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Khoury M, Gleeson JP, Sancho JM, Lacasta AM, Lindenberg K. Diffusion coefficient in periodic and random potentials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:021123. [PMID: 19792093 DOI: 10.1103/physreve.80.021123] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/07/2009] [Indexed: 05/28/2023]
Abstract
Transport and diffusion of particles on modulated surfaces is a nonequilibrium problem which is receiving a great deal of attention due to its technological applications, but analytical calculations are scarce. In earlier work, we developed a perturbative approach to begin to provide an analytic platform for predictions about particle trajectories over such surfaces. In some temperature and forcing regimes, we successfully reproduced results for average particle velocities obtained from numerical simulations. In this paper, we extend the perturbation theory to the calculation of higher moments, in particular the diffusion tensor and the skewness. Numerical simulations are used to check the domain of validity of the perturbative approach.
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Orlandi JG, Sancho JM. Theoretical study of a membrane channel gated by ATP. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2009; 29:329-336. [PMID: 19575251 DOI: 10.1140/epje/i2009-10483-9] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/24/2009] [Revised: 05/27/2009] [Accepted: 06/02/2009] [Indexed: 05/28/2023]
Abstract
We study channel transport across biomembranes. We propose a model that couples the diffusive dynamics with the gating process via a two-state ratchet mechanism. This gating process is governed by ATP binding and hydrolysis, and the process exhibits Michaelis-Menten enzymatic kinetics. The particle flow and permeability of the channel are studied both analytically and numerically in the steady-state regime, while working between fixed concentrations. The results are compared with simpler models and with experimental data. Also, a simulation framework, that allows high flexibility in parameter exploration, is introduced.
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Gomez-Marin A, Sancho JM. Brownian pump powered by a white-noise flashing ratchet. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:031108. [PMID: 18517330 DOI: 10.1103/physreve.77.031108] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/06/2007] [Indexed: 05/26/2023]
Abstract
A Brownian pump of particles powered by a stochastic flashing ratchet mechanism is studied. The pumping device is embedded in a finite region and bounded by particle reservoirs. In the steady state, we exactly calculate the spatial density profile, the concentration ratio between both reservoirs and the particle flux. We propose a simulation framework for the consistent evaluation of such observable quantities.
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Gomez-Marin A, Garcia-Ojalvo J, Sancho JM. Self-sustained spatiotemporal oscillations induced by membrane-bulk coupling. PHYSICAL REVIEW LETTERS 2007; 98:168303. [PMID: 17501471 DOI: 10.1103/physrevlett.98.168303] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/05/2006] [Indexed: 05/15/2023]
Abstract
We propose a novel mechanism leading to spatiotemporal oscillations in extended systems that does not rely on local bulk instabilities. Instead, oscillations arise from the interaction of two subsystems of different spatial dimensionality. Specifically, we show that coupling a passive diffusive bulk of dimension d with an excitable membrane of dimension d-1 produces a self-sustained oscillatory behavior. An analytical explanation of the phenomenon is provided for d=1. Moreover, in-phase and antiphase synchronization of oscillations are found numerically in one and two dimensions. This novel dynamic instability could be used by biological systems such as cells, where the dynamics on the cellular membrane is necessarily different from that of the cytoplasmic bulk.
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Lindenberg K, Sancho JM, Lacasta AM, Sokolov IM. Dispersionless transport in a washboard potential. PHYSICAL REVIEW LETTERS 2007; 98:020602. [PMID: 17358592 DOI: 10.1103/physrevlett.98.020602] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/21/2006] [Indexed: 05/14/2023]
Abstract
We study and characterize a new dynamical regime of underdamped particles in a tilted washboard potential. We find that for small friction in a finite range of forces the particles move essentially nondispersively, that is, coherently, over long intervals of time. The associated distribution of the particle positions moves at an essentially constant velocity and is far from Gaussian-like. This new regime is complementary to, and entirely different from, well-known nonlinear response and large dispersion regimes observed for other values of the external force.
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Gomez-Marin A, Sancho JM. Tight coupling in thermal Brownian motors. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:062102. [PMID: 17280105 DOI: 10.1103/physreve.74.062102] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/18/2006] [Indexed: 05/13/2023]
Abstract
We study analytically a thermal Brownian motor model and calculate exactly the Onsager coefficients. We show how the reciprocity relation holds and that the determinant of the Onsager matrix vanishes. Such a condition implies that the device is built with tight coupling. This explains why Carnot's efficiency can be achieved in the limit of infinitely slow velocities. We also prove that the efficiency at maximum power has the maximum possible value, which corresponds to the Curzon-Alhborn bound. Finally, we discuss the model acting as a Brownian refrigerator.
