From subdiffusion to superdiffusion of particles on solid surfaces

Fickian yet non-Gaussian diffusion of a quasi-2D colloidal system in an optical speckle field: experiment and simulations

We investigate a quasi-2D suspension of Brownian particles in an optical speckle field produced by holographic manipulation of a laser wavefront. This system was developed to study, in a systematic and controllable way, a distinctive instance of diffusion, called Fickian yet Non Gaussian diffusion (FnGD), observed, during the last decade, for colloidal particles in a variety of complex and biological fluids. Our setup generates an optical speckle field that behaves like a disordered set of optical traps. First, we describe the experimental setup and the dynamics of the particles, focusing on mean square displacements, displacement distributions and kurtosis. Then, we present Brownian Dynamics simulations of point-like particles in a complex energy landscape, mimicking that generated by the optical speckle field. We show that our simulations can capture the salient features of the experimental results, including the emergence of FnGD, also covering times longer than the ones so far achieved in experiments. Some deviations are observed at long time only, with the Gaussian restoring being slower in simulations than in experiments. Overall, the introduced numerical model might be exploited to guide the design of upcoming experiments targeted, for example, to fully monitor the recovery of Gaussianity.

Diffusion Coefficient of a Brownian Particle in Equilibrium and Nonequilibrium: Einstein Model and Beyond

The diffusion of small particles is omnipresent in many processes occurring in nature. As such, it is widely studied and exerted in almost all branches of sciences. It constitutes such a broad and often rather complex subject of exploration that we opt here to narrow our survey to the case of the diffusion coefficient for a Brownian particle that can be modeled in the framework of Langevin dynamics. Our main focus centers on the temperature dependence of the diffusion coefficient for several fundamental models of diverse physical systems. Starting out with diffusion in equilibrium for which the Einstein theory holds, we consider a number of physical situations outside of free Brownian motion and end by surveying nonequilibrium diffusion for a time-periodically driven Brownian particle dwelling randomly in a periodic potential. For this latter situation the diffusion coefficient exhibits an intriguingly non-monotonic dependence on temperature.

Collective movement and thermal stability of fullerene clusters on the graphene layer

Understanding the motion characteristics of fullerene clusters on the graphene surface is critical for designing surface manipulation systems. Toward this purpose, using the molecular dynamics method, we evaluated six clusters of fullerenes including 1, 2, 3, 5, 10, and 25 molecules on the graphene surface, in the temperature range of 25 to 500 K. First, the surface motion of clusters is studied at 200 K and lower temperatures, in which fullerenes remain as a single group. The trajectories of the motion as well as the diffusion coefficients indicate the reduction of surface mobility as a response to the increase of the fullerene number. The clusters show normal diffusion at the temperature of 25 K, while they follow the super-diffusion regime at higher temperatures. The separation of fullerenes occurs at 300 K and higher temperatures. Due to the increase of vdW attraction with the increase of the fullerene number, the separation of fullerenes in larger clusters occurs at higher temperatures. The thermal energy at 500 K is sufficient to divide the large C60 clusters into smaller clusters. This energy level is related to the saturation of the interaction energy experienced by individual fullerenes, which can be estimated from the potential energy analysis. The results of simulations reveal that the separation occurs at the edge of clusters. Moreover, we studied the thermal stability of multilayer fullerene clusters on graphene. The simulation results indicate the tendency of multilayer clusters to locate on the surface, which implies the wetting property of C60s on the graphene layer.

Morphology of Nanometric Overlayers Made of Porphyrin-Type Molecules Physisorbed on Cellulose Iβ Crystals and Nanocrystals

Molecular dynamics simulations based on an atomistic empirical force field have been carried out to investigate structural, thermodynamic, and dynamical properties of adlayers made of porphyrin-type molecules physisorbed on surfaces of cellulose Iβ nanocrystals. The results show that low-index surfaces provide a thermally stable, weakly perturbing support for the deposition of non-hydrogen-bonded organic molecules. At submonolayer coverage, the discoidal porphyrin molecules lay flat on the surface, forming compact 2D clusters with clear elements of ordering. The adlayer grows layer-by-layer for the smallest porphyrin species on compact cellulose surfaces, while forming 3D clusters on a first relatively ordered adlayer (Stranski-Krastanov growth) in all other cases. The adsorption energy exceeds ∼1 eV per molecule, underlying the thermal stability of the adsorbate. Entropy plays a non-negligible role, destabilizing to some extent the adlayer. The in-plane dynamics of the smallest porphyrin species, i.e., porphine, on compact surfaces shows signs of superlubricity, due to the low energy and momentum exchange between the flat admolecule and the equally flat cellulose surface.

