N. Pottier

Aging properties of an anomalously diffusing particule

We report new results about the two-time dynamics of an anomalously diffusing classical particle, as described by the generalized Langevin equation with a frequency-dependent noise and the associated friction. The noise is defined by its spectral density proportional to ωδ−1 at low frequencies, with 0<δ<1 (subdiffusion) or 1<δ<2 (superdiffusion). Using Laplace analysis, we derive analytic expressions in terms of Mittag–Leffler functions for the correlation functions of the velocity and of the displacement. While the velocity thermalizes at large times (slowly, in contrast to the standard Brownian motion case δ=1), the displacement never attains equilibrium: it ages. We thus show that this feature of normal diffusion is shared by a subdiffusive or superdiffusive motion. We provide a closed form analytic expression for the fluctuation–dissipation ratio characterizing aging.

Aging effects in free quantum Brownian motion

The two-time correlation function Cxx(t,t′) of the displacement x(t)−x(t0) of a free quantum Brownian particle with respect to its position at a given time t0 is calculated analytically in the framework of the Caldeira and Leggett ohmic dissipation model. As a result, at any temperature T,Cxx(t,t′) exhibits aging, i.e. it depends explicitly on both times t and t′ and not only on the time difference τ=t−t′, even in the limit of large age t′(t0⩽t′⩽t), in contrast with a dynamic variable in equilibrium such as the particle velocity. The equilibrium quantum fluctuation-dissipation theorem (QFDT) has to be modified in order to relate the response function χxx(t,t′) to Cxx(t,t′), since this latter quantity takes into account even those fluctuations of the displacement which take place during the waiting time tw=t′−t0. We describe the deviation from QFDT in terms of an effective inverse temperature βeff(τ,tw). The behaviour of this quantity as a function of τ for given values of T and tw is analysed. In the classical limit it is shown that βeff(τ,tw)=βD(τ)/[D(τ)+D(tw)], where D(t) denotes the time-dependent diffusion coefficient.

Quantum ohmic dissipation: Transition from coherent to incoherent dynamics

Abstract We investigate the real-time dynamics of a particle in a double well coupled to phonons with ohmic dissipation. We refine previous results about the T =0 symmetry breaking: the critical behaviour of the order parameter is displayed and the symmetry is shown to be only fully broken for infinite coupling.

Dynamical Exponents for One-Dimensional Random-Random Directed Walks

The dynamical exponents of the coordinate and of the mean square displacement are explicitly calculated in the case of a directed random walk on a one-dimensional random lattice. Moreover, it is shown that, in the dynamical phase where the coordinate increases slower thant, the latter is not a self-averaging quantity.

Exact results and self-averaging properties for random-random walks on a one-dimensional infinite lattice

We present new exact results for a one-dimensional asymmetric disordered hopping model. The lattice is taken infinite from the start and we do not resort to the periodization scheme used by Derrida. An explicit resummation allows for the calculation of the velocityV and the diffusion constantD (which are found to coincide with those given by Derrida) and for demonstrating thatV is indeed a self-averaging quantity; the same property is established forD in the limiting case of a directed walk.

Anomalous diffusion of a particle in an aging medium

We report new results about the anomalous diffusion of a particle in an aging medium. For each given age, the quasi-stationary particle velocity is governed by a generalized Langevin equation with a frequency-dependent friction coefficient proportional to |ω|δ−1 at small frequencies, with 0<δ<2. The aging properties of the medium are encoded in a frequency-dependent effective temperature Teff.(ω). The latter is modelized by a function proportional to |ω|α at small frequencies, with α<0, thus allowing for the medium to have a density of slow modes proportionally larger than in a thermal bath. Using asymptotic Fourier analysis, we obtain the behaviour at large times of the velocity correlation function and of the mean square displacement. As a result, the anomalous diffusion exponent in the aging medium appears to be linked, not only to δ as it would be the case in a thermal bath, but also to the exponent α characterizing the density of slow modes.

Out of equilibrium generalized Stokes–Einstein relation: determination of the effective temperature of an aging medium

We analyze in detail how the anomalous drift and diffusion properties of a particle evolving in an aging medium can be interpreted in terms of an effective temperature of the medium. From an experimental point of view, independent measurements of the mean-square displacement and of the mobility of a particle immersed in an aging medium such as a colloidal glass give access to an out of equilibrium generalized Stokes–Einstein relation, from which the effective temperature of the medium can eventually be deduced. We illustrate the procedure on a simple model with power-law behaviors.

Aging effects in the quantum dynamics of a dissipative free particle: Non-Ohmic case

We report the results related to the two-time dynamics of the coordinate of a quantum free particle, damped through its interaction with a fractal thermal bath (non-Ohmic coupling approximately omega(delta) with 0<delta <1 or 1<delta <2). When the particle is localized, its position does not age. When it undergoes anomalous diffusion, only its displacement may be defined. It is shown to be an aging variable. The finite temperature aging regime is self-similar. It is described by a scaling function of the ratio t(w)/tau of the waiting time to the observation time, as characterized by an exponent directly linked to delta.

Analytic study of the effect of persistence on a one-dimensional biased random walk

An analytic study of a one-dimensional biased random walk with correlations between nearest-neighbour steps is presented, both in a lattice model and in its continuous version. First, the treatment of the unbiased problem is recalled and the effect of correlations on the diffusion coefficient is discussed. Then the study is extended to the biased case. The problem is then completely determined by two independent parameters, the degree of correlations in the motion on the one hand and the value of the bias on the other. Both the velocity of the particle and its diffusion coefficient are computed. As a result, the velocity as well as the diffusion coefficient are enhanced when there are positive correlations (qualified as persistence) in the motion, and reduced in the opposite case.

Random walk in a one-dimensional random medium

Abstract The random walk of a particle in a one-dimensional random medium is examined by means of the equivalent transfer rates technique, in the discrete as well as in the continuous version of the model. The probability distributions of the (energy-dependent) equivalent transfer rates are found analytically, either by a matching procedure (in the discrete case) or exactly (in the continuous model). Both discrete and continuous models are shown to belong to the same universality class. The average probability of presence of the particle at its initial point is then computed. For a non-zero global bias, it decreases at large times according to a negative power-law with an exponent depending on disorder and bias; when there is no global bias (Sinais model) the decay at large times follows a logarithmic law.