Self-diffusion in granular gases

Anomalous diffusion of scaled Brownian tracers

A model for anomalous transport of tracer particles diffusing in complex media in two dimensions is proposed. The model takes into account the characteristics of persistent motion that an active bath transfers to the tracer; thus, the model proposed here extends active Brownian motion, for which the stochastic dynamics of the orientation of the propelling force is described by scaled Brownian motion (sBm), identified by time-dependent diffusivity of the form D_{β}∝t^{β-1}, β>0. If β≠1, sBm is highly nonstationary and suitable to describe such nonequilibrium dynamics induced by complex media. In this paper, we provide analytical calculations and computer simulations to show that genuine anomalous diffusion emerges in the long-time regime, with a time scaling of the mean-squared displacement t^{2-β}, while ballistic transport t^{2}, characteristic of persistent motion, is found in the short-time regime. We also analyze the time dependence of the kurtosis, and the intermediate scattering function of the position distribution, as well as the propulsion autocorrelation function, which defines the effective persistence time.

Diffusion of intruders in granular suspensions: Enskog theory and random walk interpretation

The Enskog kinetic theory is applied to compute the mean square displacement of impurities or intruders (modeled as smooth inelastic hard spheres) immersed in a granular gas of smooth inelastic hard spheres (grains). Both species (intruders and grains) are surrounded by an interstitial molecular gas (background) that plays the role of a thermal bath. The influence of the latter on the motion of intruders and grains is modeled via a standard viscous drag force supplemented by a stochastic Langevin-like force proportional to the background temperature. We solve the corresponding Enskog-Lorentz kinetic equation by means of the Chapman-Enskog expansion truncated to first order in the gradient of the intruder number density. The integral equation for the diffusion coefficient is solved by considering the first two Sonine approximations. To test these results, we also compute the diffusion coefficient from the numerical solution of the inelastic Enskog equation by means of the direct simulation Monte Carlo method. We find that the first Sonine approximation generally agrees well with the simulation results, although significant discrepancies arise when the intruders become lighter than the grains. Such discrepancies are largely mitigated by the use of the second Sonine approximation, in excellent agreement with computer simulations even for moderately strong inelasticities and/or dissimilar mass and diameter ratios. We invoke a random walk picture of the intruders' motion to shed light on the physics underlying the intricate dependence of the diffusion coefficient on the main system parameters. This approach, recently employed to study the case of an intruder immersed in a granular gas, also proves useful in the present case of a granular suspension. Finally, we discuss the applicability of our model to real systems in the self-diffusion case. We conclude that collisional effects may strongly impact the diffusion coefficient of the grains.

Universal cover-time distributions of random motion in bounded granular gases

The exhaustive random exploration of a complex domain is a fundamental issue in many natural, social, and engineering systems. The key characterizing quantity is the cover time, which is the time to visit every site in the system. One prototypical experimental platform is the confined granular gas, where the random motion of granular particles mimics the wandering of random walkers in a confined region. Here, we investigate the cover-time distribution of the random motion of tracer particles in granular gases confined in four containers to account for different boundary and angle effects and examine whether the cover time of the heterogeneous random motion of the granular gases can be rescaled into the universal Gumbel distribution according to a recent theory [Dong et al., arXiv:2210.05122 (2022)]. It is found that for long cover times, the experimental results are in full accord, while for short cover times, the agreement is reasonable, with noticeable deviations that can be attributed to spatial correlations of the sites in the covering process. Our results, thus, call for further theoretical investigations in order to take into full account these nonideal issues.

Anomalous diffusion, aging, and nonergodicity of scaled Brownian motion with fractional Gaussian noise: overview of related experimental observations and models

How does a systematic time-dependence of the diffusion coefficient $D (t)$ affect the ergodic and statistical characteristics of fractional Brownian motion (FBM)? Here, we examine how the behavior of the...

Temperature distribution in driven granular mixtures does not depend on mechanism of energy dissipation

We study analytically and numerically the distribution of granular temperatures in granular mixtures for different dissipation mechanisms of inelastic inter-particle collisions. Both driven and force-free systems are analyzed. We demonstrate that the simplified model of a constant restitution coefficient fails to predict even qualitatively a granular temperature distribution in a homogeneous cooling state. At the same time we reveal for driven systems a stunning result - the distribution of temperatures in granular mixtures is universal. That is, it does not depend on a particular dissipation mechanism of inter-particles collisions, provided the size distributions of particles is steep enough. The results of the analytic theory are compared with simulation results obtained by the direct simulation Monte Carlo (DSMC). The agreement between the theory and simulations is perfect. The reported results may have important consequences for fundamental science as well as for numerous application, e.g. for the experimental modelling in a lab of natural processes.

