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Solitons in a random field of force: A Langevin equation approach
Volume 115, number 1,2
SOLITONS
IN A R A N D O M
PHYSICS LETTERS A
24 March 1986
F I E L D O F FORCE: A LANGEVIN E Q U A T I O N A P P R O A C H
Fabio M A R C H E S O N I 1 lnstitut fi~r Theoretische Physik, R W T H Aachen, 5100 Aachen, FRG Received 6 January 1986; accepted for publication 21 January 1986
The perturbation approach of McLaughlin and Scott is extended to treat perturbed sine-Gordon equations accounting for losses and external fluctuations in real systems. At the lowest order a Langevin equation is obtained in the global variables of the solitonic solution, the coefficients of which depend on the adopted fluctuation statistics.
Soliton-bearing equations have been attracting increasing interest because of their remarkable mathematical properties, but also because of their generic occurrence as approximate model descriptions of a vast and diverse array of physical phenomena. However, realistic applications in most branches of physics and electronics demand the inclusion of various perturbations, leading to problems beyond those of pure integrable systems [ 1]. Such perturbations would account for a variety of spurious effects which cannot be neglected in most real systems of current interest. These effects have been often modelled by adding to the original soliton-bearing equation suitable extra terms which are assumed to be "small" and deterministic [2]. The role of certain classes of thermal fluctuations has been discussed earlier in the context of a classical statistical mechanical approach [3-5]. In the present letter we extend the perturbation technique of McLaughlin and Scott [2] to deal with perturbed soliton-bearing equations which may include fluctuations both in space and in time. We always address ourselves to the sine-Gordon (SG) equation; method and results obtained below, however, can be readily extended to the whole class of one-dimensional soliton-bearing equations. A perturbed SG equation in dimensionless units reads:
Permanent address: Dipartimento di Fisica dell'Universith, 1-06100 Perugia, Italy.
Ctt - CPxx + sin ¢ =f(¢, c~t,x , t ) ,
(1)
where ~b= ¢(x, t) and the perturbation f(¢, C t , x , t) is in principle arbitrary. McLaughlin and Scott assumed each localized solution of the SG equation to be stable under perturbation. A deterministic "small" perturbation f(¢, Ct) [2] puts some constraints - in the form of equations of motion - on the independent parameters, or global coordinates, which determine the unperturbed solution. For the kink/antikink solutions ¢±(x, t) = 4 tg -1 {exp[+(x - ut)/(1 - u2)1/2] } , (2) the perturbation procedure of ref. [2] yields the following first order equations in the global coordinates Xk(t ) and Uk(t ) Xk = U k ,
dHsG/dt = f -'~
f(dfi,dp~)c~(x, t) d x . (3, 4)
In eqs. (3), (4)Xk(t ) and Uk(t ) denote position and velocity of the kink center of mass and replace x and u in ~b±.HSG coincides with the total energy of the unperturbed solution ~±(Xk, Uk) , i.e. HsG(Uk) = Ek/(1 -u2) I/2, where E k = 8 is the kink mass (or rest energy). Higher perturbation orders can be calculated straightforwardly. Let us consider now more general cases when the perturbation f m a y include fluctuations both in space and in time. For start we study a familiar example
29
Volume 115, number 1,2
PHYSICS LETTERS A
f(~t, x, t) -- -a(~ t + ~(x, t) ,
(~(x, t)~(x', t ' ) ) = 2Qf(x - x ' ) b ( t
- t').
