Langevin equation for a free particle driven by power law type of noises

We study a generalized Langevin equation for a free particle driven by N internal noises. The mean square displacement and velocity autocorrelation function are derived in case of a mixture of Dirac delta, power law and Mittag-Leffler noises. Additionally, a frictional memory kernel of distributed order is considered. The long time limit and short time limit are analyzed, and the dominant contributions of noises on particle dynamics is discussed. Various different diffusive behaviors (subdiffusion, superdiffusion, normal diffusion, ultraslow diffusion) are obtained. The considered problem may be used in the theory of anomalous diffusion in complex environment.     

Transport in a disordered medium: Analysis and Monte Carlo simulation

We consider a classical stochastic model describing particle transport on a lattice with randomly distributed nearest-neighbor transition rates. Applying an effective medium theory to the model, we determine average properties related to the particle's dynamics ind-dimensions. In particular, we calculate the mean-square displacement, and the fourth moment of the displacement in one-, two- and three dimensions. The results compare favorably with Monte Carlo simulations of the model. We also present preliminary results for the velocity autocorrelation function.An aspect of the bond percolation problem, which is a special case of the stochastic model is investigated; the average inverse cluster size, <N _c ^–1>, is calculated. In one dimension the expression for this quantity is exact and in higher dimensions our results are very accurate not too close to the percolation concentration.     

Transport in polygonal billiards

We present in this work a numerical study of the dynamics of ensembles of point particles within a polygonal billiard chain. This billiard is a system with no exponential instability. Our numerical results suggest that some members of the family exhibit normal diffusive behavior while others present anomalous diffusion. Our conclusions are drawn from the numerical evaluation of the mean square displacement, the velocity autocorrelation function and its spectral analysis. Furthermore we analyze the properties of the incoherent scattering function. The super Burnett coefficient seems to be ill defined in all systems. The multifractal analysis of the spectrum of the velocity autocorrelation functions is also reported. Finally, we study the heat conduction in our polygonal chain.     

Influences of Temperature and Average Interparticle Distance on the Properties of Two-Dimensional Dusty Plasma

The structure and single-particle motion of a two-dimensional dusty plasma have been investigated. Pair correlation function, mean square displacement, velocity autocorrelation function, and the corresponding spectrum function have been computed by molecular dynamical simulation. The results show that the coagulation of a two-dimensional dusty plasma system is strongly affected by particle density and temperature, which are discussed in details.     

Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon

We study the asymptotic statistical behavior of the 2-dimensional periodic Lorentz gas with an infinite horizon. We consider a particle moving freely in the plane with elastic reflections from a periodic set of fixed convex scatterers. We assume that the initial position of the particle in the phase space is random with uniform distribution with respect to the Liouville measure of the periodic problem. We are interested in the asymptotic statistical behavior of the particle displacement in the plane as the timet goes to infinity. We assume that the particle horizon is infinite, which means that the length of free motion of the particle is unbounded. Then we show that under some natural assumptions on the free motion vector autocorrelation function, the limit distribution of the particle displacement in the plane is Gaussian, but the normalization factor is (t logt)^1/2 and nott ^1/2 as in the classical case. We find the covariance matrix of the limit distribution.     

Diffusive-ballistic crossover in 1D quantum walks

We show that particle transport, as characterized by the equilibrium mean square displacement, in a uniform, quantum multibaker map, is generically ballistic in the long time limit, for any fixed value of Planck's constant. However, for fixed times, the semiclassical limit leads to diffusion. Random matrix theory provides explicit analytical predictions for the mean square displacement of a particle in the system. These results exhibit a crossover from diffusive to ballistic motion, with crossover time on the order of the inverse of Planck's constant. We expect that, for a large class of 1D quantum random walks similar to the quantum multibaker, a sufficient condition for diffusion in the semiclassical limit is classically chaotic dynamics in each cell. The systems described generalize known quantum random walks and may have applications for quantum computation.     

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.     

