Felix Ritort

Exactly Solvable Phase Oscillator Models with Synchronization Dynamics

Populations of phase oscillators interacting globally through a general coupling function fsxd have been considered. We analyze the conditions required to ensure the existence of a Lyapunov functional giving close expressions for it in terms of a generating function. We have also proposed a family of exactly solvable models with singular couplings showing that it is possible to map the synchronization phenomenon into other physical problems. In particular, the stationary solutions of the least singular coupling considered, fsxd › sgnsxd, have been found analytically in terms of elliptic functions. This last case is one of the few nontrivial models for synchronization dynamics which can be analytically solved. [S0031-9007(98)07451-1] PACS numbers: 05.45. + b, 87.10. + e

Quantum phase transition in spin glasses with multi-spin interactions

We examine the phase diagram of the p-interaction spin glass model in a transverse field. We consider a spherical version of the model and compare with results obtained in the Ising case. The analysis of the spherical model, with and without quantization, reveals a phase diagram very similar to that obtained in the Ising case. In particular, using the static approximation, reentrance is observed at low temperatures in both the quantum spherical and Ising models. This is an artifact of the approximation and disappears when the imaginary time dependence of the order parameter is taken into account. The resulting phase diagram is checked by accurate numerical investigation of the phase boundaries.

Aging in the linear harmonic oscillator

The low temperature Monte Carlo dynamics of an ensemble of linear harmonic oscillators shows some entropic barriers related to the difficulty of finding the directions in configurational space which decrease the energy. This mechanism is enough to observe some typical non-equilibrium features of glassy systems like activated-type behavior and aging in the correlation function and in the response function. Due to the absence of interactions the model only displays a one-step relaxation process.

A moment-based approach to the dynamical solution of the Kuramoto model

We examine the dynamics of the Kuramoto model with a new analytical approach. By defining an appropriate set of moments the dynamical equations can be exactly closed. We discuss some applications of the formalism such as the existence of an effective Hamiltonian for the dynamics. We also show how this approach can be used to numerically investigate the dynamical behaviour of the model without finite-size effects.

Relaxation processes and entropic traps in the Backgammon model

We examine the density - density correlation function in a model recently proposed to study the effect of entropy barriers in glassy dynamics. We find that the relaxation proceeds in two steps with a fast beta process followed by alpha relaxation. The results are physically interpreted in the context of an adiabatic approximation which allows one to separate the two processes and define an effective temperature in the off-equilibrium dynamics of the model. We investigate the behaviour of the response function associated with the density and find violations of the fluctuation dissipation theorem.

Tempering simulations in the four dimensional ±J Ising spin glass in a magnetic field

We study the four dimensional (4D) ±J Ising spin glass in a magnetic field with the simulated tempering algorithm recently introduced by Marinari and Parisi. We compute numerically the order parameter function P(q) and analyze the temperature dependence of the first four cumulants of the distribution. We discuss the evidence in favor of the existence of a phase transition in a field. Assuming a well defined transition we are able to bound its critical temperature.

EXACT SOLUTIONS AND DYNAMICS OF GLOBALLY COUPLED OSCILLATORS

We analyze mean-field models of synchronization of phase oscillators with singular couplings and subject to external random forces. They are related to the Kuramoto–Sakaguchi model. Their probability densities satisfy local partial differential equations similar to the porous medium, Burgers and extended Burgers systems depending on the degree of singularity of the coupling. We show that porous medium oscillators (the most singularly coupled) do not synchronize and that (transient) synchronization is possible only at zero temperature for Burgers oscillators. The extended Burgers oscillators have a nonlocal coupling first introduced by Daido and they may synchronize at any temperature. Exact expressions for their synchronized phases and for Daidos order function are given in terms of elliptic functions.

Solvable Dynamics in a System of Interacting Random Tops

A new solvable model of synchronization dynamics is introduced. It consists of a system of long range interacting tops or magnetic moments with random precession frequencies. The model allows for an explicit study of orientational effects in synchronization phenomena as well as nonlinear processes in resonance phenomena in strongly coupled magnetic systems. A stability analysis of the incoherent solution is performed for different types of orientational disorder. A system with orientational disorder always synchronizes in the absence of noise. [S0031-9007(97)04832-1] Much theoretical effort has been devoted to the study of synchronization dynamics in simple systems composed of interacting units. In those cases there is competition between oscillations arising from the natural randomness in the members of a population and macroscopic synchronization of the population as a whole. It is widely believed that these models provide a plausible explanation for the existence of synchronization phenomena in a large variety of physical systems ranging from physics to biology [1]. A simple model which describes the emergence of synchronization phenomena in a population of phase oscillators was proposed many years ago by Kuramoto [2]. Despite extensive studies in the past, there are still some open issues such as a the study of the dynamics in the absence of external noise [3] or the onset of synchronization in the critical region [4,5]. Only very recently, a physical realization of the Kuramoto model has been found [6]. The aim of this Letter is to introduce a new model which shows a new mechanism for synchronization phenomena. The fundamental new feature of this model is that it explicitly introduces the role of orientational degrees of freedom in the synchronization dynamics. This feature is ubiquitous in nature in systems formed by units with natural magnetic (or angular) moment. Hence the proposed model in this Letter is a step towards a microscopic semiclassical theory of nonlinear phenomena in magnetic resonance processes (and, in particular, ferromagnetic resonance [7]) as well as synchronization phenomena in biomagnetism (and, in particular, responses of living organisms to external magnetic fields). The model consists of a system of N tops (or magnetic moments), each one characterized by a random natural precession vector

Quantum critical effects in mean-field glassy systems

vi, interacting ferromagnetically in a meanfield way. Randomness in magnetic systems can arise due to local inhomogeneities or crystal field anisotropies. The tops are specified by a three component unit vector

Langevin dynamics of the Lebowitz-Percus model

xi