Spreading of wave packets in the Anderson model on the Bethe Lattice

The spreading of wave packets evolving under the Anderson Hamiltonian on the Bethe Lattice is studied for small disorder. The mean square distance travelled by a particle in a timet is shown to grow ast ² for larget.The author was supported in part by the NSF Grant DMS-9208029.     

Quantum transport in a time-dependent random potential with a finite correlation time

Using an earlier density matrix formalism in momentum space we study the motion of a particle in a time-dependent random potential with a finite correlation time τ, for 0 < t ? τ. Within this domain we consider two subdomains bounded by kinetic time scales (t _c ² = 2m? ^-1 c ², c ² = σ ², ξ ², σξ, with 2σ the width of an initial wavepacket and the correlation length of the gaussian potential fluctuations), where we obtain power law scaling laws for the effect of the random potential in the mean squared displacement 〈x ²〉 and in the mean kinetic energy 〈E _kin〉. At short times, ? min (t _σ ², 1/2t _ξ ²), 〈x ²〉 and 〈E _kin〉 scale classically as t ⁴ and t ², respectively. At intermediate times, t _σξ ? t ? 2t _σ ² and 1/2t _ξ ² ? t ? t _σξ, these quantities scale quantum mechanically as t ^3/2 and as √t, respectively. These results lie in the perspective of recent studies of the existence of (fractional) power law behavior of 〈x ²〉 and 〈E _kin〉 at intermediate times. We also briefly discuss the scaling laws for 〈x ²〉 and 〈E _kin〉 at short times in the case of spatially uncorrelated potential.     

The collapse point of interacting trails in two dimensions from kinetic growth simulations

We present simulational evidence that kinetic growth trails on the square lattice are equivalent to interacting trails at their collapse temperature. As a consequence we give values for most of the canonical exponents of the trail collapse transition: these are significantly different from those proposed for interacting walks. We can also interpret our results in terms of the equivalent Lorentz lattice gas and find that this model does not display diffusion, as has been previously thought. Rather, the mean square displacement grows ast logt in timet.     

Tagged Particle Diffusion in One-Dimensional Gas with Hamiltonian Dynamics

We consider a one-dimensional gas of hard point particles in a finite box that are in thermal equilibrium and evolving under Hamiltonian dynamics. Tagged particle correlation functions of the middle particle are studied. For the special case where all particles have the same mass, we obtain analytic results for the velocity auto-correlation function in the short time diffusive regime and the long time approach to the saturation value when finite-size effects become relevant. In the case where the masses are unequal, numerical simulations indicate sub-diffusive behaviour with mean square displacement of the tagged particle growing as t/ln(t) with time t. Also various correlation functions, involving the velocity and position of the tagged particle, show damped oscillations at long times that are absent for the equal mass case.     

Fluctuations in the motions of mass and of patterns in one-dimensional driven diffusive systems

The stochastic spreading of mass fluctuations in systems described by a fluctuating Burgers equation increases ast ^2/3 with time. As a consequence the stochastic motion of a mass front, a point through which no excess mass current is flowing, is shown to increase ast ^1/3. The same is true for the stochastic displacement of mass points and shock fronts with respect to their average drift, provided the initial configuration is fixed. An additional average over the stationary distribution of the initial configuration yields stochastic displacements, increasing with time ast ^1/2.     

Elastic scattering at larget

Recent data forpp elastic scattering at large momentum transfer find a behaviour close tot ^?8, independent of energy, throughout the ISR energy range. This behaviour is in agreement with the triple-scattering model, proposed some years ago. If this interpretation is correct, it provides evidence that gluons have spin 1. The differential cross-sections forpp andp \(\bar p\) elastic scattering are predicted to be equal at high energy, for values oft that are large but much less thans. The case of πp elastic scattering at larget is considered. It is shown that a multiple-scattering mechanism gives energy-independent behaviour at sufficiently high energy. We expect the cross-section to vary ast ^?7, but this conclusion is not as clear-cut as thet ^?8 behaviour for thepp case.     

