D. Saint-James

Time behavior of the correlation functions in a simple dissipative quantum model

The force and velocity correlation functions for a particle interacting with a bath are calculated within a model allowing for finite memory effects. The relevance of a Brownian picture is delineated in view of the respective behavior of these functions and appears fully inadequate below some cross-over temperature; then, the interplay between quantum and thermal fluctuations yields some long time tails on the same time scale for both correlation functions. The real space transient diffusion coefficient is found to exceed its asymptotic Einstein value for most times in that regime. The limiting case of an infinitely short memory time is also investigated and is seen to produce weak divergences on a time scale which is small as compared to the other characteristic times.

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.

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.

Quantum ohmic dissipation: Particle on a one-dimensional periodic lattice

Abstract We investigate the dynamics of a particle on a periodic lattice, coupled to phonons with ohmic dissipation. At T = 0 a symmetry breaking appears, which corresponds to a transition between a localized and a delocalized regime. For T > 0 we recover a diffusive motion for which the diffusion coefficient is computed.

Quantum dissipation: Dynamics of a particle in a symmetric double well in the presence of non-ohmic friction

We investigate the equilibrium value and the approach to equilibrium of a particle in a symmetric double-well potential and subjected to dissipation, within the model of Caldeira and Leggett, Ann. Phys. (N.Y.) 149 (1983) 374, but without the ohmic assumption. For subohmic friction, the critical value of the coupling with the bath above which the particle localizes is found to be dependent on the ratio (tunnelling frequency)/(bandwidth of the bath); for an infinite bandwidth, the results given by Leggett et al., Rev. Mod. Phys. 59 (1987) 1, are recovered. For superohmic dissipation, the particle never localizes and its coordinate always goes to zero by following, at large times, a power law in time of the form t−ν; the exponent ν is shown to have a non-intuitive variation as a function of the non-ohmicity.

Random-Random walks : stability of dynamical phases against exponential correlations in a quenched directed model

We analyse the stability of the anomalous velocity regime when the disordered hopping rates are correlated. Our basic assumption is that the building of the lattice itself proceeds along a given Markov process; this gives rise to exponentially correlated transfer rates. It is shown that the slow drift regime (where the average coordinate increases like tμ with μ < 1) still occurs; the correlations introduce logarithmic corrections which compete with disorder when the latter becomes rather strong, thus tending to slightly speed up the motion.

Dynamical properties of a random walk in a one-dimensional random medium

Abstract The random motion of a particle in a one-dimensional continuous random medium with random forces is investigated by making use of the large-scale equivalence of the properties of the walk with those of a directed random walk on a discrete lattice. When the disorder is weak enough, a normal drift-diffusion regime takes place, whereas for strong disorder anomalous behaviours occur. It is shown that these results may be obtained by introducing a renormalized lattice with a finite spacing. In the normal phase this spacing appears explicitly in the result, while it turns out to be irrelevant in the anomalous ones. The dynamical exponents as well as the prefactors of the power-laws of the particle position and of its average dispresion are explicitly calculated in the anomalous phases.

Random-random walk on an asymmetric chain with a trapping attractive center

The random walk of a particle on an asymmetric chain in the presence of an attractive center, possibly trapping, is examined by means of the equivalent transfer rates technique. Both the situations of ordered and disordered hopping rates are studied. It is assumed that initially the particle is located on the attractive center. The (average) probability of presence of the particle at its initial point is computed as a function of time. In the ordered case this quantity decreases exponentially toward its limiting value (with in certain cases an inverse power-law prefactor), while in the presence of disorder it decreases according to a power law, with an exponent depending both on disorder and on asymmetry. When the possibility of trapping is taken into account, this model is relevant for the description of the transfer of energy in a photosynthetic system. The amount of energy conserved within the chain, as a function of time, and the average lifetime of the particle before it is captured by the trap are examined in both ordered and disordered situations.