(6)
In the discrete (lattice) representation of eqs. (1) and (5) ~(x, t) corresponds to an array of gaussian (in time) delta-correlated random forces acting on each site independently and with equal intensities. The statistical mechanical implications of such an incoherent fluctuation are well established [3,4]. The continuum limit of the corresponding lattice Fokker-Planck equation (FPE) [6] coincides with the classical partition function of the hamiltonian system from which the unperturbed SG equation derives (with/3 = ( k B T ) - I = a/Q). A strict equivalence between the two descriptions holds in the canonical ensemble (hamiltonian) limit, a -~ O, Q ~ 0 and/3 = a/Q constant. We determine now how a random field of force ~(x, t) with finite a and Q affects the motion of a single kink (or antikink). Without loss of generality we factorize ~'(x, t) as ~'(x, t) = ~ ( x ) v ( t ) ,
(7)
with (rl)x = O, ( 7)t = O, (rl(x)rl(x')) x = 5(x -- x ' ) and ( 7 ( t ) 7 ( t ' ) ) t = 2 Q f ( t - t'), the averages being taken with respect to space and time, respectively. As unperturbed solitons are invariant under global translation, we are only interested in the average of the perturbed solution trajectories over all the spatial realizations of
~(x, t). Such an average can be carried out by means of well-known theorems of the theory of stochastic processes [7]. Under suitable integrability conditions, it can be shown that, since the random process having the sample function ~"is a gaussian random process and Cx(X, t) is a continuous function o f x and t, then the random process having sample function oo
= f
t) dx
(8)
_co
is also a gaussian random process. The statistics of ~n(t) can be determined introducing factorization (6) and shifting the origin o f x arbitrarily so that
30
7
(5)
where a is a damping constant and ~"is a themlal random field of force with (~') = 0 and
24 March 1986
~n(t) = 7(t) J
r/(S)~x(S, 0) d s .
(9)
The integral in eq. (9) defines a gaussian random variable [7]. The average of ~n(t) over the spatial realizations of ~'(x, t), represented by ~(x), is therefore a gaussian random process in time, ~(t), with zero mean and autocorrelation function
(~(t)~(t')) t = 2QS(t - t') X
rgx)d~x(X, 0) dx
,
(10)
where, employing eq. (7),
f
o) dx
=f
(n(x)~(x)~e~(x, ' + o)e~i(x, o) dx dx'= e k .
-~*
(11)
If we assume a small ( a ' ~ 1) and T fixed [2], eqs. (3), (4) are still valid at the first order in a provided that the spatial average of the rhs of the second equation is taken according to the above prescription. The equations of motion (3), (4) are thus reduced to /3k = --aPk + V ~ k 7 ( t ) ,
(12)
where we used q~ = --Uk~~ andPk = UkEk/(1 -u2) 1/2 is the relativistic m o m e n t u m of the kink/antikink solution. The Langevin equation (LE)(12) for the global coordinates (Xk, uk) of the perturbed solitonic solution also provides us with a prediction for the mean kink/ antikink kinetic energy. It is easily proved that in the limit of low temperature, a/Q "~ 1, single non-relativistic kinks with mass E k = 8 have mean translational energy ½kBT. Furthermore, SG kinks obtain their degree of freedom at the expense of one degree of freedom in the phonon modes (Goldstone mode). According to the corresponding lattice FPE the thermal energy for each phonon mode is kBT. So far we have recast the main conclusions of ref. [3] in the LE formalism. We apply now our technique to a slightly more general case. Let us assume that ~'(x, t) has a correlation length X, i.e. after factoriza-
Volume 115, number 1,2 tion
PHYSICS LETTERS A
(7)
(~x(x)rlx(x'))x = (2?0 -1 e x p ( - I x - x'l/X).