Stochastic resonance in periodically driven bistable systems subjected to anomalous diffusion

The occurrence of stochastic resonance in bistable systems undergoing anomalous diffusions, which arise from density-dependent fluctuations, is investigated with an emphasis on the analytical formulation of the problem as well as a possible analytical derivation of key quantifiers of stochastic resonance. The nonlinear Fokker–Planck equation describing the system dynamics, together with the corresponding Ito–Langevin equation, is formulated. In the linear response regime, analytical expressions of the spectral amplification, of the signal-to-noise ratio and of the hysteresis loop area are derived as quantifiers of stochastic resonance. These quantifiers are found to be strongly dependent on the parameters controlling the type of diffusion; in particular, the peak characterizing the signal-to-noise ratio occurs only in close ranges of parameters. Results introduce the relevant information that, taking into consideration the interactions of anomalous diffusive systems with a periodic signal, can provide a better understanding of the physics of stochastic resonance in bistable systems driven by periodic forces.

Directional locking effects for active matter particles coupled to a periodic substrate

Directional locking occurs when a particle moving over a periodic substrate becomes constrained to travel along certain substrate symmetry directions. Such locking effects arise for colloids and superconducting vortices moving over ordered substrates when the direction of the external drive is varied. Here we study the directional locking of run-and-tumble active matter particles interacting with a periodic array of obstacles. In the absence of an external biasing force, we find that the active particle motion locks to various symmetry directions of the substrate when the run time between tumbles is large. The number of possible locking directions depends on the array density and on the relative sizes of the particles and the obstacles. For a square array of large obstacles, the active particle only locks to the x, y, and 45^{∘} directions, while for smaller obstacles, the number of locking angles increases. Each locking angle satisfies θ=arctan(p/q), where p and q are integers, and the angle of motion can be measured using the ratio of the velocities or the velocity distributions in the x and y directions. When a biasing driving force is applied, the directional locking behavior is affected by the ratio of the self-propulsion force to the biasing force. For large biasing, the behavior resembles that found for directional locking in passive systems. For large obstacles under biased driving, a trapping behavior occurs that is nonmonotonic as a function of increasing run length or increasing self-propulsion force, and the trapping diminishes when the run length is sufficiently large.

Diffusion crossing over a barrier in a random rough metastable potential

We carry out a detailed study of escape dynamics of a particle driven by a white noise over a metastable potential corrugated by spatial disorder in the form of zero-mean random correlated potential. The approach of double-averaging over test particles and statistic ensemble is proposed to calculate the escape rate in a finite-size random rough metastable potential, moreover, the interference mechanism of test particles multi-passing over the saddle point is considered. Through analyzing the dependence of the steady escape rate on various modelled potentials and parameters, we demonstrate that the obstruction induced by roughness leads to a decrease in the steady escape rate with the increase of rough intensity. We also add the random correlated potential into the vicinity of the saddle-point of metastable potentials of three kinds and show an enhancement phenomenon of escape rate similar to the previous study of a surmounting fluctuating sharp barrier.

Enhanced diffusion with abnormal temperature dependence in underdamped space-periodic systems subject to time-periodic driving

We present a study of the diffusion enhancement of underdamped Brownian particles in a one-dimensional symmetric space-periodic potential due to external symmetric time-periodic driving with zero mean. We show that the diffusivity can be enhanced by many orders of magnitude at an appropriate choice of the driving amplitude and frequency. The diffusivity demonstrates abnormal (decreasing) temperature dependence at the driving amplitudes exceeding a certain value. At any fixed driving frequency Ω normal temperature dependence of the diffusivity is restored at low enough temperatures, T<T_{TAD}(Ω)-in contrast with the problem with constant external driving. At fixed temperature at small driving frequency the diffusivity either slowly decreases with Ω, or (at stronger driving) goes through a maximum near Ω_{2}, the reciprocal superdiffusion regime termination time. At high frequencies, between Ω_{2} and a fraction of the oscillation frequency at the potential minimum, the diffusivity is shown to decrease with Ω according to a power law, with the exponent related to the transient superdiffusion exponent. This behavior is found similar for the cases of sinusoidal in time and piecewise constant periodic ("square") driving.

