the Gaussian integral w25,26x `

Hy`exp

ž

< l 0N <

y

/

R 2 d R.

2 DR

`

H0 exp

ž

VN Ž R . DR

/

G1 G2

(

s 2p

Fd kT

Ž 23 .

wwhere D EN has been approximated to 2 E0 x reveals a discrepancy which looks hard to reconcile. We

dR

Ž 25 .

with VN Ž R . s VN Ž R . y VN Ž R N .. Accordingly, the nucleation rate G 2 of Eq. Ž22. is underestimated by the factor `

Let us now compare the nucleation rates G 1 for the weak bias limit and G 2 for the two-body model. For intermediate bias values, say kT < Fd < E0 , the two determinations of the nucleation rates should coincide, at least in principle. Instead, as pointed out in Ref. w10x, the ratio

Ž 24 .

wThe activation rate over a potential barrier is known to coincide with the reciprocal of the mean-first passage time over the same barrier w25x.x Such an integral is proportional to the probability flow at the saddle-point of VN Ž R .. In the present case, however, the analytical form of the potential VN Ž R . is given explicitly in Eq. Ž17., so that instead of the approximate factor Ž24. we could have used the exact factor

kŽ F. s

5. The cross-over regime

55

Hy`exp

ž

y

2 DR

`

=

< l 0N <

H0 exp

ž

/

R2 d R

VN Ž R . DR

/

d R.

Ž 26 .

The reader can easily verify that k Ž F . ™ 1 for Fd 4 kT, whereas for smaller F values the leading contribution from the denominator is correctly estimated by approximating VN Ž R . to yŽ2p Fra MR . R, whence

(

k Ž F . s 2p

Fd kT

.

Ž 27 .

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C. Cattuto et al.r Chemical Physics 235 (1998) 51–59

The discrepancy of Eq. Ž23. is thus explained in close agreement with numerical evidence w21x. Thanks to the correcting factor Ž26., on decreasing F the two-body rate G 2 goes continuously over into the weak-bias limit rate G 1; furthermore, G 1 and the zero bias rate G 0 are analytically connected through Eq. Ž11.. The relevant cross-over F values are: Fc s kTn 0 ŽT .r2p from G 0 up to G 1 and Fk s kTrd from G 1 up to G 2 . Their physical interpretation is straightforward. For F s Fc the thermal energy kT coincides with the mechanical energy needed to pull a single Žanti.kink through a distance of the order of ŽT .. Thus, for values of F its mean free path ny1 0 smaller than Fc the notion of critical nucleus becomes untenable. This is why the zero bias rate G 0 cannot be reproduced through the two-body model of Section 4, no matter how accurately we handle it. For F s Fk the thermal energy kT equals the mechanical work made by the external force to move an Žanti.kink through a distance of the order of its size d. The linearization of the decay dynamics around the saddle-point R s R N applies for F 4 Fk . Effects due to the finite length L of the string may become important. For instance, certain imperfections have the ability to reduce considerably the activation energy of the critical nucleus by introducing an additional pinning potential. Such a mechanism is particularly effective in the dislocation-induced internal friction at low temperature w27x, where the pinning action is exerted by point-like defects. Another example is provided by the flux lines which thread through type-II superconducting films: their length is necessarily limited by the thickness of the sample. The line end-points, when not acting as tight pinning points, may ease the nucleation process depending on the choice of the boundary conditions. Heterogeneous nucleation processes may thus contribute appreciably w28x as either a bulk or a surface effect.