~bk = - a p k + 27rT(t). (13)
The case of spatially correlated noise is of obvious interest for application to condensed matter physics and electronics [ 1]. Using the procedure outlined above we arrive again at eq. (11) where the average under integration is now to be replaced by eq. (13). The second integral can be calculated explicitly for T+0
= (Ek/2X)'~(2 , (1 + ~,)/2X) < E k ,
(14)
where ~( , ) denotes Riemann's zeta function. The corresponding LE ibk = --apk + [(2X)-I ~(2, (1 + X)/2X)] 1/2x,~k "r(t) (12') reproduces LE (12) in the limit X ~ 0. From eq. (14) it is apparent that in the presence of spatially correlated fluctuation the picture of the perturbed kinks as brownian particles subject to a suitable external x-independent noise is still a predictive approximation. The translational (thermal) energy of a single kink/antikink solution is modified as a consequence of spatial coherence. Kinks are actually extended objects; their coupling with an external source of noise varies in intensity depending on some scale of length which quantifies the spatial correlation of the fluctuation. The results of eqs. (12') and (14) suggest that when X is smaller than the kink size the coherent motion of the localized solution as a whole is hindered due to the non-rigid (viscous) behaviour of the solitonic structure. A much more effective transfer of thermal energy from the heat bath to the solitonic structure can be achieved by using an almost coherent external fluctuation, i.e. a fluctuation with X much larger than the kink size. Indeed if we assume ~'(x, t) =- 7 ( t ) ,
24 March 1986
(15)
see for comparison eqs. (6) and (13), a distinct LE follows eq. (3) immediately
(16)
This LE is the counterpart of eq. (12) for a spatially coherent fluctuation. The translational (thermal) energy associated with each kink in the presence of an additive gaussian and (in time) delta-correlated noise is ~Tr2kBT. This result is only valid in the nonrelativistic limit, i.e. T ~ 0. A statistical mechanical analysis of the solutions of the perturbed SG equation (1), (5) and (15) has been carried out in refs. [3,4]. That allowed us to check the correctness of our first predictions based on the LE formalism. The last case discussed in this letter, however, has not been fully investigated in the literature yet. We prove now that the predictions based on the LE (16) are compatible with a statistical mechanical approach to the perturbed SG equation (1), (5) and (15). We start noting that the FPE associated with the discrete representation of such equation does not verify the conditions of detailed balance [6]. The corresponding equilibrium distribution function cannot be determined analytically by adopting conventional techniques. We resorted eventually to the phenomenological (approximate) method expounded in sect. 3 of ref. [3]. It is assumed that the effects of a perturbation f(q~, ~t, x, t) can be incorporated in the classical mechanical analysis of the unperturbed system by simply renormalizing the kink self-energy Zk(T ) for possible structure effects. In notation of ref. [3] such renormalization of Zk(T ) amounts to a mere additional term AZk(T ) c~
where the averages are taken over all ~(x, t) realizations both in space and in time. This quantity vanishes in the case of spatially incoherent fluctuations (5)-(7). For the system under study, instead, eq. (17) yields AZk(T)=
1~1 2 -- 1)kBTln(Ek/kBT ). --~t'~zr
(18)
Introducing Zk(T ) + AZk(T ) in the formalism of ref. [3] we easily determine the relevant statistical quantities of the system. In particular we checked that the 31
Volume 115, number 1,2
PHYSICS LETTERS A
mean translational energy of each kink/antikink component is rt2/2 times the prediction for the unperturbed SG equation. Accordingly, the average density of such components n(T) is given by
n(T) = (Ek/k B T)~2/4-1/2 n s G ( T ) ,
(19)
where nsG(T ) coincides with the corresponding predictions of refs. [3,4]. The renor0aalization of Zk(T ) (17) can be justified as follows. We note immediately that the prefactor of eq. (17) corrects the definition of kink kinetic energy to allow for nonlocal spatial correlations. This interpretation is particulary transparent in the discrete representation. A second argument in favour of our choice (17) comes from the FPE formalism. Applying the FP operator corresponding to the perturbed SG equation to the classical SG partition function P0 [¢] (or to the equilibrium distribution function of the FPE corresponding to the system (1), (5) and (6)) we obtained
F Q
.
_ .
.
.
f
oIOl.
ons and antifluxons takes place at a frequency determined by the intensity of an external dc current bias, our approach introduces quite naturally lower bounds to the oscillator line width achievable under diverse experimental circumstances [ 10]. Applications of the LE approach to the study of soliton-bearing equations in the presence of stochastic perturbation will be reported in forthcoming publications. I wish to thank the Alexander yon Humboldt Stiftung for a fellowship and Professor B.U. Felderhof for hospitality at the Institut f~ir theoretische Physik (RWTH Aachen). Professor L. V,Szquez, Professor N.G. van Kampen and Dr. W. Renz are thanked for useful discussions and the Fondazione A Della Riccia for financial support.
Note added. After submission of the present letter for publication Dr. M. Salerno brought to my attention a paper by Joergensen et al. [11] where the Langevin equation (12) is obtained. The method adopted there and subsequent discussion, however, are completely different.