Transport, diffusion, and energy studies in the Arnold-Beltrami-Childress map

We study the transport and diffusion properties of passive inertial particles described by a six-dimensional dissipative bailout embedding map. The base map chosen for the study is the three-dimensional incompressible Arnold-Beltrami-Childress (ABC) map chosen as a representation of volume preserving flows. There are two distinct cases: the two-action and the one-action cases, depending on whether two or one of the parameters (A,B,C) exceed 1. The embedded map dynamics is governed by two parameters (α,γ), which quantify the mass density ratio and dissipation, respectively. There are important differences between the aerosol (α<1) and the bubble (α>1) regimes. We have studied the diffusive behavior of the system and constructed the phase diagram in the parameter space by computing the diffusion exponents η. Three classes have been broadly classified-subdiffusive transport (η<1), normal diffusion (η≈1), and superdiffusion (η>1) with η≈2 referred to as the ballistic regime. Correlating the diffusive phase diagram with the phase diagram for dynamical regimes seen earlier, we find that the hyperchaotic bubble regime is largely correlated with normal and superdiffusive behavior. In contrast, in the aerosol regime, ballistic superdiffusion is seen in regions that largely show periodic dynamical behaviors, whereas subdiffusive behavior is seen in both periodic and chaotic regimes. The probability distributions of the diffusion exponents show power-law scaling for both aerosol and bubbles in the superdiffusive regimes. We further study the Poincáre recurrence times statistics of the system. Here, we find that recurrence time distributions show power law regimes due to the existence of partial barriers to transport in the phase space. Moreover, the plot of average particle kinetic energies versus the mass density ratio for the two-action case exhibits a devil's staircase-like structure for higher dissipation values. We explain these results and discuss their implications for realistic systems.

Using Diffusion To Characterize Interfacial Heterogeneity

We report on the use of molecular diffusional motion over a range of length scales to characterize compositional heterogeneity in monolayer structures. This work focuses on the diffusional motion of perylene in two types of films supported on functionalized silica surfaces: single-component (stearic acid) and two-component (hydrocarbon/fluorocarbon) films. Langmuir-Blodgett (LB) monolayers were deposited directly on silica or were bound to surface-modified silica by means of metal ion complexation. The LB films were characterized by their π-A isotherms and by Brewster angle microscopy (BAM) during formation and deposition. Chromophore mobility and monolayer structural heterogeneity were evaluated by comparing rotational diffusion data (fluorescence anisotropy decay imaging) and translational diffusion data (fluorescence recovery after photobleaching) on the same LB films. Our results indicate that the mobility of the chromophore depends sensitively on both metal ion identity and film composition.

Time-structuring in the evolution of 2D nanopatterns through interactions with substrate

Hydrophobic dodecanethiol capped gold nanoparticles (AuNPs) are found to self-assemble into two-dimensional patterns in monolayers of amphiphiles spread at the air-water interface of a Langmuir trough. In this communication we investigate the role of the nanoparticle-monolayer (FNMA) and monolayer-monolayer (FMMA) lipophilic attraction in influencing morphology and dynamics of AuNP cluster patterns in fatty acid monolayers. FNMA and FMMA are progressively varied by changing n, where n is the number of -CH2 groups in the alkyl tails of the amphiphilic fatty acid (CH3(CH2)nCOOH) molecules forming the monolayer. Compressibility measurements on the pristine and nanoparticle-laden monolayers show that, while the compressibility of the pristine monolayer decreases with increasing n, pointing to a progressive increase in FMMA, the effect of nanoparticles (increase in compressibility or lowering of FMMA) is discernible only for 14 < n < 22. The corresponding pattern morphology, observed with a Brewster Angle Microscope (BAM) at an in-plane resolution of 450 nm for 6 hours, reveals that there are essentially three stages in pattern evolution, lamellae of Au nanoclusters spread over the fatty-acid monolayer background (the λ state) followed by a network of nanoclusters with high node density (the ν state) and finally rings (circular/elongated) of random sizes with very low node density (the ρ state), evolving from an initial unsegregated state, without appreciable change in the average nanoparticle number density over the field of view. Increasing FNMA alongwith FMMA is found to shift a certain state to later times, thus playing the role of a viscous drag and introducing a delay in the timeline. The mean square fluctuation of BAM intensity remains flat and then decays as f(ξ) = ξ(2H) over smaller length scales, where ξ is the spatial separation and H the Hurst exponent. The study of f(ξ) over time reveals the growth of a sub-diffusive regime (H < 0.5) at the intermediate length scale, in almost all the films coinciding with the emergence of the ρ state. The growth of this sub-diffusive regime is slower for stronger FNMA and FMMA, the interactions thus acting as control parameters in dictating the time structure of the spatio-temporal patterns.

