Pinning and depinning of wave fronts are ubiquitous features of spatially
discrete systems describing a host of phenomena in physics, biology, etc. A
large class of discrete systems is described by overdamped chains of nonlinear
oscillators with nearest-neighbor coupling and subject to random external
forces. The presence of weak randomness shrinks the pinning interval and it
changes the critical exponent of the wave front depinning transition from 1/2
to 3/2. This effect is derived by means of a recent asymptotic theory of the
depinning transition, extended to discrete drift-diffusion models of transport
in semiconductor superlattices and confirmed by numerical calculations.Comment: 4 pages, 3 figures, to appear as a Rapid Commun. in Phys. Rev. 

A. Carpio

A. Luzón

A.A. Middleton

A.E. Bugrim

D.S. Fisher

G. Grüner

H.S.J. van der Zant

J. Frenkel

J.P. Keener

L. L. Bonilla

L.L. Bonilla

M. Löcher

O.M. Braun

English

arXiv

Crossref

Effects of disorder on the wave front depinning transition in spatially discrete systems

Pinning and depinning of wave fronts are ubiquitous features of spatially discrete systems describing a host of phenomena in physics, biology, etc. A large class of discrete systems is described by overdamped chains of nonlinear oscillators with nearest- neighbor coupling and subject to random external forces. The presence of weak randomness shrinks the pinning interval and it changes the critical exponent of the wave front depinning transition from 1/2 to 3/2. This effect is derived by means of a recent asymptotic theory of the depinning transition, extended to discrete drift-diffusion models of transport in semiconductor superlattices and is confirmed by numerical calculations.Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEpu

Carpio Rodríguez, Ana María

Bonilla, Luis L.