6. Inertial effects In Sections 3–5 we assumed that the SG string was overdamped, i.e. a 4 v 0 . This allowed us to neglect the radiation effects involved in the kink-kink and kink-antikink collisions and, even more notably, to identify the kink Žantikink. lifetime with its mean

free time t Žsee Section 3.. This picture changes dramatically at low damping w29x. In order to discuss the outcome of our numerical simulation in Section 7, we restrict ourselves to the more tractable case of the decay of a metastable string. We start with imposing initial conditions f Ž x,0. s arcsinŽ Frv 02 ., f t Ž x,0. s f x Ž x,0. s 0 and strong tilt, kT < Fd < E0 , so that the two-body model of Section 4 is likely to apply. Inertial effects may be incorporated systematically in the formalism of the multidimensional escape processes w26x. An explicit two-body calculation w30x for the inertial SG string led to the conclusion that Eq. Ž20. is valid at finite values of the damping constant a , too, provided that one rescales

a a™

2

ž ( 1q

1q

4v 02

a2

/

.

Ž 28 .

Thus, for vanishingly small values of arv 0 the relevant nucleation rate G 2 would read

G2 s

v0 2p d

F

1r2

2 E0

ž / ž / v 02

kT

1r2

eyD E N r kT ,

Ž 29 .

independent of a . The rescaling law Ž28. is consistent with the mean-first-passage time argument given in Section 5 w26x, so that it can be carried over into the intermediate tilt regime Fc < F < Fk , as well. It follows that the nucleation rate G 1 at very low damping ought to be rewritten as

G1 s

8v0 d

Fd

ž / kT

eyD E N r kT .

Ž 30 .

At this point, we notice that both Eqs. Ž29. and Ž30. boil down to a transition-state theory ŽTST. approximation w23x. The validity of such an approximation has been tested so far only for the 0 q 1 dimensional case Že.g. for the Brownian motion in a washboard potential w25x.: the TST fails for arv 0 0.1 and in the underdamped limit arv 0 ™ 0 the escape rates turn out to be linear in a w13,25x. For the 1 q 1 dimensional system at hand, in the absence of tilt, the failure of the TST approximation seems to occur only at extremely low damping w29x, anyway, well below the arv 0 values where our simulation shows a significant statistics. Moreover, metastable

C. Cattuto et al.r Chemical Physics 235 (1998) 51–59

57

SG strings at low damping undergo the so-called unlocking transition: for F larger than a threshold value Fth Žat zero temperature Fth , Ž2 q '2 . av 0 . a steady string flow sets in, so that the notion of pair nucleation becomes ineffective w16x. Pair nucleation dominates the flow onset in the form of a cascaded nucleation mechanism Žor avalanche formation. which is hardly describable within the two-body model. Furthermore, since Fth is linear in a , a clear-cut low-damping nucleation process is likely to be observable at very small tilts, only, where the G 1 rate formula Ž30. replaces formula Ž29..

7. Numerical simulation We ran the simulation code of Ref. w16x for small tilts, F < v 02 , and high-to-intermediate damping constant values a R v 0 , in order to verify some of the theoretical predictions of Sections 3-6. Independent Gaussian random number generators simulate the noise sources z i Ž t . at each site i s 1,2, . . . , N of the discretized SG chain Ždiscretization step D x s 1.. The integration time step D t must be taken small enough Žhere v 0 D t s 10y3 . for the simulated noises z i Ž t . to be considered delta-correlated Žor white.. The discretized version of the string Eq. Ž1. is integrated by means of a modified version of Mil’shtein’s algorithm Žsee Ref. w16x.. We set initial conditions f Ž x,0. s f x Ž x,0. s 0 and let the string evolve in time according to Eq. Ž1.: the number N Ž t . of nuclei with size not smaller than 2 RŽ F ., see Eq. Ž18., was recorded as a function of time. A good statistics was achieved by simulating chains with length L much larger than the kink mean ŽT . Žergodic assumption.. In Fig. free-path, L 4 ny1 0 2 we display an example of the curve kink density nŽ t . s N Ž t .rL versus time: its approach towards the relevant equilibrium value n 0 ŽT . obeys the exponential transient law n Ž t . s n 0 Ž 1 y ey2 t r t .