Brownian colloids in underdamped and overdamped regimes with nonhomogeneous temperature

The motion of Brownian particles when temperature is spatially dependent is studied by stochastic simulations and theoretical analysis. Nonequilibrium steady probability distributions P(st)(z,v) for both underdamped and overdamped regimes are analyzed. The existence of local kinetic energy equipartition theorem is also discussed. The transition between both regimes is characterized by a dimensionless friction parameter. This study is applied to three physical systems of colloidal particles.

Understanding and controlling regime switching in molecular diffusion

Diffusion can be strongly affected by ballistic flights (long jumps) as well as long-lived sticking trajectories (long sticks). Using statistical inference techniques in the spirit of Granger causality, we investigate the appearance of long jumps and sticks in molecular-dynamics simulations of diffusion in a prototype system, a benzene molecule on a graphite substrate. We find that specific fluctuations in certain, but not all, internal degrees of freedom of the molecule can be linked to either long jumps or sticks. Furthermore, by changing the prevalence of these predictors with an outside influence, the diffusion of the molecule can be controlled. The approach presented in this proof of concept study is very generic and can be applied to larger and more complex molecules. Additionally, the predictor variables can be chosen in a general way so as to be accessible in experiments, making the method feasible for control of diffusion in applications. Our results also demonstrate that data-mining techniques can be used to investigate the phase-space structure of high-dimensional nonlinear dynamical systems.

Hydrodynamic interactions induce anomalous diffusion under partial confinement

Under partial confinement, the motion of colloidal particles is restricted to a plane or a line but their dynamics is influenced by hydrodynamic interactions mediated by the unconfined, three-dimensional flow of the embedding fluid. We demonstrate that this dimensionality mismatch induces a characteristic divergence in the collective diffusion coefficient of the colloidal subsystem. This result, independent of the specific interparticle forces in the colloid, is solely due to the kinematical constraint on the colloidal particles, and it is different from the known divergence of transport coefficients in purely one or two-dimensional fluids.

Single-molecule diffusion in a periodic potential at a solid-liquid interface

We used single-molecule tracking experiments to observe the motion of small hydrophobic fluorescent molecules at the interface between water and a solid surface that exhibited periodic chemical patterns. The dynamics were characterized by non-ergodic, continuous time random walk statistics. The step-size distributions displayed enhanced probability of steps to periodic distances, consistent with theoretical predictions for diffusion in an atomic/molecular scale periodic potential. Surprisingly, this general behavior was observed here for surfaces exhibiting characteristic length scales three orders of magnitude larger than atomic/molecular dimensions, and may provide a new way to understand and control solid-liquid interfacial diffusion for molecular targeting applications.

Brownian particles on rough substrates: relation between intermediate subdiffusion and asymptotic long-time diffusion

Brownian particles in random potentials show an extended regime of subdiffusive dynamics at intermediate times. The asymptotic diffusive behavior is often established at very long times and thus cannot be accessed in experiments or simulations. For the case of one-dimensional random potentials with Gaussian distributed energies, we present a detailed analysis of experimental and simulation data. It is shown that the asymptotic long-time diffusion coefficient can be related to the behavior at intermediate times, namely, the minimum of the exponent that characterizes subdiffusion and hence corresponds to the maximum degree of subdiffusion. As a consequence, investigating only the dynamics at intermediate times is sufficient to predict the order of magnitude of the long-time diffusion coefficient and the time scale at which the crossover from subdiffusion to diffusion occurs, i.e., when the long-time diffusive regime and hence thermal equilibrium is established. Inversely, theoretical predictions derived for the asymptotic long-time behavior can be used to quantitatively characterize the intermediate behavior, which hardly has been studied so far.

