Multistable Spatiotemporal Dynamics in the Driven Frenkel Kontorova Lattice

Transcription

1 Commun. Theor. Phys. (Beijing, China) 36 (2001) pp c International Academic Publishers Vol. 36, No. 1, July 15, 2001 Multistable Spatiotemporal Dynamics in the Driven Frenkel Kontorova Lattice ZHENG Zhi-Gang Department of Physics, Beijing Normal University, Beijing , China (Received August 4, 2000; Revised November 22, 2000) Abstract Spatiotemporal dynamics of the damped dc-driven Frenkel Kontorova lattice is studied. Multistable topologies are shown. Intermittency of the dynamical contraction factor is found, and this behavior is a consequence of the collisions of kinks and antikinks. Fast kinks and antikinks are unstable. The transition from the localized kink to the whirling mode is found to be a temporal bifurcation cascade of generations of kink-antikink pairs and the collision-induced avalanche dynamics. Noise-induced topology transition is observed and discussed. PACS numbers: a, x Key words: Frenkel Kontorova model, kink-antikink pair, dynamical contraction factor 1 Introduction Much effort in recent years has been focused on collective behaviors and spatiotemporal dynamics of coupled nonlinear systems, which are closely related to a variety of real situations in physics, biology, chemistry, and engineering. [1 3] The significance of these studies lies that coupled systems describe many cooperative phenomena and enable one to understand essential features of complicated spatiotemporal behaviors. Among all these studies, the competition between spatial scales and time scales (so-called wavevector competition) plays a crucial role. The Frenkel Kontorova (FK) model, which describes an array of N single pendula interacting with the nearestneighboring coupling, may be one of the simplest capable of capturing the essential features of the competitive interactions. [4,5] In conservative case and normalized form, the Hamiltonian of the model can be written as N [ 1 H = 2ẋ2 i + V (x i ) + U(x i+1 x i )], (1) i=1 where x i, ẋ i = dx i /dt are the coordinate and the corresponding velocity of the i-th oscillator, V (x i ) is the potential energy of the i-th oscillator, and U(x i+1 x i ) denotes the interaction energy between the i-th and the (i + 1)-th oscillators. For a standard FK model, the potential energy is considered to be of the following simplest form V (x) = d [1 cos x], U(x) = 1 2 K(x α)2. (2) Here d, K, α are the height of the potential, the coupling strength and the spring constant, respectively. δ = α/2π is the frustration between the periodic potential and the spring constant. Equally the FK model can also be considered as a harmonic chain subject to a periodic substrate potential field. Despite its simplicity, this model has been fully investigated in relating to a number of problems, such as charge-density waves, [6] nano-tribology and surface problems, [7] and Josephson-junction arrays and ladders. [8] The ground state of the FK model has been exhaustively explored over the past few decades, [9] and Aubry gave a beautiful picture of the correspondence between the commensurate-incommensurate (CI) phase transition and the breaking of the last KAM torus in the standard map. [10] Dynamics of the FK model under various external influences has also been extensively studied recently. In the damped case and under the drive of a dc force, Watanabe et al. [8] found resonant steps of the voltage-current characteristics in relating to dynamics of Josephson-junction arrays. They interpreted these resonant steps by using the discrete sine-gordon model (also the FK lattice). Zheng et al. [11] gave a mean-field solution of resonant steps and studied the spatiotemporal dynamics of the dc-driven FK systems. For a single pendulum, in the damped case, bistability is a key behavior for moderate dc forces. This bistability results in complicated spatiotemporal dynamics for the FK lattice. This has been explored by many groups. [8,11 13] In spite of