(20)
The resulting temporal divergence of the function HSG(Uk) (the unperturbed kink energy) is balanced right by the self-energy renormalization (17). We make now some concluding remarks about the importance of our results. Our approach provides a workable counterpart to the numerical simulation of the time evolution of solitonic solutions in the presence of external fluctuations [8]. When the perturbation is taken small enough for our perturbation approach to work the life time of a single perturbed solitonic solution is still very long. The diffusional properties of simulated solutions can then be better compared with a LE description than with a canonical ensemble one. In the presence of a coherent field of force the stability of such objects is markedly affected as proved by the increase of the kink/antikink density n(T) in eq. (19). Another interesting application of our LE approach is the design of Josephson junction transmission lines. Since the socalled fluxon oscillators [2,9] are depicted as circuits where a periodic flow of flux-
32
24 March 1986
References [1] A.R. Bishop, in: Solitons in action, eds. K. Lonngren and A.C. Scott (Academic Press, New York, 1978). [2] D.W. McLaughlin and A.C. Scott, Phys. Rev. A18 (1978) 1652. [3] J.F. Currie, J.A. Krumhansl, A.R. Bishop and S.E. Trullinger, Phys. Rev. B22 (1980) 477. [4] M. BlRtiker and R. Landauer, Phys. Rev. A23 (1981) 1397. [5] W. Wonneberger, Physica 103A (1980) 543. [6] H. Risken, Springer series in synergeties, VoL 18. The Fokker-Planck equation (Springer, Berlin, 1983). [7] W.B. Davenport and W.L. Root, Random signals and noise (McGraw-Hill, New York, 1958). [8] P. Pasqualand L. V~zquez, Phys. Rev. B32 (1985) 8532; F. If, M.P. Soerensen and P.L. Christiansen, Phys. Lott. 100A (1984) 68. [9] M. Salerno, Phys. Lett. 87A (1981) 116; M. Salerno and A.C. Scott, Phys. Rev. B26 (1982) 2474. [10] F. Maxchesoni, AppL Phys. Lett. (1986), to be pubfised. [11 ] Joergonsen et al., Phys. Rev. Lett. 49 (1982) 1093.
SOLITONS
IN A R A N D O M
PHYSICS LETTERS A
24 March 1986
F I E L D O F FORCE: A LANGEVIN E Q U A T I O N A P P R O A C H
Fabio M A R C H E S O N I 1 lnstitut fi~r Theoretische Physik, R W T H Aachen, 5100 Aachen, FRG Received 6 January 1986; accepted for publication 21 January 1986
The perturbation approach of McLaughlin and Scott is extended to treat perturbed sine-Gordon equations accounting for losses and external fluctuations in real systems. At the lowest order a Langevin equation is obtained in the global variables of the solitonic solution, the coefficients of which depend on the adopted fluctuation statistics.
Soliton-bearing equations have been attracting increasing interest because of their remarkable mathematical properties, but also because of their generic occurrence as approximate model descriptions of a vast and diverse array of physical phenomena. However, realistic applications in most branches of physics and electronics demand the inclusion of various perturbations, leading to problems beyond those of pure integrable systems [ 1]. Such perturbations would account for a variety of spurious effects which cannot be neglected in most real systems of current interest. These effects have been often modelled by adding to the original soliton-bearing equation suitable extra terms which are assumed to be "small" and deterministic [2]. The role of certain classes of thermal fluctuations has been discussed earlier in the context of a classical statistical mechanical approach [3-5]. In the present letter we extend the perturbation technique of McLaughlin and Scott [2] to deal with perturbed soliton-bearing equations which may include fluctuations both in space and in time. We always address ourselves to the sine-Gordon (SG) equation; method and results obtained below, however, can be readily extended to the whole class of one-dimensional soliton-bearing equations. A perturbed SG equation in dimensionless units reads:
Permanent address: Dipartimento di Fisica dell'Universith, 1-06100 Perugia, Italy.
