VOLUME 46, NUMBER 1 PHYSICAL REVIEW LETTERS 5 JANUARY 1981

Gunther and Imry Respond: Recent ly , 1 we showed that at t e m p e r a t u r e s so low that kBT i s much l e s s than the r e s t energy of a kink,2 with kinks a s ­sumed to have infinite lifetime, a given particle in an infinite s ine-Gordon chain has a mean -squa re d isp lacement given by

(x2)=ns(4Dst/n)^\ (1)

where Ds is the kink diffusion constant and ns is the total density of kinks. This result was subse­quently confirmed by Buttiker and Landauer.3

A more difficult problem is the effect of the finite lifetime of kinks on the mean-square dis­placement (x2). In Ref. 1, we considered this problem by making an approximation, that kinks and antikinks are not correlated, being, in par­ticular, created and annihilated independently of one another. Assuming that dying kinks are r e ­placed by kinks which are created uniformly in time and in space along the chain, with no correla­tions over time scales much greater than the life­time rs of a kink, we obtained

(x2) = ns(Ds/rs)^2t. (2)

The change from tl/2 to t behavior is dramatic. In the preceding Comment,4 Buttiker and Lan-

dauer state that the approximation which we made, of uncorrelated kinks and antikinks, is critically incorrect, and that, in fact, when one takes into account creation and annihilation in pairs, our Eq. (1) remains valid over time scales much greater than that originally envisaged by us (namely t£rs). This is supported by the pheno-menological theory of Ref. 3, which makes use of the fact that pair creation and annihilation do not change Jpdx, where p = (density of kinks -density of antikinks), and that p and its co r res ­ponding current density are what govern particle motion.

We would like to point out that linear behavior can also be obtained under conditions much more general than those assumed in Ref. 1.

Let us assume that there is no long-time mem­ory between kink annihilations and future kink creations, Then the effect of kink motion during any given time interval T » T S is essentially in­dependent of the corresponding effect during any future time interval r.5 In this case, we expect a given particle to undergo effectively a random walk, with

<*2> = & V T , for t»T, (3)

where b is the root-mean-square displacement of x during a time interval r„6

76

Thus the argument of Refs, 3 and 4 implies that the independence of events occurring at times separated by intervals much greater than TS does not hold0 According to Refs. 3 and 4, this is due to the kinks' and antikinks' being created and annihilated in pairsc The system should thus ex­hibit long-time memory.7 It would be extremely interesting to study in detail how this long-time memory effect can come about in this dissipative system.

This work was supported in part by the XL S. -Israel Binational Science Foundation.

Leon Gunther Physics Department Tufts University, Medford, Mass. 02155

Yoseph Imry IBM Thomas J. Watson Research Center Yorktown Heights, New York 10598 and Tel-Aviv University Ramat Aviv, I s r a e l ^

Received 12 December 1980 PACS numbers: 05.40,+j, 63.75,+ z, 64.60.My

a)Permanent address. 1L. Gunther and Y. Imry, Phys. Rev. Lett. 44, 1225

(1980). 2The term "soliton" was used in Ref. 1. 3M. Buttiker and R. Landauer, J . Phys. C 13, L325

(1980). 4M. Buttiker and R. Landauer, preceding Comment

[Phys. Rev. Lett. 4£, 75(C) (1981)]. 5In the actual damped sine-Gordon chain, the position

where a pair is annihilated will certainly create a "hot spot" due to the energy of the pair. There will there­fore temporarily be an increased likelihood that a new pair will be created at that position. We assume that the vast majority of annihilated pairs either are recreated so rapidly at the position of annihilation that we might as well not regard an annihilation as having taken place or have their energies dissipated away without signifi­cantly affecting the position where future pair creations take place.

6It is important to note that the choice of r is not e s ­sential as long as T»TS. If, for example, T is doubled, b2 will be doubled too, resulting in no change in (*2) .

7It should be understood however, that such a memory effect is of a different nature than the "long-time tail" suggested by Schneider and Stoll I Phys. Rev. Lett. 41, 1429 (1980)], which results in tA/z behavior. This i ^ 3

behavior has been amended by Schneider and Stoll [Phys Rev. B £2, 395 (1980)] to tllz behavior for the Hamil-tonian system and t1'2 behavior for the overdamped system. We emphasize that the t 2 behavior of Refs. 3 and 4 should hold at long enough times for any amount of damping, however small.