Particle dynamics in two-dimensional random-energy landscapes: experiments and simulations

The dynamics of individual colloidal particles in random potential energy landscapes was investigated experimentally and by Monte Carlo simulations. The value of the potential at each point in the two-dimensional energy landscape follows a Gaussian distribution. The width of the distribution, and hence the degree of roughness of the energy landscape, was varied and its effect on the particle dynamics studied. This situation represents an example of Brownian dynamics in the presence of disorder. In the experiments, the energy landscapes were generated optically using a holographic setup with a spatial light modulator, and the particle trajectories were followed by video microscopy. The dynamics is characterized using, e.g., the time-dependent diffusion coefficient, the mean squared displacement, the van Hove function, and the non-Gaussian parameter. In both experiments and simulations the dynamics is initially diffusive, showing an extended subdiffusive regime at intermediate times before diffusive motion is recovered at very long times. The dependence of the long-time diffusion coefficient on the width of the Gaussian distribution agrees with theoretical predictions. Compared to the dynamics in a one-dimensional potential energy landscape, the localization at intermediate times is weaker and the diffusive regime at long times reached earlier, which is due to the possibility to avoid local maxima in two-dimensional energy landscapes.

First-passage properties of molecular spiders

Molecular spiders are synthetic catalytic DNA-based nanoscale walkers. We study the mean first-passage time for abstract models of spiders moving on a finite two-dimensional lattice with various boundary conditions and compare it with the mean first-passage time of spiders moving on a one-dimensional track. We evaluate by how much the slowdown on newly visited sites, owing to catalysis, can improve the mean first-passage time of spiders and show that in one dimension, when both ends of the track are an absorbing boundary, the performance gain is lower than in two dimensions, when the absorbing boundary is a circle; this persists even when the absorbing boundary is a single site.

Internal degrees of freedom and transport of benzene on graphite

In this paper, the chaotic internal degrees of freedom of a benzene molecule adsorbed on a graphite substrate, their interplay with thermal noise, and their effects on the diffusion and drift are investigated analytically by making use of the presence of two different time scales as well as by molecular-dynamics simulations. The effects of thermal noise are investigated, and it is found that noise does not significantly alter the dynamics of the internal degrees of freedom yet does affect the friction and diffusion of the center of mass. Qualitative and quantitative theoretical predictions for the friction and diffusion of the molecule on the substrate are made and are compared to molecular-dynamics simulations. Contributions to the friction and diffusion from the finite heat bath as well as the slow dynamics of the center of mass are formally identified. It is shown that the torsion in benzene, which dominates the nonlinear coupling, significantly affects the friction of the molecule on the surface. The results compare favorably with recent results from He-neutron spin echo experiments on this system. Based on the analytical and numerical results, some suggestions are made for experimental conditions under which the effects of internal degrees of freedom might be observable.

Weak disorder: anomalous transport and diffusion are normal yet again

We carry out a detailed study of the motion of particles driven by a constant external force over a landscape consisting of a periodic potential corrugated by a small amount of spatial disorder. We observe anomalous behavior in the form of subdiffusion and superdiffusion and even subtransport over very long time scales. Recent studies of transport over slightly random landscapes have focused only on parameters leading to normal behavior, and while enhanced diffusion has been identified when the external force approaches the critical value associated with the transition from locked to running solutions, the regime of anomalous behavior had not been recognized. We provide a qualitative explanation for the origin of these anomalies, and make connections with a continuous time random walk approach.