Ctt - CPxx + sin ¢ =f(¢, c~t,x , t ) ,
(1)
where ~b= ¢(x, t) and the perturbation f(¢, C t , x , t) is in principle arbitrary. McLaughlin and Scott assumed each localized solution of the SG equation to be stable under perturbation. A deterministic "small" perturbation f(¢, Ct) [2] puts some constraints - in the form of equations of motion - on the independent parameters, or global coordinates, which determine the unperturbed solution. For the kink/antikink solutions ¢±(x, t) = 4 tg -1 {exp[+(x - ut)/(1 - u2)1/2] } , (2) the perturbation procedure of ref. [2] yields the following first order equations in the global coordinates Xk(t ) and Uk(t ) Xk = U k ,
dHsG/dt = f -'~
f(dfi,dp~)c~(x, t) d x . (3, 4)
In eqs. (3), (4)Xk(t ) and Uk(t ) denote position and velocity of the kink center of mass and replace x and u in ~b±.HSG coincides with the total energy of the unperturbed solution ~±(Xk, Uk) , i.e. HsG(Uk) = Ek/(1 -u2) I/2, where E k = 8 is the kink mass (or rest energy). Higher perturbation orders can be calculated straightforwardly. Let us consider now more general cases when the perturbation f m a y include fluctuations both in space and in time. For start we study a familiar example
29
Volume 115, number 1,2
PHYSICS LETTERS A
f(~t, x, t) -- -a(~ t + ~(x, t) ,
(~(x, t)~(x', t ' ) ) = 2Qf(x - x ' ) b ( t
- t').
(6)
In the discrete (lattice) representation of eqs. (1) and (5) ~(x, t) corresponds to an array of gaussian (in time) delta-correlated random forces acting on each site independently and with equal intensities. The statistical mechanical implications of such an incoherent fluctuation are well established [3,4]. The continuum limit of the corresponding lattice Fokker-Planck equation (FPE) [6] coincides with the classical partition function of the hamiltonian system from which the unperturbed SG equation derives (with/3 = ( k B T ) - I = a/Q). A strict equivalence between the two descriptions holds in the canonical ensemble (hamiltonian) limit, a -~ O, Q ~ 0 and/3 = a/Q constant. We determine now how a random field of force ~(x, t) with finite a and Q affects the motion of a single kink (or antikink). Without loss of generality we factorize ~'(x, t) as ~'(x, t) = ~ ( x ) v ( t ) ,
(7)
with (rl)x = O, ( 7)t = O, (rl(x)rl(x')) x = 5(x -- x ' ) and ( 7 ( t ) 7 ( t ' ) ) t = 2 Q f ( t - t'), the averages being taken with respect to space and time, respectively. As unperturbed solitons are invariant under global translation, we are only interested in the average of the perturbed solution trajectories over all the spatial realizations of
~(x, t). Such an average can be carried out by means of well-known theorems of the theory of stochastic processes [7]. Under suitable integrability conditions, it can be shown that, since the random process having the sample function ~"is a gaussian random process and Cx(X, t) is a continuous function o f x and t, then the random process having sample function oo
= f
t) dx
(8)
_co
is also a gaussian random process. The statistics of ~n(t) can be determined introducing factorization (6) and shifting the origin o f x arbitrarily so that
30
7
(5)
where a is a damping constant and ~"is a themlal random field of force with (~') = 0 and
24 March 1986
~n(t) = 7(t) J
r/(S)~x(S, 0) d s .
(9)
The integral in eq. (9) defines a gaussian random variable [7]. The average of ~n(t) over the spatial realizations of ~'(x, t), represented by ~(x), is therefore a gaussian random process in time, ~(t), with zero mean and autocorrelation function
(~(t)~(t')) t = 2QS(t - t') X
rgx)d~x(X, 0) dx
,
(10)
where, employing eq. (7),
f
o) dx
=f
(n(x)~(x)~e~(x, ' + o)e~i(x, o) dx dx'= e k .
