Bernardo Sánchez-Rey

Thermodynamic description in a simple model for granular compaction

A simple model for the dynamics of a granular system under tapping is studied. The model can be considered as a particularization for short taps of a more general one-dimensional lattice model with facilitated dynamics. The steady state reached by the system is discussed and the results are shown to be consistent with the thermodynamic granular theory developed by Edwards and coworkers. In particular, the basic assumption of the theory, i.e., that the probability distribution depends only on the volume of the configuration, is verified.

Demonstration of the stability or instability of multibreathers at low coupling

Abstract Whereas there exists a mathematical proof for one-site breathers stability, and an unpublished one for two-site breathers, the methods for determining the stability properties of multibreathers rely on numerical computation of the Floquet multipliers or on the weak nonlinearity approximation leading to discrete nonlinear Schrodinger equations. Here we present a set of multibreather stability theorems (MST) that provides a simple method to determine multibreathers stability in Klein–Gordon systems. These theorems are based in the application of degenerate perturbation theory to Aubry’s band theory. We illustrate them with several examples.

Bright and dark breathers in Fermi-Pasta-Ulam lattices

In this paper we study the existence and linear stability of bright and dark breathers in one-dimensional FPU lattices. On the one hand, we test the range of validity of a recent breathers existence proof [G. James, {\em C. R. Acad. Sci. Paris}, 332, Ser. 1, pp. 581 (2001)] using numerical computations. Approximate analytical expressions for small amplitude bright and dark breathers are found to fit very well exact numerical solutions even far from the top of the phonon band. On the other hand, we study numerically large amplitude breathers non predicted in the above cited reference. In particular, for a class of asymmetric FPU potentials we find an energy threshold for the existence of exact discrete breathers, which is a relatively unexplored phenomenon in one-dimensional lattices. Bright and dark breathers superposed on a uniformly stressed static configuration are also investigated.

A Dynamical Monte Carlo Algorithm for Master Equations with Time-Dependent Transition Rates

A Monte Carlo algorithm for simulating master equations with time-dependent transition rates is described. It is based on a waiting time image, and takes into account that the system can become frozen when the transition rates tend to zero fast enough in time. An analytical justification is provided. The algorithm reduces to the Bortz-Kalos-Lebowitz one when the transition rates are constant. Since the exact evaluation of waiting times is rather involved in general, a simple and efficient iterative method for accurately calculating them is introduced. As an example, the algorithm is applied to a one-dimensional Ising system with Glauber dynamics. It is shown that it reproduces the exact analytical results, being more efficient than the direct implementation of the Metropolis algorithm

Simple model with facilitated dynamics for granular compaction

A simple lattice model is used to study compaction in granular media. As in real experiments, we consider a series of taps separated by large enough waiting times. The relaxation of the density exhibits the characteristic inverse logarithmic law. Moreover, we have been able to identify analytically the relevant time scale, leading to a relaxation law independent of the specific values of the parameters. Also, an expression for the asymptotic density reached in the compaction process has been derived. The theoretical predictions agree fairly well with the results from the Monte Carlo simulation.

Hysteresis in vibrated granular media

Some general dynamical properties of models for compaction of granular media based on master equations are analyzed. In particular, a one-dimensional lattice model with short-ranged dynamical constraints is considered. The stationary state is consistent with Edwards theory of powders. The system is submitted to processes in which the tapping strength is monotonically increased and decreased. In such processes the behavior of the model resembles the reversible–irreversible branches which have been recently observed in experiments. This behavior is understood in terms of the general dynamical properties of the model, and related to the hysteresis cycles exhibited by structural glasses in thermal cycles. The existence of a “normal” solution, i.e., a special solution of the master equation which is monotonically approached by all the other solutions, plays a fundamental role in the understanding of the hysteresis effects.

Approximation of solitons in the discrete NLS equation

Abstract We study four different approximations for finding the profile of discrete solitons in the one-dimensional Discrete Nonlinear Schrödinger (DNLS) Equation. Three of them are discrete approximations (namely, a variational approach, an approximation to homoclinic orbits and a Green-function approach), and the other one is a quasi-continuum approximation. All the results are compared with numerical computations.

Analytical approach to soliton ratchets in asymmetric potentials

We use soliton perturbation theory and collective coordinate ansatz to investigate the mechanism of soliton ratchets in a driven and damped asymmetric double sine-Gordon equation. We show that, at the second order of the perturbation scheme, the soliton internal vibrations can couple effectively, in presence of damping, to the motion of the center of mass, giving rise to transport. An analytical expression for the mean velocity of the soliton is derived. The results of our analysis confirm the internal mode mechanism of soliton ratchets proposed in [Phys. Rev. E 65, 025602(R) (2002)].

Escape Dynamics in the Discrete Repulsive Model

Abstract We study deterministic escape dynamics of the discrete Klein–Gordon model with a repulsive quartic on-site potential. Using a combination of analytical techniques, based on differential and algebraic inequalities and selected numerical illustrations, we first derive conditions for collapse of an initially excited single-site unit, for both the Hamiltonian and the linearly damped versions of the system and showcase different potential fates of the single-site excitation, such as the possibility to be “pulled back” from outside the well or to “drive over” the barrier some of its neighbors. Next, we study the evolution of a uniform (small) segment of the chain and, in turn, consider the conditions that support its escape and collapse of the chain. Finally, our path from one to the few and finally to the many excited sites is completed by a modulational stability analysis and the exploration of its connection to the escape process for plane wave initial data. This reveals the existence of three distinct regimes, namely modulational stability, modulational instability without escape and, finally, modulational instability accompanied by escape. These are corroborated by direct numerical simulations. In each of the above cases, the variations of the relevant model parameters enable a consideration of the interplay of discreteness and nonlinearity within the observed phenomenology.

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

We discuss the existence of breathers and lower bounds on their power, in nonlinear Schrödinger lattices with nonlinear hopping. Our methods extend from a simple variational approach to fixed-point arguments, deriving lower bounds for the power which can serve as a threshold for the existence of breather solutions. Qualitatively, the theoretical results justify non-existence of breathers below the prescribed lower bounds of the power which depend on the dimension, the parameters of the lattice as well as of the frequency of breathers. In the case of supercritical power nonlinearities we investigate the interplay of these estimates with the optimal constant of the discrete interpolation inequality. Improvements of the general estimates, taking into account the localization of the true breather solutions are derived. Numerical studies in the one-dimensional lattice corroborate the theoretical bounds and illustrate that in certain parameter regimes of physical significance, the estimates can serve as accurate predictors of the breather power and its dependence on the various system parameters.