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Gleeson JP, Sancho JM, Lacasta AM, Lindenberg K. Analytical approach to sorting in periodic and random potentials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:041102. [PMID: 16711782 DOI: 10.1103/physreve.73.041102] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/26/2005] [Revised: 03/14/2006] [Indexed: 05/09/2023]
Abstract
There has been a recent revolution in the ability to manipulate micrometer-sized objects on surfaces patterned by traps or obstacles of controllable configurations and shapes. One application of this technology is to separate particles driven across such a surface by an external force according to some particle characteristic such as size or index of refraction. The surface features cause the trajectories of particles driven across the surface to deviate from the direction of the force by an amount that depends on the particular characteristic, thus leading to sorting. While models of this behavior have provided a good understanding of these observations, the solutions have so far been primarily numerical. In this paper we provide analytic predictions for the dependence of the angle between the direction of motion and the external force on a number of model parameters for periodic as well as random surfaces. We test these predictions against exact numerical simulations.
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Gomez-Marin A, Sancho JM. Heat fluctuations in Brownian transducers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:045101. [PMID: 16711864 DOI: 10.1103/physreve.73.045101] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/04/2006] [Indexed: 05/09/2023]
Abstract
The heat fluctuation probability distribution function in Brownian transducers operating between two heat reservoirs is studied. We find, both analytically and numerically, that the recently proposed fluctuation theorem for heat exchange [C. Jarzynski and D. K. Wojcik, Phys. Rev. Lett. 92, 230602 (2004)] has to be applied carefully when the coupling mechanism between both baths is considered. We also conjecture how to extend such a relation when an external work is present.
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Lacasta AM, Sancho JM, Sagués F. Exported oscillator competition: a concept to analyze complex rhythms. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:016206. [PMID: 16486257 DOI: 10.1103/physreve.73.016206] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/23/2005] [Indexed: 05/06/2023]
Abstract
We study the interaction between two independent nonlinear oscillators competing through a neutral excitable element. The first oscillator, completely deterministic, acts as a normal pacemaker sending pulses to the neutral element which fires when it is excited by these pulses. The second oscillator, endowed with some randomness, though unable to make the excitable element to beat, leads to the occasional suppression of its firing. The missing beats or errors are registered and their statistics analyzed in terms of the noise intensity and the periods of both oscillators. This study is inspired in some complex rhythms such as a particular class of heart arrhythmia.
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Ciudad A, Lacasta AM, Sancho JM. Physical analysis of a processive molecular motor: the conventional kinesin. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:031918. [PMID: 16241493 DOI: 10.1103/physreve.72.031918] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2005] [Indexed: 05/05/2023]
Abstract
We present a model that allows for the derivation of the experimentally accessible observables: spatial steps, mean velocity, stall force, useful power, efficiency and randomness, etc. as a function of the [adenosine triphosphate] concentration and an external load F. The model presents a minimum of adjustable parameters and the theoretical predictions compare well with the available experimental results.
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Lacasta AM, Sancho JM, Romero AH, Lindenberg K. Sorting on periodic surfaces. PHYSICAL REVIEW LETTERS 2005; 94:160601. [PMID: 15904204 DOI: 10.1103/physrevlett.94.160601] [Citation(s) in RCA: 46] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2005] [Indexed: 05/02/2023]
Abstract
Particles moving on crystalline surfaces and driven by external forces or flow fields can acquire velocities along directions that deviate from that of the external force. This effect depends upon the characteristics of the particles, most notably particle size or particle index of refraction, and can therefore be (and has been) used to sort different particles. We introduce a simple model for particles subject to thermal fluctuations and moving in appropriate potential landscapes. Numerical results are compared to recent experiments on landscapes produced with holographic optical tweezers and microfabricated technology. Our approach clarifies the relevance of different parameters, the direction and magnitude of the external force, particle size, and temperature.
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Gomez-Marin A, Sancho JM. Symmetric Brownian motor. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:021101. [PMID: 15783312 DOI: 10.1103/physreve.71.021101] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/27/2004] [Indexed: 05/24/2023]
Abstract
In this paper, we present a model of a symmetric Brownian motor which changes the sign of its velocity when the temperature gradient is inverted. The velocity, external work, and efficiency are studied as a function of the temperatures of the baths and other relevant parameters. The motor shows a current reversal when another parameter (a phase shift) is varied. Analytical predictions and results from numerical simulations are performed and agree very well. Generic properties of this type of motor are discussed.