a number of studies of spatiotemporal dynamics of the FK system, a topological description of the array is still lacking. It is not clear about the configurational dynamics of the chain varying with external parameters and the manifestation of the topology during dynamical transitions. Studies of these problems may shed light on collective behaviors of coupled systems with spatiotemporal competitions. The configurational behaviors of the FK model have been studied in relating to the groundstate problem and the CI phase transitions. [9,10] As for the topology of the dynamical array under various influences, no detailed studies have been found till now. [14] It is our task in this paper to explore the multistabilityinduced spatiotemporal topological dynamics. Due to the presence of multistability, not only the collective motion exhibits rich features, the topology of the lattice also possesses various structures. We numerically show the The project supported by National Natural Science Foundation of China ( ), the Special Funds for Major State Basic Research Projects (G ), and the Foundation for University Key Teachers by the Ministry of Education of China

2 38 ZHENG Zhi-Gang Vol. 36 existence of antikinks. The antikink moves in opposite direction to the kink, which leads to the collision of kinks and antikinks. We find that this microscopic dynamics leads to an intermittent behavior of the macroscopic parameter, the dynamical contraction factor. For a large force, the localized kink changes into an extended kink (whirling mode) by a cascade of metastable instabilities. We also study the noise effect and a noise-induced topology transition, i.e., the kink kink-antikink transition is found. 2 The Damped dc-driven FK Lattice In the damped case, when the FK array is driven by a constant external force and subject to thermal noises, in dimensionless form, the equation of motion can be written by setting d = 1 as ẋ i = v i, v i = γv i sinx i + K(x i+1 2x i + x i 1 ) + F + ξ i (t), (3) where i = 1,2,...,N, γ is the damping coefficient, and F the constant external force. The thermal noise ξ i (t) on the i-th site is usually assumed to be noncorrelated between time and lattice, [15] ξ i (t) = 0, ξ i (t + τ)ξ j (t) = 2γDδ i,j δ(τ). (4) Note that the frustration δ does not appear in Eq. (3), but this parameter may play a very significant role. Under a dc drive F, the array may possess a nonzero current defined by v = 1 N 1 T lim v j (t)dt. (5) N T T 0 j=1 For system (3), several regimes are distinguished in the absence of thermal noises. Figure 1a gives a qualitative diagram. When F < F c1, the lattice is pinned. A small force cannot overcome the Pierels Nabarro barrier. [16] The configuration of the array is a localized kink (LK). We call this regime the pinned LK regime. When F c1 < F < F c2, the localized kink can move along the lattice in a stick-slip motion. In this moving LK regime, due to the discreteness and the competition between the coupling and the periodic potential, the array oscillates during its drift on the substrate, leading to radiation of phonon waves. Under certain conditions, the kink motion and radiated phonon waves may become phase-locked, forming quantized velocities. In the interval F c2 < F < F c3, the moving LK may coexist with the moving kink-antikink pairs (KAPs). Little was studied previously due to the complexity in this regime. This is the focus of the present paper. When F > F c3, the localized kink becomes extended, the motion of the array is uniform. We call this the moving extended kink (EK) regime. It is an interesting issue to study in what a way the LK loses its stability. This will also be addressed. Fig. 1 (a) A diagram for different regimes of motions and topologies; (b) The evolution of the average kinetic energy ε p(t) with a random initial condition when F = 0, ε p(t) exp( γt) as t ; (c) The evolution of the DCF β(t). In the long-term limit, the system evolves to the ground state, thus β(t) β MF.