Mechanism of diffusive transport in molecular spider models

Recent advances in single-molecule chemistry have led to designs for artificial multipedal walkers that follow tracks of chemicals. We investigate the motion of a class of walkers, called molecular spiders, which consist of a rigid chemically inert body and several flexible enzymatic legs. The legs can reversibly bind to chemical substrates on a surface and through their enzymatic action convert them to products. The legs can also reversibly bind to products, but at a different rate. Antal and Krapivsky have proposed a model for molecular spider motion over regular one-dimensional lattices [T. Antal and P. L. Krapivsky, Phys. Rev. E 76, 021121 (2007).]. In the model the legs hop from site to site under constraints imposed by connection to a common body. The first time a leg visits a site, the site is an uncleaved substrate, and the leg hops from this site only once it has cleaved it into a product. This cleavage happens at a rate r<1, slower than dissociation from a product site, r=1. The effect of cleavage is to slow down the hopping rate for legs that visit a site for the first time. Along with the constraints imposed on the legs, this leads to an effective bias in the direction of unvisited sites that decreases the average time needed to visit n sites. The overall motion, however, remains diffusive in the long time limit. We have reformulated the Antal-Krapivsky model as a continuous-time Markov process and simulated many traces of this process using kinetic Monte Carlo techniques. Our simulations show a previously unpredicted transient behavior wherein spiders with small r values move superdiffusively over significant distances and times. We explain this transient period of superdiffusive behavior by describing the spider process as switching between two metastates: a diffusive state D wherein the spider moves in an unbiased manner over previously visited sites, and a boundary state B wherein the spider is on the boundary between regions of visited and unvisited sites and experiences a bias in the direction of unvisited sites. We show that while the spider remains in the B state it moves ballistically in the direction of unvisited sites, and while the spider is in the D state it moves diffusively. The relative amount of time the spider spends in the two states determines how superdiffusively the spider moves. We show that the B state is Markovian, but the D state is non-Markovian because the duration of a D period depends on how many sites have been visited previously. As time passes the spider spends progressively more time in the D state (moving diffusively) and less time in the B state (moving ballistically). This explains both the transient superdiffusive motion and the eventual decay to diffusive motion as t→∞.

Diffusive motion of C60 on a graphene sheet

The motion of a C60 molecule over a graphene sheet at finite temperature is investigated both theoretically and computationally. We show that a graphene sheet generates a van der Waals laterally periodic potential, which directly influences the motion of external objects in its proximity. The translational motion of a C60 molecule near a graphene sheet is found to be diffusive in the lateral directions, while in the perpendicular direction, the motion may be described as diffusion in an effective harmonic potential which is determined from the distribution function of the position of the C60 molecule. We also examine the rotational diffusion of C60 and show that its motion over the graphene sheet is not a rolling motion.

Transient anomalous subdiffusion: effects of specific and nonspecific probe binding with actin gels

When signaling molecules diffuse through the cytosol, they encounter a wide variety of obstacles that hinder their mobility in space and time. Some of those factors include, but are not limited to, interactions with mobile and immobile targets or obstacles. Besides finding a crowded environment inside the cell, macromolecules assemble into molecular complexes that drive specific biological functions adding additional complexity to their diffusion. Thus, simple models of diffusion often fail to explain mobility through the cell interior, and new approaches are needed. Here we used fluorescent correlation spectroscopy to measure diffusion of three molecules of similar size with different surface properties diffusing in actin gels. The fluorescent probes were (a) quantum dots, (b) yellow-green fluorescent spheres, and (c) the beta isoform of Ca(2+) calmodulin-dependent protein kinase II tagged with green fluorescent protein. We compared various models for fitting the autocorrelation function (ACF) including single component, two-component, and anomalous diffusion. The two-component and anomalous diffusion models were superior and were largely indistinguishable based on a goodness of fit criteria. To better resolve differences between these two models, we modified the ACF to observe temporal variations in diffusion. We found in both simulated and experimental data a transient anomalous subdiffusion between two freely diffusing regimes produced by binding interactions of the diffusive tracers with actin gels.

Dispersionless motion in a periodically rocked periodic potential

Recently, dispersionless (coherent) motion of (noninteracting) massive Brownian particles, at intermediate time scales, was reported in a sinusoidal potential with a constant tilt. The coherent motion persists for a finite length of time before the motion becomes diffusive. We show that such coherent motion can be obtained repeatedly by applying an external zero-mean square-wave drive of appropriate period and amplitude instead of a constant tilt. Thus, the cumulative duration of coherent motion of particles is prolonged. Moreover, by taking an appropriate combination of periods of the external field, one can postpone the beginning of the coherent motion and can even have coherent motion at a lower value of position dispersion than in the constant tilt case.