Dynamic compensation temperature in the kinetic spin-2 Ising model in an oscillating magnetic field on alternate layers of a hexagonal lattice

Eigenstate bases are used to study electrical conductivity in graphene in the presence of short-range diagonal disorder and inter-valley scattering. For the first time, the behavior of graphene in a moderate and weak disorderd regime is presented. For disorder strength, W / t ≥  5, the density of states is flat. A connection is then established with the work of Abrahams et al. using Microscopic Renormalization Group (MRG) approach. For disorder strength, W / t = 5, results are in good agreement. For low disorder strength, W / t = 2, energy-resolved current matrix elements squared for different locations of the Fermi energy from the band centre is studied. Explicit dependence of the current matrix elements on Fermi energy is shown. It is found that states close to the band centre are more extended and fall off nearly as 1/E_l ² as one moves away from the band centre. Further studies on current matrix elements versus disorder strength suggests a cross-over from weakly localized to a very weakly localized system. Using the Kubo-Greenwood formula, conductivity and mobility is calculated. For low disorder strength, conductivity is in a good qualitative agreement with the experiments, even for the on-site disorder. The intensity plots of the eigenstates also reveal clear signatures of puddle formation for very small carrier concentration. We also make comparision with square lattice and find that graphene is more easily localized when subject to disorder.     

A Renormalization Group Classification of Nonstationary and/or Infinite Second Moment Diffusive Processes

Anomalous diffusion processes are often classified by their mean square displacement. If the mean square displacement grows linearly in time, the process is considered classical. If it grows like t ^β with β<1 or β>1, the process is considered subdiffusive or superdiffusive, respectively. Processes with infinite mean square displacement are considered superdiffusive. We begin by examining the ways in which power-law mean square displacements can arise; namely via non-zero drift, nonstationary increments, and correlated increments. Subsequently, we describe examples which illustrate that the above classification scheme does not work well when nonstationary increments are present. Finally, we introduce an alternative classification scheme based on renormalization groups. This scheme classifies processes with stationary increments such as Brownian motion and fractional Brownian motion in the same groups as the mean square displacement scheme, but does a better job of classifying processes with nonstationary increments and/or processes with infinite second moments such as α-stable Lévy motion. A numerical approach to analyzing data based on the renormalization group classification is also presented.     

Brownian motion in a fluid in elongational flow

Brownian motion of a spherical particle in stationary elongational flow is studied. We derive the Langevin equation together with the fluctuation-dissipation theorem for the particle from nonequilibrium fluctuating hydrodynamics to linear order in the elongation-rate-dependent inverse penetration depths. We then analyze how the velocity autocorrelation function as well as the mean square displacement are modified by the elongational flow. We find that for times small compared to the inverse elongation rate the behavior is similar to that found in the absence of the elongational flow. Upon approaching times comparable to the inverse elongation rate the behavior changes and one passes into a time domain where it becomes fundamentally different. In particular, we discuss the modification of thet ^–3/2 long-time tail of the velocity autocorrelation function and comment on the resulting contribution to the mean square displacement. The possibility of defining a diffusion coefficient in both time domains is discussed.     

Computer simulation study of the Levy flight process

The random walk simulation of a Levy flight shows a linear relation between the mean square displacement 〈r²〉 and time. We have analyzed different aspects of this linearity. It is shown that the restriction of jump length to a maximum value (l_m) affects the diffusion coefficient, even though it remains constant for l_m greater than 1464. So, this factor has no effect on the linearity. In addition, it is shown that the number of samples does not affect the results. We have demonstrated that the relation between the mean square displacement and time remains linear in a continuous space, while continuous variables just reduce the diffusion coefficient. The results also imply that the movement of a Levy flight particle is similar to the case in which the particle moves in each time step with an average jump length 〈l〉. Finally, it is shown that the non-linear relation of the Levy flight will be satisfied if we use a time average instead of an ensemble average. The difference between the time average and ensemble average results shows that the Levy distribution may be a non-ergodic distribution.     