Luzón, Ana

Docta Complutense

arXiv:cond-mat/0201318v1  [cond-mat.stat-mech]  17 Jan 2002Effects of disorder on the wave front depinning transition in spatially discrete systemsA. Carpio1, L. L. Bonilla2 and A. Luzón31Departamento de Matemática Aplicada, Universidad Complutense de Madrid,28040 Madrid, Spain2Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid,Avenida de la Universidad 30, 28911 Leganés, Spain3Escuela Técnica Superior de Ingenieros de Montes, Universidad Politécnica de Madrid,28040 Madrid, Spain(7 December 2001)Pinning and depinning of wave fronts are ubiquitous features of spatially discrete systems de-scribing a host of phenomena in physics, biology, etc. A large class of discrete systems is describedby overdamped chains of nonlinear oscillators with nearest-neighbor coupling and subject to randomexternal forces. The presence of weak randomness shrinks the pinning interval and it changes thecritical exponent of the wave front depinning transition from 1/2 to 3/2. This effect is derived bymeans of a recent asymptotic theory of the depinning transition, extended to discrete drift-diffusionmodels of transport in semiconductor superlattices and confirmed by numerical calculations.05.45.-a; 82.40.Bj; 45.05.+xPhenomena in many different fields may be describedby means of spatially discrete systems: motion of dislo-cations in crystals [1], atoms adsorbed on a periodic sub-strate [2], arrays of coupled diode resonators [3], weaklycoupled semiconductor superlattices (SL) [4,5], sliding ofcharge density waves (CDW) [6], superconductor Joseph-son array junctions [7], propagation of nerve impulsesalong myelinated fibers [8,9], pulse propagation throughcardiac cells [9], calcium release waves in living cells [10],etc. In many of these systems, disorder due to differencesin the parameters of individual elements is important be-cause it has a strong impact in the collective behavior. Adistinctive example of collective behavior in discrete sys-tems (not shared by continuous ones) is the phenomenonof wave front pinning: for values of a control parame-ter in a certain interval, wave fronts joining two differentconstant states fail to propagate [9]. When the controlparameter surpasses a threshold, the wave front depinsand starts moving [8]. The existence of such thresholdsis an intrinsically discrete fact, which is lost in contin-uum aproximations. Recently, a theory of front depin-ning and motion near threshold has been proposed bysome of us for one-dimensional (1D) nonlinear spatiallydiscrete reaction-diffusion (RD) systems [11]. In our the-ory, propagation failure and front depinning are charac-terized by studying the behavior of a few sites, providedthe effects of spatial discretization are sufficiently strong[5,11].In this paper, we consider the effect of weak disorderon the wave front depinning transition in spatially dis-crete 1D systems. Applications will include discrete RDsystems subject to a random field, sliding CDW and do-main motion in SL. In these examples, the main effect ofdisorder is to soften the transition, changing the criticalexponent from 1/2 to 3/2. The latter value was obtainedby D. Fisher in a mean field model of sliding CDW usingscaling arguments [12].We consider chains of diffusively coupled overdampedoscillators in a potential V , subject to a random forcefield F + γξn:dundt= un+1 − 2un + un−1 + F − Ag(un) + γξn. (1)Here g(u) = V ′(u) presents a ‘cubic’ nonlinearity, suchthat Ag(u) − F has three zeros, U1(F/A) < U2(F/A) <U3(F/A) in a certain force interval (g′(Ui(F/A)) > 0 fori = 1, 3, g′(U2(F/A)) < 0). The fluctuating part of theforce field is γξn, where γ ≥ 0 characterizes the disorderstrength and ξn is a zero mean random variable takingvalues on an interval (−1, 1) with equal probability. Anexample of a model described by Eq. (1) is (except forthe mean field approximation which we do not make)D. Fisher’s modification of the Fukuyama-Lee model ofsliding CDW [12]. In it, un = θn − χn, g(u) = sin u,γξn = χn+1 − 2χn + χn−1, where θn is the CDW phaseat the site n and χn is a random variable