Transport and diffusion in the embedding map

We study the transport properties of passive inertial particles in two-dimensional (2D) incompressible flows. Here, the particle dynamics is represented by the four-dimensional dissipative embedding map of the 2D area-preserving standard map which models the incompressible flow. The system is a model for impurity dynamics in a fluid and is characterized by two parameters, the inertia parameter alpha and the dissipation parameter gamma . The aerosol regime, where the particles are denser than the fluid, and the bubble regime, where they are less dense than the fluid, correspond to the parameter regimes alpha>1 and alpha<1 , respectively. Earlier studies of this system show a rich phase diagram with dynamical regimes corresponding to periodic orbits, chaotic structures, and mixed regimes. We obtain the statistical characterizers of transport for this system in these dynamical regimes. These are the recurrence time statistics, the diffusion exponent, and the distribution of jump lengths. The recurrence time distribution shows a power-law tail in the dynamical regimes, where there is preferential concentration of particles in sticky regions of the phase space, and an exponential decay in mixing regimes. The diffusion exponent shows behavior of three types-normal, subdiffusive, and superdiffusive, depending on the parameter regimes. Phase diagrams of the system are constructed to differentiate different types of diffusion behavior, as well as the behavior of the absolute drift. We correlate the dynamical regimes seen for the system at different parameter values with the transport properties observed at these regimes and in the behavior of the transients. This system also shows the existence of a crisis and unstable dimension variability at certain parameter values. The signature of the unstable dimension variability is seen in the statistical characterizers of transport. We discuss the implications of our results for realistic systems.

Diffusion in two-dimensional colloidal systems on periodic substrates

We study the diffusive behavior of two-dimensional charged colloidal suspensions subjected to a sinusoidal substrate by means of Brownian dynamics simulations. We mainly focus on the dependence of the mean-square displacement on the substrate strength. Our findings show a variation in the particle diffusion due to a substrate-induced distortion of the dynamic cage of nearest-neighbor colloids. This mechanism leads to a transition from normal diffusion at short times to subdiffusion on intermediate time scales. However, at long times normal diffusion is recovered. We also show that the variation in the long-time self-diffusion coefficient may be associated with the freezing and re-entrant melting transitions.

Brownian particles in random and quasicrystalline potentials: how they approach the equilibrium

We study the Brownian motion of an ensemble of single colloidal particles in a random square and a quasicrystalline potential when they start from non-equlibrium. For both potentials, Brownian dynamics simulations reveal a widespread subdiffusive regime before the diffusive long-time limit is reached in thermal equilibrium. We develop a random trap model based on a distribution for the depths of trapping sites that reproduces the results of the simulations in detail. Especially, it gives analytic formulas for the long-time diffusion constant and the relaxation time into the diffusive regime. Aside from detailed differences, our work demonstrates that quasicrystalline potentials can be used to mimic aspects of random potentials.

Aging correlation functions of the interrupted fractional Fokker-Planck propagator

The authors explore aging in a general semi-Markov process with arbitrary waiting time distributions and discuss the role of trapping phenomenon in glasses in determining the waiting time distributions. In certain limits, they obtain a two-time propagator for the fractional Fokker-Planck equation through coordination, but one can generalize the construction to incorporate cutoffs in the aging of the system, i.e., the interrupted aging. The construction allows the incorporation of cutoffs in the aging of the system, and the exploration of signatures of aging effects in systems that eventually achieve equilibrium. Cutoffs in aging effects are illustrated for interrupted fractional diffusion in a harmonic potential and for the on-time probability of a quantum dot system.

Dispersionless transport in a washboard potential

We study and characterize a new dynamical regime of underdamped particles in a tilted washboard potential. We find that for small friction in a finite range of forces the particles move essentially nondispersively, that is, coherently, over long intervals of time. The associated distribution of the particle positions moves at an essentially constant velocity and is far from Gaussian-like. This new regime is complementary to, and entirely different from, well-known nonlinear response and large dispersion regimes observed for other values of the external force.

Target problem with evanescent subdiffusive traps

We calculate the survival probability of a stationary target in one dimension surrounded by diffusive or subdiffusive traps of time-dependent density. The survival probability of a target in the presence of traps of constant density is known to go to zero as a stretched exponential whose specific power is determined by the exponent that characterizes the motion of the traps. A density of traps that grows in time always leads to an asymptotically vanishing survival probability. Trap evanescence leads to a survival probability of the target that may go to zero or to a finite value indicating a probability of eternal survival, depending on the way in which the traps disappear with time.

Anomalous diffusion in periodic potentials under self-similar colored noise

We present numerical studies of anomalous diffusion in periodic potentials by simulating a generalized Langevin equation. It is proved that the particle driven by a thermal colored noise with the spectral density vanishing at zero frequency allows superdiffusive motion. It is found that the system subjected to sub- or superohmic damping exhibits two motion modes in a corrugated plane: running oscillated state and mixed running and oscillating states, respectively. Induced, the anomalous power can be enhanced up twice for the latter case and thus a wide range of diffusive regimes is observed with changing titled force.