3 No. 1 Multistable Spatiotemporal Dynamics in the Driven Frenkel Kontorova Lattice 39 To describe the topological behavior of the array, the following contraction factor can be introduced β(t) = 1 N cos x j (t), (6) N j=1 and a time average of β(t) can also be defined 1 β(t) = lim T T T 0 β(t)dt. (7) It is interesting that the formula of β(t) is much similar to the definition of the mean-field contraction factor β MF = 1 N j [cos x j ], which was introduced to describe the topological effect of the ground state, [11] where {x j } represents the ground-state configuration of the FK model. Here β(t) depicts the dynamical and macroscopic features of the topology of the lattice. Therefore we call β(t) the dynamical contraction factor (DCF). Usually the value of β lies between 0 and 1. The larger β is, the stronger the localization of the kink is. When δ 0, β 1 corresponding to a very strong localization of the kink (2π-kink). For very small F, one has β(t) β MF. When F becomes very large, the array moves uniformly with the velocity v = F/γ along the periodic potential. One may approximately write the steady solution as x j (t) = ψ + αj + F γ t, where ψ is an arbitrary constant phase. By inserting it into the definition of DCF, we may easily get β(t) = Re{ 1 N N j=1 exp[ix j(t)]} = 0. In the present paper we adopt γ = 0.1 (the underdamped case) and moderate couplings (K = 1.0, 2.0). Periodic boundary conditions are considered, i.e., x j+n (t) = x j (t) + 2πM, where M is the number of geometric kinks (twists) trapped in the array. Thus the frustration δ = M/N and the static length of the harmonic chain should be α = 2πδ. From below we discuss the commensurate case by setting N = 100, M = 1, then we have δ = 0.01, a = 0.02π. For open boundary conditions and other cases of frustrations, the conclusions addressed in this work do not qualitatively change. 3 Antikinks and Macroscopic Intermittency In the absence of the dc drive force, the lattice is pinned, i.e., no macroscopic directed motion can be observed. In Fig. 1b, we give the evolution of the kinetic energy ε p (t) = 1 N N i=1 (v2 i /2) with a random initial condition when F = 0. Obviously ε p (t) decays to zero due to the damping term in Eqs (3), and ε p (t) exp( γt) as t. The evolution of the DCF β(t) is shown in Fig. 1c. In the long-term limit, the system evolves to the ground state, thus β(t) β MF. Under the drive of a moderate force F c2 < F < F c3, the spatiotemporal patterns of the underdamped system strongly depend on initial conditions, or more precisely, on initial fluctuations. ε p approximately gives the fluctuation of the array. If we add a dc force F = 0.3 from the moment labeled in Fig. 1b, i.e., from different levels of fluctuations, one may find different topological dynamics, as shown in Figs 2. In all patterns points are plotted whenever x i (t) passes 2nπ with n an integer. In Fig. 2a, the dc force is applied from t 0 = 150 (the undriven system has almost evolved to the ground state), the localized kink is depinned. The motion starts first from the tail of the kink (Most particles lie in 0 and 2π wells, those a few particles out of these wells are called the tail of the kink), i.e., the particle at the tail of the kink first loses its stability (Note dots propagating opposite to the lattice). The snapshots of the moving kink at different moments are shown in Fig. 3a. The oscillation at the tail of the kink can be seen. This oscillatory tail is the excited phonon superimposed on the lattice oscillation. If one applies the dc force at earlier time, e.g., at t 0 = 20, the strong initial fluctuation leads to a different spatiotemporal pattern, as shown in Fig. 2b. One can observe two kinks propagating opposite to the lattice and a new bundle moving along the lattice. This new bundle is an antikink. Thus a kink+kap solution [labeled as the 1-KAP solution] forms. This is shown in Fig. 3b, where the profiles of the lattice at different moments are shown. All the three profiles possess a kink-like in Fig. 3a and a hump or anti-hump, where these humps are composed of a kink and an antikink. For stronger