taking valueswith equal probability on (0, 2π).Provided g(u) is odd with respect to U2(0) and γ = 0,there is a symmetric interval |F | ≤ Fc where the wavefronts joining the stable zeros U1(F/A) and U3(F/A)are pinned. For |F | > Fc, there are smooth travelingwave fronts, un(t) = u(n − ct), with u(−∞) = U1 andu(∞) = U3. The velocity c(A, F ) depends on A and Fand it satisfies cF < 0 and |c| ∝ (|F | −Fc)12 as |F | → Fc[11]. Examples are the overdamped Frenkel-Kontorova(FK) model (g = sinu) [13] and the quartic double wellpotential (V = (u2 − 1)2/4). Less symmetric nonlin-earities yield a non-symmetric pinning interval and ouranalysis of the depinning transition applies to them withtrivial modifications [5].Let us recall the main features of the active point the-ory of the wave front depinning transition in the absenceof disorder [11,5]. Except when A is too small (the con-tinuum limit in which Fc → 0), the stable wave front1profile differs appreciably from either U1 or U3 at finitelymany points, un, n = −L, . . . ,−1, 0, 1, . . . , M , called theactive points. At n < −L, un ≈ U1(F/A) and at n > M ,un ≈ U3(F/A). We shall reconstruct the wave front pro-file u(n−ct) by analyzing the behavior of the active pointsun(t), n = −L, . . . ,−1, 0, 1, . . . , M , as the front moves forF > Fc > 0 (the case F < 0 can be obtained by usingsymmetry considerations). The arbitrary phase of the(translation invariant) wave front will be fixed by impos-ing that the solution of the system of active points attime t = 0 (and F slightly larger than Fc) be equal to itsstationary solution at F = Fc, un(A, Fc), n = −L, . . . , Mup to terms of order (F − Fc). Fc is obtained from thecondition that the matrix of the coefficients in the systemof active points linearized about the stationary solutionhas a zero eigenvalue. Provided Vn (V2−L + . . . + V2M =1) is the corresponding eigenvector, an outer approxi-mation to the solution un(t) is un(t) ∼ un(A, Fc) +ϕ(t)Vn, where the amplitude ϕ obeys the equationdϕ/dt = α(F − Fc) + βϕ2, in which α =∑Mi=−L Vi +A−1 [V−L/g′(U1(Fc/A)) + VM/g′(U3(Fc/A))] > 0, β =−(A/2)∑Mi=−L g′′(ui(A, Fc))V 3i > 0. The solutionof this equation such that ϕ(0) = 0 [equivalent toun(0) = un(A, Fc) for n = −L, . . . , M ] is ϕ = [α(F −Fc)/β]12 tan(√αβ(F − Fc)t). The amplitude ϕ blows upat times t = ±tb, tb = π/(2√αβ(F − Fc)). The inversewidth of this time interval yields an approximation forthe wave front velocity, |c| ∼√αβ(F − Fc)/π. At theblow up times, the previous outer approximation to un(t)has to be matched to an appropriate inner solution. Atthe later blow up time tb, the appropriate inner solutionis the solution of the active point system at F = Fc withthe boundary conditions that un = un(A, Fc) as t → −∞and un = un+1(A, Fc) as t → ∞. At the earlier blow uptime −tb, the inner solution obeys the same system ofequations at F = Fc, but the boundary conditions areun = un−1(A, Fc) as t → −∞ and un = un(A, Fc) ast → ∞ [14].Effects of disorder. How does weak disorder modifythis picture of the wave front depinning transition? Ourmain idea is to find a dominant balance of the disor-der effects with nonlinearities and (F − Fc) near the de-pinning transition. Given our active point constructionsketched above, the dominant balance is struck provided(F − Fc) = O(γ) as γ → 0. The amplitude Equationbecomesdϕdt= α(F − Fc) + γM∑n=−LVnξn + βϕ2, (2)and the matching condition is the same as before. Thesolution of Eq. (2) blows up at the end of time intervalsof duration 1/|cR|, where|cR| =1π√√√√αβ(F − Fc) + γβM∑n=−LVR+nξR+n, (3)provided the argument of the square root is positive.Otherwise the motion of the wave front stops and it be-comes pinned. Notice that we have chosen now uR(t) asthe central active point that was distinguished with thesubscript zero in our previous formulas. After the blowup, uR jumps to uR+1(A, Fc), approximately, and it re-mains there until a time 1/|cR+1| has elapsed. Then itjumps to uR+2(A, Fc) approximately, and so on.The magnitude of interest in these systems is usuallyan average velocity |cR| over sufficiently many points.For