Rafael Morgado

The Fluctuation-Dissipation Theorem fails for fast superdiffusion

We study anomalous diffusion for one-dimensional systems described by a generalized Langevin equation. We show that superdiffusive systems can be divided into two classes: normal and fast. For fast superdiffusion we prove that the Fluctuation-Dissipation Theorem does not hold. As a result, the system acquires an effective temperature. This effective temperature is a signature of metastability found in many complex systems such as spin-glass and granular material.

Khinchin theorem and anomalous diffusion.

A recent Letter [M. H. Lee, Phys. Rev. Lett. 98, 190601 (2007)] has called attention to the fact that irreversibility is a broader concept than ergodicity, and that therefore the Khinchin theorem [A. I. Khinchin, (Dover, New York, 1949)] may fail in some systems. In this Letter we show that for all ranges of normal and anomalous diffusion described by a generalized Langevin equation the Khinchin theorem holds.

Non-exponential relaxation for anomalous diffusion

We study the relaxation process in normal and anomalous diffusion regimes for systems described by a generalized Langevin equation (GLE). We demonstrate the existence of a very general correlation function which describes the relaxation phenomena. Such function is even; therefore, it cannot be an exponential or a stretched exponential. However, for a proper choice of the parameters, those functions can be reproduced within certain intervals with good precision. We also show the passage from the non-Markovian to the Markovian behaviour in the normal diffusion regime. For times longer than the relaxation time, the correlation function for anomalous diffusion becomes a power law for broad-band noise.

Synchronization in the presence of memory

We study the effect of memory on synchronization of identical chaotic systems driven by common external noises. Our examples show that while in general the synchronization transition becomes more difficult to meet when the memory range increases, for intermediate ranges the synchronization tendency of systems can be enhanced. Generally the synchronization transition is found to depend on the memory profile and range and the ratio of noise strength to memory amplitude, which indicates a possibility of optimizing synchronization by memory. We also point out a close link between dynamics with memory and noise, and recently discovered synchronizing properties of networks with delayed interactions.

Multistage Markov Chain Modeling of the Genetic Algorithm and Convergence Results

The genetic algorithm (GA) has been widely used to solve combinatorial global optimization problems. Despite the successes that GA encounters in practical applications, there exist few precise results on its behavior. In this article, we formulate a fully rigorous mathematical modeling of GA as a multistage Markov chain and derive convergence results. Variations that include the simulated annealing algorithm and the GA with superindividual are considered.

Stochastic description of the dynamics of a random-exchange Heisenberg chain

Abstract We study the diffusion process in a Heisenberg chain with correlated spatial disorder, with a power spectrum in the momentum space behaving as k − β , using a stochastic description. It establishes a direct connection between the fluctuation in the spin-wave density of states and the noise density of states. For continuous ranges of the exponent β , we find superdiffusive and ballistic spin-wave motions. Both diffusion exponents predicted by the stochastic procedure agree with the ones calculated using the Hamiltonian dynamics.

Spatio-temporal conjecture for diffusion

We present here a conjecture about the equivalence between the noise density of states of a system governed by a generalized Langevin equation and the fluctuation in the energy density of states in a Hamiltonian system. We present evidence of this for a disordered Heisenberg system.

Superdiffusive conduction: AC conductivity with correlated noise

We present evidence of the existence of a superdiffusive regime in systems with correlated disorder for which localization is suppressed. An expression for anomalous electrical conductivity at low frequencies is found by using a generalized Langevin equation whose memory function accounts for the interactions between the carriers. New mechanisms inducing a superdiffusive conductivity are discussed and experimental possibilities for observing that phenomenon in nanotubes and super-lattices are presented.