fluctuations, more antikinks can be excited. These are shown in Figs 2c and 2d for t 0 = 10 and 5, corresponding to the 2-KAP solution and 3-KAP solution, which can be identified from the profiles in Figs 3c and 3d. It can be easily found that due to the property of the solution, the numbers of kinks and antikinks have a simple relation N k N a = M, (8) where N k,a denote numbers of kinks and antikinks, and M the number of twists (geometric kinks) trapped in the array. All the solutions we found above obey this relation (M = 1). We have also tested cases of M 1 and found the validity of this relation. It is not quite clear whether there is solution breaking this conservation relation. Now let us resort to the investigation of the macroscopic behavior of the array. This can be well described by the DCF β(t). In Figs 4, the evolutions of the DCF corresponding to Figs 2 are given. In Fig. 4a, the DCF exhibits a small-amplitude periodic oscillation. This oscillation is the result of the generation of phonon waves. Figure 4a shows this laminar feature. Intermittent motion can be observed in Fig. 4b, for a large fluctuation of initial conditions, where pulses are superimposed on the laminar background. This macroscopic intermittency occurs more frequently in Figs 4c and 4d, and an irregular macroscopic behavior can be observed in Fig. 4d. This phenomenon is much similar to the intermittency route to turbulence, where the laminar phase is destroyed gradually with increasing fluctuations. [17] This turbulence takes place on a macroscopic scale. By contrasting Figs 4 and the spatiotemporal patterns in Figs 2, one may find that the macroscopic intermittency originates from the

4 40 ZHENG Zhi-Gang Vol. 36 emergence of antikinks. Due to the opposite motion of kinks and antikinks, they may collide when they meet. Due to the discreteness of the lattice, the collision may excite additional phonons, which result in a macroscopic intermittency of the system. For large fluctuations of initial conditions, more antikinks may be excited and this in turn leads to more drastic collisions and more frequent intermittencies on a large scale. This strong intermittency also leads to a topology change: the localized kink becomes gradually extended, i.e., β(t) decreases. For a large enough dc force, all modes of solutions we proposed above will lose their stability and the LK is fully extended. This results in a topological transition. Fig. 2 Spatiotemporal patterns of the system under a dc force F = 0.3 and initial conditions labeled in Fig. 1a. (a) t 0 = 150, (b) t 0 = 20, (c) t 0 = 10, and (d) t 0 = 5. For (b), (c) and (d), antikinks are observed. Fig. 3 Profiles of the position of particles corresponding to initial conditions in Figs 2a 2d at different moments. t = 125, 150, and 175 for squared, circled, crossed lines, respectively. (a) corresponds to a moving localized kink; (b) a kink plus a kink-antikink pair; (c) a kink plus two kink-antikink pairs; (d) a kink plus three kink-antikink pairs.

5 No. 1 Multistable Spatiotemporal Dynamics in the Driven Frenkel Kontorova Lattice 41 Fig. 4 Evolution of the DCF with initial conditions corresponding to Figs 2a 2d. Intermittency behaviors are observed in (b) (d), and these intermittencies correspond to collisions of kinks and antikinks in Figs 2a 2d. 4 KAP Instability and Topology Transitions 4.1 Instability of Fast Kinks and KAPs: Temporal Bifurcation As shown in the last section, the LK solution loses its stability when the initial fluctuation increases, then the 1-KAP solution, 2-KAP solution,..., N-KAP solution, and so on, all lose their stabilities when further increasing initial fluctuations. This is an intrinsic consequence of the parametric instability of fast kink and KAPs. This means a kink or KAP with high propagating velocity is unstable. [18] A KAP solution with more KAPs needs a stronger excitation. The EK (extended kink) solution is unstable in the regime F (F c2,f c3 ). Only when F > F c3 will all KAP solutions become unstable. In Fig. 5a, we give the spatiotemporal pattern for F = 0.7. The system evolves from the ground state, thus one may