Characterisation of a three-dimensional Brownian motor in optical lattices

We present here a detailed study of the behaviour of a three dimensional Brownian motor based on cold atoms in a double optical lattice [P. Sj?lund et al., Phys. Rev. Lett. 96, 190602 (2006)]. This includes both experiments and numerical simulations of a Brownian particle. The potentials used are spatially and temporally symmetric, but combined spatiotemporal symmetry is broken by phase shifts and asymmetric transfer rates between potentials. The diffusion of atoms in the optical lattices is rectified and controlled both in direction and speed along three dimensions. We explore a large range of experimental parameters, where irradiances and detunings of the optical lattice lights are varied within the dissipative regime. Induced drift velocities in the order of one atomic recoil velocity have been achieved.     

Transient anomalous dispersion in random walkers

A simple model of dispersive tracers which display a transient anomalous regime is presented. It is based on an ensemble of random walkers belonging to two independent populations characterized by different Lagrangian decorrelation times. Apart from short-time ballistic and long-time diffusive behavior, the dispersion shows anomalous scaling at intermediate times over a wide range of variability for the free parameters of the model.     

Brownian motion at short time scales

Brownian motion has played important roles in many different fields of science since its origin was first explained by Albert Einstein in 1905. Einstein's theory of Brownian motion, however, is only applicable at long time scales. At short time scales, Brownian motion of a suspended particle is not completely random, due to the inertia of the particle and the surrounding fluid. Moreover, the thermal force exerted on a particle suspended in a liquid is not a white noise, but is colored. Recent experimental developments in optical trapping and detection have made this new regime of Brownian motion accessible. This review summarizes related theories and recent experiments on Brownian motion at short time scales, with a focus on the measurement of the instantaneous velocity of a Brownian particle in a gas and the observation of the transition from ballistic to diffusive Brownian motion in a liquid.     

Velocity autocorrelation function of a Brownian particle

In this article, we present molecular dynamics study of the velocity autocorrelation function (VACF) of a Brownian particle. We compare the results of the simulation with the exact analytic predictions for a compressible fluid from [T.S. Chow, J.J. Hermans, Physica 65, 156 (1973)] and an approximate result combining the predictions from hydrodynamics at short and long times. The physical quantities which determine the decay were determined from separate bulk simulations of the Lennard-Jones fluid at the same thermodynamic state point. We observe that the long-time regime of the VACF compares well the predictions from the macroscopic hydrodynamics, but the intermediate decay is sensitive to the viscoelastic nature of the solvent.     

Spatial diffusion in a periodic optical lattice: revisiting the Sisyphus effect

We numerically study the spatial diffusion of an atomic cloud experiencing Sisyphus cooling in a three-dimensional linlin optical lattice in a broad range of lattice parameters. In particular, we investigate the dependence on the size of the lattice sites which changes with the angle between the laser beams. We show that the steady-state temperature is largely independent of the lattice angle, but that the spatial diffusion changes significantly. It is shown that the numerical results fulfill the Einstein relations of Brownian motion in the jumping regime as well as in the oscillating regime. We finally derive an effective Brownian motion model from first principles which gives good agreement with the simulations. Received 8 August 2001 and Received in final form 6 November 2001     

Fluctuations and correlations in wave transport through complex media

Intensity correlations between waves reflected and transmitted through random media are analyzed within the framework of the random matrix theory of transport. We introduce a new kind of correlations between transmitted and reflected intensities. For transmitted waves, the spatial correlation function can be expressed as the sum of three terms, with distinctive spatial dependences. This result coincides with the one obtained in the diffusive regime from perturbative calculations, but holds all the way from quasi-ballistic transport to localization.     

The localization transition of the two-dimensional Lorentz model

We investigate the dynamics of a single tracer particle performing Brownian motion in a two-dimensional course of randomly distributed hard obstacles. At a certain critical obstacle density, the motion of the tracer becomes anomalous over many decades in time, which is rationalized in terms of an underlying percolation transition of the void space. In the vicinity of this critical density the dynamics follows the anomalous one up to a crossover time scale where the motion becomes either diffusive or localized. We analyze the scaling behavior of the time-dependent diffusion coefficient D(t) including corrections to scaling. Away from the critical density, D(t) exhibits universal hydrodynamic long-time tails both in the diffusive as well as in the localized phase.     

Central Limit Theorem for Branching Brownian Motions in Random Environment

We introduce a model of branching Brownian motions in time-space random environment associated with the Poisson random measure. We prove that, if the randomness of the environment is moderated by that of the Brownian motion, the population density satisfies a central limit theorem and the growth rate of the population size is the same as its expectation with strictly positive probability. We also characterize the diffusive behavior of our model in terms of the decay rate of the replica overlap. On the other hand, we show that, if the randomness of the environment is strong enough, the growth rate of the population size is strictly less than its expectation almost surely. To do this, we use a connection between our model and the model of Brownian directed polymers in random environment introduced by Comets and Yoshida. Partly supported by the Global COE program at Department of Mathematics and Research Institute for Mathematical Sciences, Kyoto University.     

长时相干效应与分形布朗运动

 我们研究了阻尼布朗粒子，在具有幂律长时相干C(t)~t^-β（0<β<1，1<β<2）的无规涨落力作用下的运动情况。我们发现它是作分形布朗运动，而不是作普通的布朗运动，而且，找出了分形布朗运动的有效Fokker-Planck方程，以及相应的精确解。于是第一次把长时相干效应和分形布朗运动建立了定量的联系。     

Freezing and melting of a colloidal adsorbate on a 1D quasicrystalline substrate

Using Monte Carlo simulations and an extended Landau-Alexander-McTague theory, we demonstrate that colloids in a one-dimensional quasicrystalline potential order in triangular and rhombic-alpha crystalline phases. Increasing the strength of the potential further, a new type of light-induced melting is discovered that has its origin in the nonperiodicity of the potential. In contrast to reentrant melting in periodic potentials, the quasicrystalline potential melts the crystalline phases even when they already exist at zero potential.     