Two-state random walk model of diffusion. 2. Oscillatory diffusion

Continuing our study of interrupted diffusion, we consider the problem of a particle executing a random walk interspersed with localized oscillations during its halts (e.g., at lattice sites). Earlier approaches proceedvia approximation schemes for the solution of the Fokker-Planck equation for diffusion in a periodic potential. In contrast, we visualize a two-state random walk in velocity space with the particle alternating between a state of flight and one of localized oscillation. Using simple, physically plausible inputs for the primary quantities characterising the random walk, we employ the powerful continuous-time random walk formalism to derive convenient and tractable closed-form expressions for all the objects of interest: the velocity autocorrelation, generalized diffusion constant, dynamic mobility, mean square displacement, dynamic structure factor (in the Gaussian approximation), etc. The interplay of the three characteristic times in the problem (the mean residence and flight times, and the period of the ‘local mode’) is elucidated. The emergence of a number of striking features of oscillatory diffusion (e.g., the local mode peak in the dynamic mobility and structure factor, and the transition between the oscillatory and diffusive regimes) is demonstrated.     

Approach to field-induced stationary state in a gas of hard rods

The Boltzmann equation describing one-dimensional motion of a charged hard rod in a neutral hard rod gas at temperatureT = 0 is solved. Under the action of a constant and uniform field the charged particle attains a stationary state. In the long time limit the velocity autocorrelation function decays via damped oscillations. In the reference system moving with the mean particle velocity the decay of fluctuations in the position space is governed (in the hydrodynamic limit) by the diffusion equation. Both the stationary current and the diffusion coefficient are proportional to the square root of the field. It is conjectured that this result also holds forT > 0 in a strong field limit.On leave from the Institute of Theoretical Physics, University of Warsaw, Hoza 69, 00-081 Warsaw, Poland.     

Velocity correlations in dense granular flows observed with internal imaging

We show that the velocity correlations in uniform dense granular flows inside a silo are similar to the hydrodynamic response of an elastic hard-sphere liquid. The measurements are made using a fluorescent refractive-index-matched interstitial fluid in a regime where the flow is dominated by grains in enduring contact and fluctuations scale with the distance traveled, independent of flow rate. The velocity autocorrelation function of the grains in the bulk shows a negative correlation at short time and slow oscillatory decay to zero similar to simple liquids. Weak spatial velocity correlations are observed over several grain diameters. The mean square displacements show an inflection point indicative of caging dynamics. The observed correlations are qualitatively different at the boundaries.     

Lorentz lattice gases: Basic theory

We present several ballistic models of the Lorentz gas in two-dimensional lattices with deterministic and stochastic deflection rules, and their corresponding Liouville equations. Boltzmann-level-equation results are obtained for the diffusion coefficient and velocity autocorrelation function for models with stochastic deflection rules. The long-time behavior of the mean square displacement is briefly discussed and the possibility of abnormal diffusion indicated. Even if the diffusion coefficient exists, its low-density limit may not be given correctly by the Boltzmann equation.     

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.     

A comparison of constant energy,constant temperature and constant pressure ensembles in molecular dynamics simulations of atomic liquids

Dynamic and static properties of the LJ fluid at a density and temperature close to the triple point are determined and compared for molecular dynamics computer simulations using (N, V, E), (N, V, T), (N, P, H) and (N, P, T) ensembles. As expected, the mean values of thermodynamic properties for all the different ensembles show good agreement. For the velocity autocorrelation function and the time dependence of the mean square displacement it is shown that the constant temperature and/or constant pressure algorithms used produce results which are identical, within statistical accuracy, to those obtained using the constant energy ensemble. The equations of motion are presented in a readily implementable form.     