-~*
(11)
If we assume a small ( a ' ~ 1) and T fixed [2], eqs. (3), (4) are still valid at the first order in a provided that the spatial average of the rhs of the second equation is taken according to the above prescription. The equations of motion (3), (4) are thus reduced to /3k = --aPk + V ~ k 7 ( t ) ,
(12)
where we used q~ = --Uk~~ andPk = UkEk/(1 -u2) 1/2 is the relativistic m o m e n t u m of the kink/antikink solution. The Langevin equation (LE)(12) for the global coordinates (Xk, uk) of the perturbed solitonic solution also provides us with a prediction for the mean kink/ antikink kinetic energy. It is easily proved that in the limit of low temperature, a/Q "~ 1, single non-relativistic kinks with mass E k = 8 have mean translational energy ½kBT. Furthermore, SG kinks obtain their degree of freedom at the expense of one degree of freedom in the phonon modes (Goldstone mode). According to the corresponding lattice FPE the thermal energy for each phonon mode is kBT. So far we have recast the main conclusions of ref. [3] in the LE formalism. We apply now our technique to a slightly more general case. Let us assume that ~'(x, t) has a correlation length X, i.e. after factoriza-
Volume 115, number 1,2 tion
PHYSICS LETTERS A
(7)
(~x(x)rlx(x'))x = (2?0 -1 e x p ( - I x - x'l/X).
~bk = - a p k + 27rT(t). (13)
The case of spatially correlated noise is of obvious interest for application to condensed matter physics and electronics [ 1]. Using the procedure outlined above we arrive again at eq. (11) where the average under integration is now to be replaced by eq. (13). The second integral can be calculated explicitly for T+0
= (Ek/2X)'~(2 , (1 + ~,)/2X) < E k ,
(14)
where ~( , ) denotes Riemann's zeta function. The corresponding LE ibk = --apk + [(2X)-I ~(2, (1 + X)/2X)] 1/2x,~k "r(t) (12') reproduces LE (12) in the limit X ~ 0. From eq. (14) it is apparent that in the presence of spatially correlated fluctuation the picture of the perturbed kinks as brownian particles subject to a suitable external x-independent noise is still a predictive approximation. The translational (thermal) energy of a single kink/antikink solution is modified as a consequence of spatial coherence. Kinks are actually extended objects; their coupling with an external source of noise varies in intensity depending on some scale of length which quantifies the spatial correlation of the fluctuation. The results of eqs. (12') and (14) suggest that when X is smaller than the kink size the coherent motion of the localized solution as a whole is hindered due to the non-rigid (viscous) behaviour of the solitonic structure. A much more effective transfer of thermal energy from the heat bath to the solitonic structure can be achieved by using an almost coherent external fluctuation, i.e. a fluctuation with X much larger than the kink size. Indeed if we assume ~'(x, t) =- 7 ( t ) ,
24 March 1986
(15)
see for comparison eqs. (6) and (13), a distinct LE follows eq. (3) immediately
(16)
This LE is the counterpart of eq. (12) for a spatially coherent fluctuation. The translational (thermal) energy associated with each kink in the presence of an additive gaussian and (in time) delta-correlated noise is ~Tr2kBT. This result is only valid in the nonrelativistic limit, i.e. T ~ 0. A statistical mechanical analysis of the solutions of the perturbed SG equation (1), (5) and (15) has been carried out in refs. [3,4]. That allowed us to check the correctness of our first predictions based on the LE formalism. The last case discussed in this letter, however, has not been fully investigated in the literature yet. We prove now that the predictions based on the LE (16) are compatible with a statistical mechanical approach to the perturbed SG equation (1), (5) and (15). We start noting that the FPE associated with the discrete representation of such equation does not verify the conditions of detailed balance [6]. The corresponding equilibrium distribution function cannot be determined analytically by adopting conventional techniques. We resorted eventually to the phenomenological (approximate) method expounded in sect. 3 of ref. [3]. It is assumed that the effects of a perturbation f(q~, ~t, x, t) can be incorporated in the classical mechanical analysis of the unperturbed system by simply renormalizing the kink self-energy Zk(T ) for possible structure effects. In notation of ref. [3] such renormalization of Zk(T ) amounts to a mere additional term AZk(T ) c~
where the averages are taken over all ~(x, t) realizations both in space and in time. This quantity vanishes in the case of spatially incoherent fluctuations (5)-(7). For the system under study, instead, eq. (17) yields AZk(T)=
1~1 2 -- 1)kBTln(Ek/kBT ). --~t'~zr
(18)
Introducing Zk(T ) + AZk(T ) in the formalism of ref. [3] we easily determine the relevant statistical quantities of the system. In particular we checked that the 31
Volume 115, number 1,2
PHYSICS LETTERS A
mean translational energy of each kink/antikink component is rt2/2 times the prediction for the unperturbed SG equation. Accordingly, the average density of such components n(T) is given by
n(T) = (Ek/k B T)~2/4-1/2 n s G ( T ) ,
(19)
where nsG(T ) coincides with the corresponding predictions of refs. [3,4]. The renor0aalization of Zk(T ) (17) can be justified as follows. We note immediately that the prefactor of eq. (17) corrects the definition of kink kinetic energy to allow for nonlocal spatial correlations. This interpretation is particulary transparent in the discrete representation. A second argument in favour of our choice (17) comes from the FPE formalism. Applying the FP operator corresponding to the perturbed SG equation to the classical SG partition function P0 [¢] (or to the equilibrium distribution function of the FPE corresponding to the system (1), (5) and (6)) we obtained
F Q
.
