Normal and anomalous diffusion of Brownian particles on disordered potentials

In this work we study the transition from normal to anomalous diffusion of Brownian particles on disordered potentials. The potential model consists of a series of "potential hills" (defined on unit cell of constant length) whose heights are chosen randomly from a given distribution. We calculate the exact expression for the diffusion coefficient in the case of uncorrelated potentials for arbitrary distributions. We particularly show that when the potential heights have a Gaussian distribution (with zero mean and a finite variance) the diffusion of the particles is always normal. In contrast when the distribution of the potential heights are exponentially distributed we show that the diffusion coefficient vanishes when system is placed below a critical temperature. We calculate analytically the diffusion exponent for the anomalous (subdiffusive) phase by using the so-called "random trap model". We test our predictions by means of Langevin simulations obtaining good agreement within the accuracy of our numerical calculations.Comment: 15 pages, 4 figure

Resonant Response in Non-equilibrium Steady States

The time-dependent probability density function of a system evolving towards a stationary state exhibits an oscillatory behavior if the eigenvalues of the corresponding evolution operator are complex. The frequencies \omega_n, with which the system reaches its stationary state, correspond to the imaginary part of such eigenvalues. If the system is further driven by a small and oscillating perturbation with a given frequency \omega, we formally prove that the linear response to the probability density function is enhanced when \omega = \omega_n. We prove that the occurrence of this phenomenon is characteristic of systems that reach a non-equilibrium stationary state. In particular we obtain an explicit formula for the frequency-dependent mobility in terms of the of the relaxation to the stationary state of the (unperturbed) probability current. We test all these predictions by means of numerical simulations considering an ensemble of non-interacting overdamped particles on a tilted periodic potential.Comment: 9 pages, 10 figures, submitted to Physical Review

A metric property of umbilic points

In the space $\mathbb U^4$ of cubic forms of surfaces, regarded as a $G$-space and endowed with a natural invariant metric, the ratio of the volumes of those representing umbilic points with negative to those with positive indexes is evaluated in terms of the asymmetry of the metric, defined here. A connection of this ratio with that reported by Berry and Hannay (1977) in the domain of Statistical Physics, is discussed.Comment: 8 pages, 1 figur

Stabilizing Entangled States with Quasi-Local Quantum Dynamical Semigroups

We provide a solution to the problem of determining whether a target pure state can be asymptotically prepared using dissipative Markovian dynamics under fixed locality constraints. Beside recovering existing results for a large class of physically relevant entangled states, our approach has the advantage of providing an explicit stabilization test solely based on the input state and constraints of the problem. Connections with the formalism of frustration-free parent Hamiltonians are discussed, as well as control implementations in terms of a switching output-feedback law.Comment: 11 pages, no figure