example, this magnitude is proportional to the cur-rent due to a sliding CDW and it is important to knowits behavior near the depinning field and the magnitudethereof. We shall argue that the average velocity is ap-proximately given by the following equation:|c| ≡ 1NN∑R=1|cR| = 〈|cR(ξ)|〉, (4)〈|cR(ξ)|〉 ≡12π∫ 1−1{αβ(F − Fc) + γβσξ}12+ dξ. (5)Here N ≫ (L + M + 1) is sufficiently large, σ = 1, and{x}12+ is√x if x > 0 and zero otherwise. The idea ofthe proof is as follows. Let us assume that A is so largethat L = M = 0 and there is only one active point.Then the central limit theorem applied to Eq. (3) withVR = 1 yields Eqs. (4) - (5). Let us assume now thatthere are two active points. The previous argument failsbecause now the sum in Eq. (3) comprises two terms in-stead of one. Then the terms in the arithmetic mean areno longer independent: when uR = u1 for instance, Eq.(3) contains ξ1 and ξ2. After the blow up time, uR = u2and Eq. (3) contains ξ2 and ξ3, etc. However, we cangroup the sums appearing in the arithmetic average ofEq. (4) in two groups containing only independent ran-dom variables: R = 2r−1 and R = 2r, with r = 1, 2, . . ..The variable V1ξ1 + V2ξ2 has zero mean and correlation2σ2/3, where σ2 = V 21 + V22 = 1. This correlation isexactly the same as that of the variable ξ1. Then thecentral limit theorem applied to each group (of sums of‘dimer’ random variables) gives one half the integral inEq. (5), and the sum of these two halves yields Eq. (4). Ifwe have more active points, we just have to subdivide thearithmetic mean in as many subgroups as active pointsand use the previous argument to prove Eq. (4).The elementary integral in Eq. (5) yields c = 0 ifF < Fc − γσ/α (recall that σ = 1),|c| =√βσ3πγ{|ασ (F − Fc) + γ|32 , if |F − Fc| ≤ γσα ,|ασ (F − Fc) + γ|32 − |ασ (F − Fc) − γ|32(6)if (F − Fc) > γσ/α. Clearly we have a new critical fieldF ∗c = Fc−γσ/α, and a new critical exponent 3/2 (insteadof 1/2 for the case without disorder), |c| ∝ (F − F ∗c )3/2.2We have compared our theory to the direct numerical so-lution of Eq. (1) in Fig. 1, obtaining an excellent agree-ment of theoretical predictions and numerical simulation.Similarly, we can analyze the effect of weak disorder inthe doping of the wells on the motion of wave fronts in dccurrent biased semiconductor SL. If the total current den-sity is close to a pinning value, the displacement currentis almost zero except at certain times during wave frontmotion at which the wave front jumps from one well tothe adjacent one. The average velocity given by Eq. (4) isproportional to the arithmetic mean of time averages ofthe displacement current over the time interval betweenmaxima thereof. The average velocity is basically themean velocity at which the wave front traverses N wells.To calculate it, we take advantage of our theory of wavefront motion in current biased SL [5]. The equationswe use are those in Ref. [5] except that the dimensionlessPoisson equation is now Ei−Ei−1 = ν(ni−1−γξi) whereγ and ξ defined as in Eq. (1) represent the disorder in welldoping. In the SL equations, the roles of the force F andthe parameter A are taken by the total current density Jand the dimensionless doping ν. If γ = 0 and ν surpassesa certain minimal value, there are two critical values ofthe current, J1 and J2, such that a wave front is pinnedif J1 ≤ J ≤ J2 and it moves at a constant velocity c(J, ν)otherwise. The velocity c is positive if J < J1 and neg-ative if J > J2. Furthermore, near the critical currents,|c| ∝ |J − Jc|1/2 (Jc is either J1 or J2) [5]. How does thedisorder correct this picture? The effect of disorder is toadd a term γD(Ej)ξj+1−γ[v(Ej)+D(Ej)]ξj to the totalcurrent density J in the dimensionless discrete Ampèreequation. Then the theory in Ref. [5] yields an equationsimilar to Eq. (3) for the front velocity:|cR| =1π√√√√αβ(J − Jc) − γβM∑n=−LU †R+nWR+n, (7)WR+n = (vR+n + DR+n) ξR+n − DR+nξR+n+1. (8)Here: (i) U †R+n is the left eigenvector corresponding tothe zero eigenvalue of the linearized equations aboutthe stationary solution at J = Jc (chosen so that∑Mn=−L U†R+nUR+n = 1; UR+n is the right eigenvector),(ii) we define vR+n = v(ER+n), etc., and (iii) α and βare given by Eqs. (20) and (21) in Ref. [5]. The val-ues ER+n are those of the stationary electric field profileat J = Jc. The noise term in Eq. (8) can be writtenas∑[U †R+n(vR+n +DR+n)−U†R+n−1DR+n−1]ξR+n, pro-vided we take U †R+n = 0 for n < −L and n > M + 1.The noise term has zero average and a correlation 2σ2/3,with σ2 =∑[U †R+n(vR+n + DR+n) − U†R+n−1DR+n−1]2.Using the previously mentioned argument of splitting thearithmetic mean in groups of independent random vari-ables, we can show that Eqs. (4) and (5) hold provided σin Eq. (5) is given by the previous formula and (F − Fc)is replaced by (J − Jc). Comparisons of the resultingaverage front velocity c with the results of numericallysolving the SL model are shown in Figures 2 and 3 forcurrents close to J1 and J2, respectively.In conclusion, we have shown that weak disorderchanges qualitatively the wave front depinning transitionin overdamped one-dimensional discrete models. Disor-der shrinks the pinning interval and it changes the criticalexponent for the velocity from 1/2 to 3/2. Whether thesefeatures are robust and hold for strong disorder remainsto be seen. An interesting indication is that the criti-cal exponent 3/2 is obtained independently of the noisestrength in mean field models of sliding CDW [12].This work was supported by the Spanish DGES grantPB98-0142-C04-01 and by the Third Regional ResearchProgram of the Autonomous Region of Madrid (StrategicGroups Action).[1] F.R.N. Nabarro, Theory of Crystal Dislocations (OxfordUniversity Press, Oxford, 1967).[2] P. M. Chaikin and T. C. Lubensky, Principles of con-densed matter physics (Cambridge University Press,Cambridge, 1995). Chapter 10.[3] M. Löcher, G.A. Johnson and E.R. Hunt, Phys. Rev.Lett. 77, 4698 (1996).[4] L.L. Bonilla, J. Galán, J.A. Cuesta, F.C. Mart́ınez and J.M. Molera, Phys. Rev. B 50, 8644 (1994); L. L. Bonilla,G. Platero and D. Sánchez, Phys. Rev. B 62, 2786 (2000);A. Wacker, in Theory and transport properties of semi-conductor nanostructures, edited by E. Schöll (Chapmanand Hall, New York, 1998). Chapter 10.[5] A. Carpio, L. L. Bonilla and G. Dell’Acqua, Phys. Rev.E 64, 036204 (2001).[6] G. Grüner, Rev. Mod. Phys. 60, 1129 (1988); A.A. Mid-dleton, Phys. Rev. Lett. 68, 670 (1992).[7] H. S. J. van der Zant, T. P. Orlando, S. Watanabe andS. H. Strogatz, Phys. Rev. Lett. 74, 174 (1995).[8] J. P. Keener, SIAM J. Appl. Math. 47, 556 (1987).[9] J.P. Keener and J. Sneyd, Mathematical Physiology(Springer, New York, 1998). Chapter 9.[10] A. E. Bugrim, A.M. Zhabotinsky and I.R. Epstein, Bio-phys. J. 73, 2897 (1997).[11] A. Carpio and L. L. Bonilla, Phys. Rev. Lett. 86, 6034(2001).[12] D. S. Fisher, Phys. Rev. Lett. 50, 1486 (1983); Phys.Rev. B 31, 1396 (1985).[13] J. Frenkel and T. Kontorova, Phys. Z. SowjUn. 13, 1(1938); O. M. Braun and Yu. S. Kivshar, Phys. Rep.306, 1 (1998).[14] A. Carpio and L. L. Bonilla, unpublished.36 6.05 6.1 6.15 6.2 6.25 6.300.050.10.150.20.250.30.35F|c(F)|(a)Numerical AveragePredicted without noisePredited with noise6.05 6.1 6.15 6.2 6.250.40.60.811.21.41.6Fln(|c(F)|)/ln(F)(b)FIG. 1. (a) Average velocity |c| as a function of F forA = 10, Fc = 6.102281 and γ = 0.1. (b) Graph of ln|c|/lnFshowing the crossover to the critical exponent 3/2.0.1786 0.1788 0.179 0.1792 0.1794 0.179601234567x 10-4Jc(J)RealizationsNumerical averagePredicted without noisePredicted with noiseFIG. 2. Dimensionless average wave front velocity for the9/4 SL with dimensionless parameters ν = 3, J1 = 0.179203,γ = 0.01.0.79 0.7902 0.7904 0.7906 0.7908 0.791−0.016−0.014−0.012−0.01−0.008−0.006−0.004−0.0020Jc(J)Realizations           Numerical average      Predicted without noisePredicted with noise   FIG. 3. Same as Fig. 2 with parameters ν = 40,J2 = 0.790203, γ = 0.01.4

Physical Review E

Effects of disorder on the wave front depinning transition in spatially
  discrete systems

Effects of disorder on the wave front depinning transition in spatially discrete systems

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