_ .
.
.
f
oIOl.
ons and antifluxons takes place at a frequency determined by the intensity of an external dc current bias, our approach introduces quite naturally lower bounds to the oscillator line width achievable under diverse experimental circumstances [ 10]. Applications of the LE approach to the study of soliton-bearing equations in the presence of stochastic perturbation will be reported in forthcoming publications. I wish to thank the Alexander yon Humboldt Stiftung for a fellowship and Professor B.U. Felderhof for hospitality at the Institut f~ir theoretische Physik (RWTH Aachen). Professor L. V,Szquez, Professor N.G. van Kampen and Dr. W. Renz are thanked for useful discussions and the Fondazione A Della Riccia for financial support.
Note added. After submission of the present letter for publication Dr. M. Salerno brought to my attention a paper by Joergensen et al. [11] where the Langevin equation (12) is obtained. The method adopted there and subsequent discussion, however, are completely different.
(20)
The resulting temporal divergence of the function HSG(Uk) (the unperturbed kink energy) is balanced right by the self-energy renormalization (17). We make now some concluding remarks about the importance of our results. Our approach provides a workable counterpart to the numerical simulation of the time evolution of solitonic solutions in the presence of external fluctuations [8]. When the perturbation is taken small enough for our perturbation approach to work the life time of a single perturbed solitonic solution is still very long. The diffusional properties of simulated solutions can then be better compared with a LE description than with a canonical ensemble one. In the presence of a coherent field of force the stability of such objects is markedly affected as proved by the increase of the kink/antikink density n(T) in eq. (19). Another interesting application of our LE approach is the design of Josephson junction transmission lines. Since the socalled fluxon oscillators [2,9] are depicted as circuits where a periodic flow of flux-
32
24 March 1986
References [1] A.R. Bishop, in: Solitons in action, eds. K. Lonngren and A.C. Scott (Academic Press, New York, 1978). [2] D.W. McLaughlin and A.C. Scott, Phys. Rev. A18 (1978) 1652. [3] J.F. Currie, J.A. Krumhansl, A.R. Bishop and S.E. Trullinger, Phys. Rev. B22 (1980) 477. [4] M. BlRtiker and R. Landauer, Phys. Rev. A23 (1981) 1397. [5] W. Wonneberger, Physica 103A (1980) 543. [6] H. Risken, Springer series in synergeties, VoL 18. The Fokker-Planck equation (Springer, Berlin, 1983). [7] W.B. Davenport and W.L. Root, Random signals and noise (McGraw-Hill, New York, 1958). [8] P. Pasqualand L. V~zquez, Phys. Rev. B32 (1985) 8532; F. If, M.P. Soerensen and P.L. Christiansen, Phys. Lott. 100A (1984) 68. [9] M. Salerno, Phys. Lett. 87A (1981) 116; M. Salerno and A.C. Scott, Phys. Rev. B26 (1982) 2474. [10] F. Maxchesoni, AppL Phys. Lett. (1986), to be pubfised. [11 ] Joergonsen et al., Phys. Rev. Lett. 49 (1982) 1093.