Analytical approach to sorting in periodic and random potentials

There has been a recent revolution in the ability to manipulate micrometer-sized objects on surfaces patterned by traps or obstacles of controllable configurations and shapes. One application of this technology is to separate particles driven across such a surface by an external force according to some particle characteristic such as size or index of refraction. The surface features cause the trajectories of particles driven across the surface to deviate from the direction of the force by an amount that depends on the particular characteristic, thus leading to sorting. While models of this behavior have provided a good understanding of these observations, the solutions have so far been primarily numerical. In this paper we provide analytic predictions for the dependence of the angle between the direction of motion and the external force on a number of model parameters for periodic as well as random surfaces. We test these predictions against exact numerical simulations.

Time-dependent diffusion in a random correlated potential

Diffusive behavior of a particle in a two-dimensional random correlated potential with Gaussian distribution and exponential correlation is investigated via Langevin simulation. Our results show that superdiffusion appears only in the early period of the time of evolution and there does not exist an intermediate time for the occurrence of the whole issue from subdiffusion to superdiffusion. Whether the asymptotic situation of the particle could be arrived before the simulation stops is strongly influenced by the finite-size effect of the random correlated potential simulated. By applying the random correlated potential to the decay of a metastable system, we find that the escape rate of a particle is decreased by hill effect of the random potential.

Trapping reactions with subdiffusive traps and particles characterized by different anomalous diffusion exponents

A number of results for reactions involving subdiffusive species, all with the same anomalous exponent , have recently appeared in the literature and can often be understood in terms of a subordination principle whereby time in ordinary diffusion is replaced by tgamma. However, very few results are known for reactions involving different species characterized by different anomalous diffusion exponents. Here we study the reaction dynamics of a (sub)diffusive particle surrounded by a sea of (sub)diffusive traps in one dimension. We find rigorous results for the asymptotic survival probability of the particle in most cases, with the exception of the case of a particle that diffuses normally while the anomalous diffusion exponent of the traps is smaller than 2/3.

Effect of hydrodynamic flow on low-field spin-lattice relaxation in liquids in the nanoscopic vicinity of solid surfaces: theory and Monte Carlo simulations of model pore spaces

It is shown that slow hydrodynamic flow with velocities of a few millimeters per second reduces the spin-lattice relaxation rate of fluids confined to pores of a diamagnetic, polar, solid material. The effect is predicted by an analytical theory and Monte Carlo simulations of model pore spaces. Adsorbate molecules diffusing in the vicinity of pore surfaces can perform adsorption, desorption, and readsorption cycles, effectively leading to displacements along the surface (also termed "bulk mediated surface diffusion" or BMSD). Since the surface determines the orientation of the adsorbed molecule relative to the external magnetic field, desorption at one site and readsorption at another site of a nonplanar surface will cause molecular reorientation. This is the basis of the "reorientation mediated by translational displacements" (RMTD) relaxation mechanism. If hydrodynamic flow is superimposed on diffusion, the RMTD process will be accelerated in a sort of rotational analog to translational hydrodynamic (or Taylor-Aris) dispersion. This reveals itself by a prolongation of spin-lattice relaxation times at low frequencies. The flow-relaxation effect takes place in the vicinity of the pore surfaces on the order of nanometers. The conclusions are (i) the BMSD and RMTD relaxation mechanism of fluids in porous materials is corroborated, (ii) hydrodynamic dispersion affects molecular displacements at surfaces, and (iii) interfacial slip in the sense of a molecular hopping, i.e., a desorption-readsorption process takes place.

Sorting on periodic surfaces

Particles moving on crystalline surfaces and driven by external forces or flow fields can acquire velocities along directions that deviate from that of the external force. This effect depends upon the characteristics of the particles, most notably particle size or particle index of refraction, and can therefore be (and has been) used to sort different particles. We introduce a simple model for particles subject to thermal fluctuations and moving in appropriate potential landscapes. Numerical results are compared to recent experiments on landscapes produced with holographic optical tweezers and microfabricated technology. Our approach clarifies the relevance of different parameters, the direction and magnitude of the external force, particle size, and temperature.