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Alonso S, Sancho JM, Sagués F. Suppression of scroll wave turbulence by noise. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:067201. [PMID: 15697558 DOI: 10.1103/physreve.70.067201] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/13/2004] [Revised: 05/03/2004] [Indexed: 05/24/2023]
Abstract
Rotating scroll waves are dynamical spatiotemporal structures characteristic of three-dimensional active media. It is well known that, under low excitability conditions, scroll waves develop an intrinsically unstable dynamical regime that leads to a highly disorganized pattern of wave propagation. Such a "turbulent" state bears some resemblance to fibrillation states in cardiac tissue. We show here that this unstable regime can be controlled by using a spatially distributed random forcing superimposed on a control parameter of the system. Our results are obtained from numerical simulations but an explicit analytical argument that rationalizes our observations is also presented.
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Lacasta AM, Sancho JM, Romero AH, Sokolov IM, Lindenberg K. From subdiffusion to superdiffusion of particles on solid surfaces. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:051104. [PMID: 15600587 DOI: 10.1103/physreve.70.051104] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/29/2004] [Indexed: 05/24/2023]
Abstract
We present a numerical and partially analytical study of classical particles obeying a Langevin equation that describes diffusion on a surface modeled by a two-dimensional potential. The potential may be either periodic or random. Depending on the potential and the damping, we observe superdiffusion, large-step diffusion, diffusion, and subdiffusion. Superdiffusive behavior is associated with low damping and is in most cases transient, albeit often long. Subdiffusive behavior is associated with highly damped particles in random potentials. In some cases subdiffusive behavior persists over our entire simulation and may be characterized as metastable. In any case, we stress that this rich variety of behaviors emerges naturally from an ordinary Langevin equation for a system described by ordinary canonical Maxwell-Boltzmann statistics.
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Sancho JM, Lacasta AM, Lindenberg K, Sokolov IM, Romero AH. Diffusion on a solid surface: anomalous is normal. PHYSICAL REVIEW LETTERS 2004; 92:250601. [PMID: 15244990 DOI: 10.1103/physrevlett.92.250601] [Citation(s) in RCA: 60] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/22/2003] [Indexed: 05/21/2023]
Abstract
We present a numerical study of classical particles diffusing on a solid surface. The particles' motion is modeled by an underdamped Langevin equation with ordinary thermal noise. The particle-surface interaction is described by a periodic or a random two-dimensional potential. The model leads to a rich variety of different transport regimes, some of which correspond to anomalous diffusion such as has recently been observed in experiments and Monte Carlo simulations. We show that this anomalous behavior is controlled by the friction coefficient and stress that it emerges naturally in a system described by ordinary canonical Maxwell-Boltzmann statistics.
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Romero AH, Lacasta AM, Sancho JM. Modelization of surface diffusion of a molecular dimer. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:051105. [PMID: 15244806 DOI: 10.1103/physreve.69.051105] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/2004] [Indexed: 05/24/2023]
Abstract
A simple model for a dimer molecular diffusion on a crystalline surface, as a function of temperature, is presented. The dimer is formed by two particles coupled by a quadratic potential. The dimer diffusion is modeled by an overdamped Langevin equation in the presence of a two-dimensional periodic potential. Numerical simulation's results exhibit some dynamical properties observed, for example, in Si2 diffusion on a silicon [100] surface. They can be used to predict the value of the effective friction parameter. Comparison between our model and experimental measurements is presented.
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Reigada R, Hillary RM, Bees MA, Sancho JM, Sagués F. Plankton blooms induced by turbulent flows. Proc Biol Sci 2003; 270:875-80. [PMID: 12737667 PMCID: PMC1691315 DOI: 10.1098/rspb.2002.2298] [Citation(s) in RCA: 42] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Plankton play an important role in the ecology of the ocean and the climate because of their participation in the global carbon cycle at the base of the food chain. However, damaging plankton blooms can sometimes occur and are initially characterized by sudden transient increases in the phytoplankton population. They are thought to be driven by several effects, such as seasonal variations in temperature and salinity, and nutrient mixing. Furthermore, phytoplankton and zooplankton have different buoyancy properties, leading to a differential response in turbulent environments. In this paper, we investigate this effect in a model of advected plankton dynamics. We find that, over a range of parameter values, flows of marine species subjected to inertial/viscous forces naturally lead to patchiness and, in turn, periodically sustained plankton blooms.
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Carrillo O, Ibañes M, García-Ojalvo J, Casademunt J, Sancho JM. Intrinsic noise-induced phase transitions: beyond the noise interpretation. PHYSICAL REVIEW E 2003; 67:046110. [PMID: 12786439 DOI: 10.1103/physreve.67.046110] [Citation(s) in RCA: 49] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/02/2002] [Indexed: 11/07/2022]
Abstract
We discuss intrinsic noise effects in stochastic multiplicative-noise partial differential equations, which are qualitatively independent of the noise interpretation (Itô vs Stratonovich), in particular in the context of noise-induced ordering phase transitions. We study a model which, contrary to all cases known so far, exhibits such ordering transitions when the noise is interpreted not only according to Stratonovich, but also to Itô. The main feature of this model is the absence of a linear instability at the transition point. The dynamical properties of the resulting noise-induced growth processes are studied and compared in the two interpretations and with a reference Ginzburg-Landau-type model. A detailed discussion of a different numerical algorithm valid for both interpretations is also presented.