Early-stage aggregation in three-dimensional charged granular gas

Neutral grains made of the same dielectric material can attain considerable charges due to collisions and generate long-range interactions. We perform molecular dynamic simulations in three dimensions for a dilute, freely cooling granular gas of viscoelastic particles that exchange charges during collisions. As compared to the case of clustering of viscoelastic particles solely due to dissipation, we find that the electrostatic interactions due to collisional charging alter the characteristic size, morphology, and growth rate of the clusters. The average cluster size grows with time as a power law, whose exponent is relatively larger in the charged gas than the neutral case. The growth of the average cluster size is found to be independent of the ratio of characteristic Coulomb to kinetic energy, or equivalently, of the typical Bjerrum length. However, this ratio alters the crossover time of the growth. Both simulations and mean-field calculations based on Smoluchowski's equation suggest that a suppression of particle diffusion due to the electrostatic interactions helps in the aggregation process.

Velocity Distribution of a Homogeneously Driven Two-Dimensional Granular Gas

The theory of homogeneously driven granular gases of hard particles predicts that the stationary state is characterized by a velocity distribution function with overpopulated high-energy tails as compared to the exponential decay valid for molecular gases. While this fundamental theoretical result was confirmed by numerous numerical simulations, an experimental confirmation is still missing. Using self-rotating active granular particles, we find a power-law decay of the velocity distribution whose exponent agrees well with the theoretic prediction.

Aging underdamped scaled Brownian motion: Ensemble- and time-averaged particle displacements, nonergodicity, and the failure of the overdamping approximation

We investigate both analytically and by computer simulations the ensemble- and time-averaged, nonergodic, and aging properties of massive particles diffusing in a medium with a time dependent diffusivity. We call this stochastic diffusion process the (aging) underdamped scaled Brownian motion (UDSBM). We demonstrate how the mean squared displacement (MSD) and the time-averaged MSD of UDSBM are affected by the inertial term in the Langevin equation, both at short, intermediate, and even long diffusion times. In particular, we quantify the ballistic regime for the MSD and the time-averaged MSD as well as the spread of individual time-averaged MSD trajectories. One of the main effects we observe is that, both for the MSD and the time-averaged MSD, for superdiffusive UDSBM the ballistic regime is much shorter than for ordinary Brownian motion. In contrast, for subdiffusive UDSBM, the ballistic region extends to much longer diffusion times. Therefore, particular care needs to be taken under what conditions the overdamped limit indeed provides a correct description, even in the long time limit. We also analyze to what extent ergodicity in the Boltzmann-Khinchin sense in this nonstationary system is broken, both for subdiffusive and superdiffusive UDSBM. Finally, the limiting case of ultraslow UDSBM is considered, with a mixed logarithmic and power-law dependence of the ensemble- and time-averaged MSDs of the particles. In the limit of strong aging, remarkably, the ordinary UDSBM and the ultraslow UDSBM behave similarly in the short time ballistic limit. The approaches developed here open ways for considering other stochastic processes under physically important conditions when a finite particle mass and aging in the system cannot be neglected.

Underdamped scaled Brownian motion: (non-)existence of the overdamped limit in anomalous diffusion

It is quite generally assumed that the overdamped Langevin equation provides a quantitative description of the dynamics of a classical Brownian particle in the long time limit. We establish and investigate a paradigm anomalous diffusion process governed by an underdamped Langevin equation with an explicit time dependence of the system temperature and thus the diffusion and damping coefficients. We show that for this underdamped scaled Brownian motion (UDSBM) the overdamped limit fails to describe the long time behaviour of the system and may practically even not exist at all for a certain range of the parameter values. Thus persistent inertial effects play a non-negligible role even at significantly long times. From this study a general questions on the applicability of the overdamped limit to describe the long time motion of an anomalously diffusing particle arises, with profound consequences for the relevance of overdamped anomalous diffusion models. We elucidate our results in view of analytical and simulations results for the anomalous diffusion of particles in free cooling granular gases.

Anomalous self-diffusion in a freely evolving granular gas near the shearing instability

The self-diffusion coefficient of a granular gas in the homogeneous cooling state is analyzed near the shearing instability. Using mode-coupling theory, it is shown that the coefficient diverges logarithmically as the instability is approached, due to the coupling of the diffusion process with the shear modes. The divergent behavior, which is peculiar in granular gases and disappears in the elastic limit, does not depend on any other transport coefficient. The theoretical prediction is confirmed by molecular dynamics simulation results for two-dimensional systems.