Fig. 2. Kink density versus time for kT s 2 and F s 0.3. The string parameters are: v 02 s1, c 02 s 0.1, a s10; the main simulation parameters are D x s1, D t s10y3 , Ls10 4 , with periodic boundary conditions. The solid line represents the fitting law Ž31.; the shift of the time origin accounts for the string thermalization transient, where the temperature is raised from zero up to the chosen value T.

vals large with respect to the phonon time scale vy1 0 , but small compared to the kink life-time t . In Fig. 3 we show the dependence of 1rt on the bias amplitude F for several values of kT. Two features are clearly visible: Ž1. G s 2 n 0rt grows almost linearly with F – but only for not too small Frv 02 values! This is in agreement with the kinetic model prediction G 1 of Section 3. Unfortunately, the two-body model prediction G 2 of Section 4, G 2 A 'F , is not easily

Ž 31 .

fairly well. The numerical computation of the kink density requires special caution in order not to mistake local fluctuations Žbelonging to the phonon sector of the SG theory. for nucleating kink-antikink pairs. In particular, the number of barrier crossings to either direction has been averaged over time inter-

Fig. 3. Reciprocal of the kink life-time t versus F at kT s1, kT s 2, kT s 3. Both the string and the simulation parameters are as in Fig. 2. The error magnitude on the fitting parameter t is smaller than 15%.

58

C. Cattuto et al.r Chemical Physics 235 (1998) 51–59

Fig. 4. Reciprocal of the kink life-time t versus the damping constant a for kT s 2 and three values of F. The remaining string and simulation parameters are as in Fig. 2.

accessible numerically, since the simultaneous requirements Frv 02 4 kTrd and F < v 02 would imply extremely long computer runs at very low temperatures. Ž2. As F approaches zero the nucleation rate Žor the kink life-time. tends to a constant value: according to the kinetic model of Section 3 this corresponds to the purely diffusive regime of the nucleation process, described by the rate G 0 . The crossover between the two limits G 0 and G 1 of G Ž F . is well located by the critical value Fc s kTn 0 ŽT .r2p discussed in Section 5. Finally we investigated the dependence of the pair nucleation rate on the damping constant a for v 0 Q a F ` Žhigh-to-intermediate damping.. As expected, the TST theory approximation of Section 6 is consistent with the outcome of our simulations. In particular, the nucleation rate Žor the kink life-time. retains its dependence on the external bias, no matter what the value of arv 0 . For instance, for the F values of Fig. 4 the curves G Ž a . fall within the zero-bias limit of Section 3. A detailed discussion of our simulation work will be presented in a forthcoming article.

8. Conclusions We conclude by hinting at a few affordable extensions of our theory of nucleation in one dimensional strings. First of all, we notice that both the kinetic and the two-body model can be implemented for two more soliton-bearing strings of wide use in physics,

namely the f 4 and the double-quadratic strings w14x. A detailed analysis of the relevant nucleation process requires the knowledge of the Žanti.kink shape and the phonon spectrum in the presence of a kink Žantikink.; constraints in the Žanti.kink statistics must be introduced to account for the lack of transport currents in the f 4 theory. Analytical expressions for the nuclation rates in the overdamped limit may be easily obtained following the procedures of Sections 3 and 4. Pair nucleation may be facilitated by the presence of point-like defects which are ineliminable even in extremely pure materials w1–3x. For instance, a defect such as an impurity tends to bind to a string-like dislocation; correspondingly, the dislocation bulges in the direction of the defect by producing a nucleus with activation energy smaller than D EN Ž F .. As a matter of fact, even in the zero bias limit F s 0, the nucleation process can still occur through a two-body mechanism, as long as the f ™ yf symmetry gets broken locally by a random distribution of defects w27x. The role of spatial disorder as a control parameter of the nucleation process is matter for ongoing research.

Acknowledgements Work supported in part by the Istituto Nazionale di Fisica Nucleare ŽINFN., VIRGO Project.

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