Brownian motion in crystals with topological defects

The diffusion behaviour of a Brownian particle in a crystal with randomly distributed topological defects is analyzed by means of the continuum theory of defects combined with the theory of diffusion on manifolds. A path-integral representation of the diffusion process is used for the calculation of cumulants of the particle position averaged over a defect ensemble. For a random distribution of disclinations the problem of Brownian motion reduces to a known random-drift problem. Depending on the properties of the disclination ensemble, this yields various types of subdiffusional behaviour. In a random array of parallel screw dislocations one finds a normal, but anisotropic, diffusion behaviour of the mean-square displacement. Moreover, the process turns out to be non-Gaussian, and reveals long-time tails in the higher-order cumulants.Dedicated to Professor Herbert Wagner on the occasion of his 60th birthday     

Universality and diversity in a phonon-transmission histogram of isotope-disordered carbon nanotubes

Universal fluctuations in phonon transmission and other features of phonon-transmission histograms are investigated by performing numerical simulations of coherent-phonon transport in isotope-disordered carbon nanotubes. Interestingly, the phonon-transmission fluctuation in the diffusive regime is universal, irrespective of the average phonon transmission, the tube chirality, and the concentrations, and masses of isotopes. We also find that the histogram, which has a Gaussian distribution in the diffusive regime, has a log-normal distribution in the localization regime.     

Chaos out of internal noise in the collective dynamics of diffusively coupled cells

We study the transition from stochasticity to determinism in calcium oscillations via diffusive coupling of individual cells that are modeled by stochastic simulations of the governing reaction-diffusion equations. As expected, the stochastic solutions gradually converge to their deterministic limit as the number of coupled cells increases. Remarkably however, although the strict deterministic limit dictates a fully periodic behavior, the stochastic solution remains chaotic even for large numbers of coupled cells if the system is set close to an inherently chaotic regime. On the other hand, the lack of proximity to a chaotic regime leads to an expected convergence to the fully periodic behavior, thus suggesting that near-chaotic states are presently a crucial predisposition for the observation of noise-induced chaos. Our results suggest that chaos may exist in real biological systems due to intrinsic fluctuations and uncertainties characterizing their functioning on small scales.     

Center-of-mass tomographic approach to quantum dynamics

In the tomography representation we propose a new approach, which describes the dynamics of quantum particles by the Kolmogorov equations for non-negative propagators. To solve the Kolmogorov equations we use a diffusive Markovian random processes described by the related nonlinear stochastic Langevin equations. As a result the dynamics of quantum particles is described by the proposed numerical scheme combining both Langevin dynamics and Monte Carlo methods. We test the developed approach by applying it to the wave packet dynamics in harmonic potentials and to particle tunneling through a barrier.     

Demonstration of a controllable three-dimensional Brownian motor in symmetric potentials

We demonstrate a Brownian motor, based on cold atoms in optical lattices, where isotropic random fluctuations are rectified in order to induce controlled atomic motion in arbitrary directions. In contrast to earlier demonstrations of ratchet effects, our Brownian motor operates in potentials that are spatially and temporally symmetric, but where spatiotemporal symmetry is broken by a phase shift between the potentials and asymmetric transfer rates between them. The Brownian motor is demonstrated in three dimensions and the noise-induced drift is controllable in our system.     

A Brownian motion version of the directed polymer problem

Consider a Brownian particle in three dimensions in a random environment. The environment is determined by a potential random in space and time. It is shown that at small noise the large-time behavior of the particle is diffusive. The diffusion constant depends on the environment. This work generalizes previous results for random walk in a random environment. In these results the diffusion constant does not depend on the environment.     

Anisotropic diffusion of a magnetically torqued ellipsoidal microparticle

We study the dynamics of an overdamped paramagnetic ellipsoidal particle confined above a plane and subjected to an external rotating magnetic field. Without magnetic forcing, the Brownian ellipsoid exhibits a crossover from a short time anisotropic diffusion to a long time isotropic one. Application of an external static or rotating magnetic field enables controlling and varying the crossover time depending on the field frequency and amplitude. We combine analytical results and numerical simulations in order to explore the diffusive properties of the forced ellipsoid.     

Friction tensor for a pair of Brownian particles: Spurious finite-size effects and molecular dynamics estimates

In this work, we show thatin any finite system, the binary friction tensor for two Brownian particlescannot be directly estimated from an evaluation of the microscopic Green-Kubo formula, involving the time integral of force-force autocorrelation functions. This pitfall is associated with a subtle inversion of the thermodynamic and long-time limits and leads to spurious results for the estimates of the friction matrix based on molecular dynamics simulations. Starting from a careful analysis of the coupled Langevin equations for two interacting Brownian particles, we derive a method to circumvent these effects and extract the binary friction tensor from the correlation function matrix of the instantaneous forces exerted by the bath particles on the fixed Brownian particles, and from the relaxation of the total momentum of the bath in afinite system. The general methodology is applied to the case of two hard or soft Brownian spheres in a bath of light particles. Numerical estimates of the relevant correlation functions and of the resulting self and mutual components of the matrix of friction tensors are obtained by molecular dynamics simulations for various spacings between the Brownian particles. This paper is dedicated to B. Jancovici on the occassion of his 65th birthday.