Environment-dependent continuous time random walk

A generalized continuous time random walk model which is dependent on environmental damping is proposed in which the two key parameters of the usual random walk theory: the jumping distance and the waiting time, are replaced by two new ones: the pulse velocity and the flight time. The anomalous diffusion of a free particle which is characterized by the asymptotical mean square displacement <x²(t)>～t^α is realized numerically and analysed theoretically, where the value of the power index α is in a region of 0 < α < 2. Particularly, the damping leads to a sub-diffusion when the impact velocities are drawn from a Gaussian density function and the super-diffusive effect is related to statistical extremes, which are called rare-though-dominant events.     

Extrema statistics of Wiener-Einstein processes in one,two, and three dimensions

The maxima and first-passage-time statistics of Wiener-Einstein processes are evaluated analytically in one, two, and three dimensions. We show that the mean square maximum displacement has the same time dependence as the mean square displacement, i.e., it grows linearly with time. The ratio of the mean square maximum to the mean square displacement is shown to decrease with increasing dimensionality. We also calculate the mean first passage time for the process to attain a given absolute displacement and find that it grows as the square of the displacementand is independent of the dimensionality of the process. In addition, we evaluate the dispersion of maxima and of first passage times and discuss their dependence on dimensionality.Supported in part by the National Science Foundation under Grant CHE 75-20624.     

Statistical properties of intermittent diffusion in chaotic systems

Intermittent diffusion arises through tangent bifurcations from drifting periodic orbits in dynamical systems. We show the existence of infinite sequences of parameter values where intermittent diffusion sets in. These sequences are found to converge geometrically and their rate of convergence is determined. In continuous-time approximations we calculate the velocity autocorrelation function, its power spectrum, and the meansquare displacement. The spectrum exhibits excess noise (^–2) at low frequencies. The mean-square displacement grows liket ² below a crossover time.Intermittent diffusion occurs e.g. in driven Josephson junctions, where the excess noise can be detected easily. We show that quantities like the disorder parameter for the transition to intermittent chaos and the diffusion coefficient can be obtained directly from the voltage power spectrum.Work supported by Deutsche Forschungsgemeinschaft     

Anisotropic transport of microalgae Chlorella vulgaris in microfluidic channel

In this work, we study the regional dependence of transport behavior of microalgae Chlorella vulgaris inside microfluidic channel on applied fluid flow rate. The microalgae are treated as spherical naturally buoyant particles. Deviation from the normal diffusion or Brownian transport is characterized based on the scaling behavior of the mean square displacement(MSD) of the particle trajectories by resolving the displacements in the streamwise(flow) and perpendicular directions.The channel is divided into three different flow regions, namely center region of the channel and two near-wall boundaries and the particle motions are analyzed at different flow rates. We use the scaled Brownian motion to model the transitional characteristics in the scaling behavior of the MSDs. We find that there exist anisotropic anomalous transports in all the three flow regions with mixed sub-diffusive, normal and super-diffusive behavior in both longitudinal and transverse directions.     

Single-file diffusion in a box

We study diffusion of (fluorescently) tagged hard-core interacting particles of finite size in a finite one-dimensional system. We find an exact analytical expression for the tagged particle probability density function using a Bethe ansatz, from which the mean square displacement is calculated. The analysis shows the existence of three regimes of drastically different behavior for short, intermediate, and large times. The results are in excellent agreement with stochastic simulations (Gillespie algorithm).     

Ballistic diffusion induced by non-Gaussian noise

In this letter,we have analyzed the diffusive behavior of a Brownian particle subject to both internal Gaussian thermal and external non-Gaussian noise sources.We discuss two time correlation functions C(t) of the non-Gaussian stochastic process,and find that they depend on the parameter q,indicating the departure of the non-Gaussian noise from Gaussian behavior:for q ≤ 1,C(t) is fitted very well by the first-order exponentially decaying curve and approaches zero in the longtime limit,whereas for q 1,C(t) can be approximated by a second-order exponentially decaying function and converges to a non-zero constant.Due to the properties of C(t),the particle exhibits a normal diffusion for q ≤ 1,while for q 1 the non-Gaussian noise induces a ballistic diffusion,i.e.,the long-time mean square displacement of the free particle reads [x(t)-x(t)]2∝t2.