Ergodical time evolution of a Gaussian wave packet in quantum dots

The time evolution of a Gaussian wave packet (GWP) confined in a quantum dot is numerically studied. The quantum dots are modelled by a two-dimensional square box and by the potential x⁴ + y⁴. For the case of an incommensurate energy spectrum the time evolution of observables has no global period. As a result this leads to ergodic phase portraits with a finite phase volume. For the spatially wide GWP the distribution function of quantum observables may be approximated as a Gaussian one. For the case of commensurate transition frequencies in the quantum well the time evolution of observables is periodical and the phase portraits have a zero phase volume.     

On the field dependence of random walks in the presence of random fields

Numerical simulations and scaling arguments are used to study the field dependence of a random walk in a one-dimensional system with a bias field on each site. The bias is taken randomly with equal probability to be +E or ?E. The probability density¯P(x, t) is found to scale asymptotically as $$\left\{ {[A(E)]^{\beta /2} /\ln ^2 t} \right\}\exp \left( { - \left\{ {x[A(E)]^{\beta /2} /\ln ^2 t} \right\}^\alpha } \right)$$ withA(E)=ln[(1+E)/(1-E)],β=4.25, and α=1.25. The mean square displacement scales as \(\langle x^2 \rangle \sim [A(E)]^{ - \beta } F[tA^\beta (E)]\) , where F(u)～ln⁴ u asymptotically.     

Quasi-ballistic evolution of a quantum particle in a random potential with arbitrary correlation time

The evolution of the mean kinetic energy and of the mean square displacement of a quantum particle in a time-dependent random potential is studied by perturbation theory in the near ballistic regime. Convenient general formulas are derived for an arbitrary correlation time, τ, of the disorder. These formulas are studied analytically near the limit of perfect dynamic disorder (τ=0) and for static disorder (τ=∞), where detailed comparison is made with earlier results. This work is the first to relate the limits of perfect dynamic disorder and of static disorder via a unified treatment for finite τ.     

Classical Motion in Force Fields with Short Range Correlations

We study the long time motion of fast particles moving through time-dependent random force fields with correlations that decay rapidly in space, but not necessarily in time. The time dependence of the averaged kinetic energy 〈p ²(t)〉/2 and mean-squared displacement 〈q ²(t)〉 is shown to exhibit a large degree of universality; it depends only on whether the force is, or is not, a gradient vector field. When it is, 〈p ²(t)〉～t ^2/5 independently of the details of the potential and of the space dimension. The stochastically accelerated particle motion is then superballistic in one dimension, with 〈q ²(t)〉～t ^12/5, and ballistic in higher dimensions, with 〈q ²(t)〉～t ². These predictions are supported by numerical results in one and two dimensions. For force fields not obtained from a potential field, the power laws are different: 〈p ²(t)〉～t ^2/3 and 〈q ²(t)〉～t ^8/3 in all dimensions d≥1.     

Monte Carlo simulation of film growth in a phase separating binary alloy model

Using a kinetic Monte Carlo method, we simulate binary film (A_0.5B_0.5/A) growth on an L×L square lattice with the focus on the domain growth behaviour. We compute the average domain area, A(t), as a measure of domain size. For a sufficiently large system, we find that A(t) grows with a power law in time with A(t)∼t^2/3 after the initial transient time. This implies that the dynamic exponent for domain growth with non-conserved order parameter is z=3, a value which was theoretically predicted for the conserved order parameter case. Further analysis reveals that such a power-law behaviour emerges because the order parameter is approximately conserved after the early stage of growth.     

Vacuum Radiation,Entropy, and Molecular Chaos

Vacuum radiation causes a particle to make a random walk about its dynamical trajectory. In this random walk the root mean square change in spatial coordinate is proportional to t ^1/2, and the fractional changes in momentum and energy are proportional to t ^−1/2, where t is time. Thus the exchange of energy and momentum between a particle and the vacuum tends to zero over time. At the end of a mean free path the fractional change in momentum of a particle in a gas is very small. However, at the end of the mean free path each particle undergoes an interaction that magnifies the preceding change, and the net result is that the momentum distribution of the particles in a gas is randomized in a few collision times. In this way the random action of vacuum radiation and its subsequent magnification by molecular interaction produces entropy increase. This process justifies the assumption of molecular chaos used in the Boltzmann transport equation.     