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Bena I, Saxena A, Tsironis GP, Ibañes M, Sancho JM. Confinement of discrete breathers in inhomogeneously profiled nonlinear chains. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:037601. [PMID: 12689204 DOI: 10.1103/physreve.67.037601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/04/2002] [Revised: 12/11/2002] [Indexed: 05/24/2023]
Abstract
We investigate numerically the scattering of a moving discrete breather on a pair of junctions in a Fermi-Pasta-Ulam chain. These junctions delimit an extended region with different masses of the particles. We consider (i) a rectangular trap, (ii) a wedge shaped trap, and (iii) a smoothly varying convex or concave mass profile. All three cases lead to DB confinement, with the ease of trapping depending on the profile of the trap. We also study the collision and trapping of two DBs within the profile as a function of trap width, shape, and approach time at the two junctions. The latter controls whether one or both DBs are trapped.
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Buceta J, Ibañes M, Sancho JM, Lindenberg K. Noise-driven mechanism for pattern formation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:021113. [PMID: 12636659 DOI: 10.1103/physreve.67.021113] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/08/2002] [Indexed: 05/24/2023]
Abstract
We extend the mechanism for noise-induced phase transitions proposed by Ibañes et al. [Phys. Rev. Lett. 87, 020601 (2001)] to pattern formation phenomena. In contrast with known mechanisms for pure noise-induced pattern formation, this mechanism is not driven by a short-time instability amplified by collective effects. The phenomenon is analyzed by means of a modulated mean field approximation and numerical simulations.
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Lacasta AM, Sagués F, Sancho JM. Coherence and anticoherence resonance tuned by noise. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:045105. [PMID: 12443246 DOI: 10.1103/physreve.66.045105] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/07/2001] [Indexed: 05/24/2023]
Abstract
We present numerical evidence and a theoretical analysis of the appearance of anticoherence resonance induced by noise, not predicted in former analysis of coherence resonance. We have found that this phenomenon occurs for very small values of the intensity of the noise acting on an excitable system, and we claim that this is a universal signature of a nonmonotonous relaxational behavior near its oscillatory regime. Moreover, we demonstrate that this new phenomenon is totally compatible with the standard situation of coherence resonance appearing at intermediate values of noise intensity.
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Bena I, Saxena A, Sancho JM. Interaction of a discrete breather with a lattice junction. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:036617. [PMID: 12366290 DOI: 10.1103/physreve.66.036617] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/12/2002] [Indexed: 05/23/2023]
Abstract
We study the scattering of a moving discrete breather (DB) on a junction in a Fermi-Pasta-Ulam chain consisting of two segments with different masses of the particles. We consider four distinct cases: (i) a light-heavy (abrupt) junction in which the DB impinges on the junction from the segment with lighter mass, (ii) a heavy-light junction, (iii) an up mass ramp in which the mass in the heavier segment increases continuously as one moves away from the junction point, and (iv) a down mass ramp. Depending on the mass difference and DB characteristics (frequency and velocity), the DB can either reflect from, or transmit through, or get trapped at the junction or on the ramp. For the heavy-light junction, the DB can even split at the junction into a reflected and a transmitted DB. The latter is found to subsequently split into two or more DBs. For the down mass ramp the DB gets accelerated in several stages, with accompanying radiation (phonons). These results are rationalized by calculating the Peierls-Nabarro barrier for the various cases. We also point out implications of our results in realistic situations such as electron-phonon coupled chains.
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49
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Reigada R, Sagués F, Sancho JM. Binary reaction of nonpassively advected particles immersed in turbulent flows. J Chem Phys 2002. [DOI: 10.1063/1.1483293] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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50
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Báscones R, García-Ojalvo J, Sancho JM. Pulse propagation sustained by noise in arrays of bistable electronic circuits. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:061108. [PMID: 12188704 DOI: 10.1103/physreve.65.061108] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/15/2002] [Indexed: 05/23/2023]
Abstract
One-dimensional arrays of nonlinear electronic circuits are shown to support propagation of pulses when operating in a locally bistable regime, provided the circuits are under the influence of a global noise. These external random fluctuations are applied to the parameter that controls the transition between bistable and monostable dynamics in the individual circuits. As a result, propagating fronts become destabilized in the presence of noise, and the system self-organizes to allow the transmission of pulses. The phenomenon is also observed in weakly coupled arrays, when propagation failure arises in the absence of noise.
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