easily observe the instability process of the LK. This pattern indicates a cascade of temporal bifurcations. When t < t 1, as labeled in Fig. 5a, the LK starts to move, and this forms the LK solution. As the velocity increases with time, the LK solution loses its stability at t t 1, replaced by a 1-KAP solution. Consequently for t t 2 the 1-KAP solution is replaced by the 2-KAP solution. Observable bifurcation points are labeled in Fig. 5a from t 1 to t 5. Moreover, collisions of generated antikinks and kinks lead to an avalanche-like extension of the LK solution. This extension process is measured by β(t) in Fig. 5b. The DCF experiences strong oscillation during the first several bifurcations and quickly decreases to zero. It is interesting to study the details of this transition. In Fig. 5c we plot the evolutions of the configurations of the array. Each line represents a profile (i.e., snapshot of the position of each particle) of the lattice at some time. We draw lines with a time interval t = 5. It can be found that the motion starts from the tail of the kink. The particles around the tail gradually melt and start moving. Most of the particles in the array, however, are pinned in potential wells and remain unmoved. We call this a traffic jam. One may define a quantity to study the fluctuation of the array during the motion, (t) = max{x i (t),i = 1,...,N} min{x i (t),i = 1,2,...,n}. (9) This quantity measures the degree of some particles deviating from the localized kink. The larger is, the higher the degree of the traffic jam is. In Fig. 5d the evolution of (t) is given corresponding to parameters in Fig. 5a. Obviously, there are two processes, which are separated by a crossover at t 60. This value corresponds to the global propagation of the KAP throughout the chain, i.e., a structural crisis occurs. Before this crossover, (t) increases with time, indicating that some particles move and the others are jammed, called a fusion or de-nucleation process, i.e., the kink and KAPs are generated and propagate to all particles in the array. This just corresponds to the aforementioned bifurcation cascade from LK to KAPs, as shown in Fig. 5a. The second stage comes when t > 60, the structural crisis leads to a global propagation of the kink and KAPs, and (t) reaches its maximum and decreases to 2π, which implies that the fluctuation in the array is gradually eliminated. In the long-term limit, the

6 42 ZHENG Zhi-Gang Vol. 36 profile of the array relaxes to the EK, x j (t) = ψ+αj+ F γ t. Fig. 5 (a) Spatiotemporal dynamics under the dc drive F = 0.7. The initial condition is the ground state. Cascade of temporal bifurcations from kink to KAPs and structural crisis are observed. (b) The evolution of the DCF corresponding to (a). (c) The evolution of the configuration profile. Traffic jam can be observed. (d) The evolution of (t). 4.2 Noise-Induced Topology Transition Finally, we discuss the role of thermal noise on the topology dynamics. Effect of noises on collective transport has been fully explored in recent years. Noise also plays an important role in changing topologies of the system. Fig. 6 (a) The average DCF varying against the noise intensity D for K = 1.0 and 2.0. Drastic changes can be found at D c = 1.0 and 1.6 for K = 1.0 and 2.0, respectively. (b) The average velocity of the array varies with the noise intensity for K = 1.0. (c) and (d) The spatiotemporal patterns for different noise intensities.

7 No. 1 Multistable Spatiotemporal Dynamics in the Driven Frenkel Kontorova Lattice 43 In Fig. 6a, we plot the average DCF varying against the noise intensity D for K = 1.0 and 2.0. All motions start from the ground state. Drastic changes can be found at D c = 1.0 and 1.6 for K = 1.0 and 2.0, respectively, where β decreases sharply to a very low (but nonzero) value. This implies a qualitative change in the topology of the system. By checking the average velocity of the array, it is found that at D c, v jumps to a high value. We give an example in Fig. 6b for the case K = 1.0. In Figs 6c and 6d the spatiotemporal patterns for different noise intensities are given. For D = 0, the topology of the array is a localized kink, as shown in Fig. 6c. For small noise D < D c, the localized kink still exists, while there is a disordered-excitation background, induced by thermal noises. These thermal excitations lead to a slight linear decrease of the average DCF, as seen in Fig. 6a. Above the threshold, antikinks are excited, leading to frequent kinkantikink collisions, which form a different solution from that for D < D c. Therefore β jumps to a small value. Theoretical analysis of this transition is still an open and challenging problem. 