Space use by foragers consuming renewable resources

Through the analysis of unbiased random walks on fractal trees and continuous time random walks, we show that even if a process is characterized by a mean square displacement (MSD) growing linearly with time (standard behaviour) its diffusion properties can be not trivial. In particular, we show that the following scenarios are consistent with a linear increase of MSD with time: (i) the high-order moments, ?x(t) ^q ? for q > 2 and the probability density of the process exhibit multiscaling; (ii) the random walk on certain fractal graphs, with non integer spectral dimension, can display a fully standard diffusion; (iii) positive order moments satisfying standard scaling does not imply an exact scaling property of the probability density.     

Random time averaged diffusivities for Lévy walks

We investigate a Lévy walk alternating between velocities ±v ₀ with opposite sign. The sojourn time probability distribution at large times is a power law lacking its mean or second moment. The first case corresponds to a ballistic regime where the ensemble averaged mean squared displacement (MSD) at large times is ?x ²? ∝ t ², the latter to enhanced diffusion with ?x ²? ∝ t ^ν, 1 < ν < 2. The correlation function and the time averaged MSD are calculated. In the ballistic case, the deviations of the time averaged MSD from a purely ballistic behavior are shown to be distributed according to a Mittag-Leffler density function. In the enhanced diffusion regime, the fluctuations of the time averages MSD vanish at large times, yet very slowly. In both cases we quantify the discrepancy between the time averaged and ensemble averaged MSDs.     

Transport of a Passive Tracer by an Irregular Velocity Field

The trajectories of a passive tracer in a turbulent flow satisfy the ordinary differential equation x′(t)=V(t,x(t)), where V(t,x) is a stationary random field, the so-called Eulerian velocity. It is a nontrivial question to define the dynamics of the tracer in the case when the realizations of the Eulerian field are only spatially Hölder regular because the ordinary differential equation in question lacks then uniqueness. The most obvious approach is to regularize the dynamics, either by smoothing the velocity field (the so-called ε-regularization), or by adding a small molecular diffusivity (the so-called κ-regularization) and then pass to the appropriate limit with the respective regularization parameter. The first procedure corresponds to the Prandtl number Pr=∞, while the second to Pr=0. In the present paper we consider a two parameter family of Gaussian, Markovian time correlated fields V(t,x), with the power-law spectrum. Using the infinite dimensional stochastic calculus we show the existence and uniqueness of the law of the trajectory process corresponding to a given field V(t,x) for a certain regime of parameters characterizing the spectrum of the field. Additionally, this law is the limit, in the sense of the weak convergence of measures, of the laws obtained as a result of any of the described regularizations. The so-called Kolmogorov point, that corresponds to the parameters characterizing the relaxation time and energy spectrum of a turbulent, three dimensional flow, belongs to the boundary of the parameter regime considered in the paper.     

Crossover from diffusive to ballistic transport in periodic quantum maps

We derive an expression for the mean square displacement (MSD) of a particle whose motion is governed by a uniform, periodic, quantum multi-baker map. The expression is a function of both time, t, and Planck’s constant, h, and allows a study of both the long time, t→∞, and semi-classical, h→0, limits taken in either order. We evaluate the expression using random matrix theory as well as numerically, and observe good agreement between both sets of results. The long time limit shows that particle transport is generically ballistic for any fixed value of Planck’s constant. However, for fixed times, the semi-classical limit leads to diffusion. The mean square displacement for non-zero Planck’s constant, and finite time, exhibits a crossover from diffusive to ballistic motion, with crossover time on the order of the inverse of Planck’s constant. We argue that these results are generic for a large class of 1D quantum random walks, similar to the quantum multi-baker, and that a sufficient condition for diffusion in the semi-classical limit is classically chaotic dynamics in each cell. Some connections between our work and the other literature on quantum random walks are discussed. These walks are of some interest in the theory of quantum computation.