5 Conclusion To summarize, in this paper we dealt with the topology dynamics of the FK array under the external dc force. Multistable spatiotemporal dynamics with the same parameters are found starting from different initial conditions. Antikinks are excited for large initial fluctuations. KAP solutions are numerically found. We also introduce a dynamical contraction factor (DCF) to describe the macroscopic feature of the array. Macroscopic intermittency is found for its evolutions, and whose microscopic mechanism is identified as the kink-antikink collisions. It is shown that fast kinks and KAPs are unstable. This results in a total instability of all LK and KAP solutions as F > F c3, replaced by the EK. The transition from an LK to an EK (whirling mode) is considered as two processes. The first is a cascade of bifurcations from LK to KAPs. Traffic jam is shown in this regime. The process ends with a structural crisis, corresponding to a global motion of the array. The second process is a relaxation to the EK, implying that the fluctuations in the array are gradually eliminated. Noise effect on the stability of the LK is studied, and a noise-induced LK-KAP transition is found. Many questions still remain open in the present work. The first problem is a theoretical identification of the existence and stability of KAP solutions. It is difficult but of theoretical importance to prove this problem. The second issue relates to a thorough analysis of the noise-induced phase transition. Topological dynamics of the FK model under various influences is currently under detailed study. References [1] S.H. Strogatz, Nonlinear Dynamics and Chaos, Addison- Wesley, Reading, MV (1994). [2] A.T. Winfree, The Geometry of Biological Time, Springer, Berlin (1980). [3] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer, Berlin (1984). [4] J. Frenkel and T. Kontorova, Phys. Z. Sowjet. 13 (1938) 1. [5] L. Floria and J. Mazo, Adv. Phys. 45 (1996) 505. [6] D.S. Fisher, Phys. Rev. Lett. 50 (1983) 1486; Phys. Rev. B31 (1985) 1396; L. Sneddon and K. Cox, Phys. Rev. Lett. 58 (1987) 1903; L. Sneddon and S. Liu, Phys. Rev. B43 (1991) 5798; S. Coppersmith, Phys. Rev. B30 (1984) 410; S. Coppersmith and D. Fisher, Phys. Rev. A38 (1988) [7] B.N.J. Persson and E. Tosatti, Physics of Sliding Friction, Kluwer Academic Publishers, Netherlands (1996); Y. Braiman, F. Family and H.G.E. Hentschel, Phys. Rev. E53 (1996) R3005. [8] S. Watanabe, H. Zant, S. Strogatz and T. Orlando, Physica D97 (1996) 429; A. Ustinov, M. Cirillo and B. Malomed, Phys. Rev. B47 (1993) 8357; ibid. 51 (1995) 3081; L. Floria and F. Falo, Phys. Rev. Lett. 68 (1992) 2713; B.R. Trees and N. Hussain, Phys. Rev. E61 (2000) [9] W. Selke, Phys. Rep. 170 (1988) 213. [10] S. Aubry, Phys. Rep. 103 (1984) 12; M. Peyrard and S. Aubry, J. Phys. C16 (1983) [11] Z. ZHENG, B. HU and G. HU, Phys. Rev. E57 (1998) 1139; Phys. Rev. B58 (1998) 5453; Commun. Theor. Phys. (Beijing, China) 33 (2000) 191. [12] T. Strunz and F.J. Elmer, Phys. Rev. E58 (1998) 1601; ibid. E58 (1998) [13] O.M. Braun, A.R. Bishop and J. Roder, Phys. Rev. Lett. 79 (1997) [14] Z. ZHENG and B. HU, Phys. Rev. E62 (2000) [15] Z. ZHENG, B. HU and G. HU, Phys. Rev. E58 (1998) [16] M. Peyrard and M. Kruskal, Physica D14 (1984) 88; J. Currie, S. Trullinger, A. Bishop and J. Krumhansi, Phys. Rev. B15 (1977) 5567; R. Boesch, C.R. Willis and M. El-Batanouny, Phys. Rev. B40 (1989) [17] E. Ott, Chaos in Dynamical Systems, Cambridge Univ. Press, New York (1993). [18] L. Consoli, H. Knops and A. Fasolino, Phys. Rev. Lett. 85 (2000) 302.