Quantifying non-ergodic dynamics of force-free granular gases

We demonstrate how non-ergodicity arises in simple mechanistic systems such as force free, dissipative granular gases. This behaviour results from the strong non-stationarity of the process mirrored in the continuous decay of the gas temperature.

Stochastic behavior of the coefficient of normal restitution

We consider the collision of a rough sphere with a plane by detailed analysis of the collision geometry. Using stochastic methods, the effective coefficient of restitution may be described as a fluctuating quantity whose probability density follows an asymmetric Laplace distribution. This result agrees with recent experiments by Montaine et al. [Phys. Rev. E 84, 041306 (2011)].

Intermediate regimes in granular Brownian motion: superdiffusion and subdiffusion

Brownian motion in a granular gas in a homogeneous cooling state is studied theoretically and by means of molecular dynamics. We use the simplest first-principles model for the impact-velocity dependent restitution coefficient, as it follows for the model of viscoelastic spheres. We reveal that for a wide range of initial conditions the ratio of granular temperatures of Brownian and bath particles demonstrates complicated nonmonotonic behavior, which results in a transition between different regimes of Brownian dynamics: It starts from the ballistic motion, switches later to a superballistic one, and turns at still later times into subdiffusion; eventually normal diffusion is achieved. Our theory agrees very well with the molecular dynamics results, although extreme computational costs prevented us from detecting the final diffusion regime. Qualitatively, the reported intermediate diffusion regimes are generic for granular gases with any realistic dependence of the restitution coefficient on the impact velocity.

Long-time tails and cage effect in driven granular fluids

We study the velocity autocorrelation function of a driven granular fluid in the stationary state in three dimensions. As the critical volume fraction of the glass transition in the corresponding elastic system is approached, we observe pronounced cage effects in the velocity autocorrelation function as well as a strong decrease of the diffusion constant, depending on the inelasticity. At moderate densities the velocity autocorrelation function is shown to decay algebraically in time, like t(-3/2), if momentum is conserved locally, and like t(-1), if momentum is not conserved by the driving. A simple scaling argument supports the observed long-time tails.

Linear response and hydrodynamics for granular fluids

A formal derivation of linear hydrodynamics for a granular fluid is given. The linear response to small spatial perturbations of a homogeneous reference state is studied in detail, using methods of nonequilibrium statistical mechanics. A transport matrix for macroscopic excitations in the fluid is defined in terms of the response functions. An expansion in the wave vector to second order allows identification of all phenomenological susceptibilities and transport coefficients through Navier-Stokes order in terms of appropriate time correlation functions. The transport coefficients in this representation are the generalization to granular fluids of the familiar Helfand and Green-Kubo relations for normal fluids. The analysis applies to a variety of collision rules. Important differences in both the analysis and results from those for normal fluids are identified and discussed. A scaling limit is described corresponding to the conditions under which idealized inelastic hard sphere models can apply. Further details and interpretation are provided in the paper following this one, by specialization to the case of smooth, inelastic hard spheres with constant coefficient of restitution.

Maier-Saupe model of liquid crystals: isotropic-nematic phase transitions and second-order statistics studied by Shiino's perturbation theory and strongly nonlinear Smoluchowski equations

We study the first- and second-order statistical properties of a dynamical Maier-Saupe model for liquid crystals that is given in terms of a nonlinear Smoluchowski equation. Using Shiino's perturbation theory, we analyze the first-order statistics and give a rigorous proof of the emergence of a phase transition from a uniform distribution to a nonuniform distribution, reflecting phase transitions from isotropic to nematic phases, as observed in nematic liquid crystals. Using the concept of strongly nonlinear Fokker-Planck equations, the second-order statistics of the dynamical Maier-Saupe model is studied and an analytical expression for the short-time autocorrelation function of the orientation of the crystal molecules is derived.

Self-diffusion in granular gases: Green-Kubo versus Chapman-Enskog

We study the diffusion of tracers (self-diffusion) in a homogeneously cooling gas of dissipative particles, using the Green-Kubo relation and the Chapman-Enskog approach. The dissipative particle collisions are described by the coefficient of restitution epsilon which for realistic material properties depends on the impact velocity. First, we consider self-diffusion using a constant coefficient of restitution, epsilon=const, as frequently used to simplify the analysis. Second, self-diffusion is studied for a simplified (stepwise) dependence of epsilon on the impact velocity. Finally, diffusion is considered for gases of realistic viscoelastic particles. We find that for epsilon=const both methods lead to the same result for the self-diffusion coefficient. For the case of impact-velocity dependent coefficients of restitution, the Green-Kubo method is, however, either restrictive or too complicated for practical application, therefore we compute the diffusion coefficient using the Chapman-Enskog method. We conclude that in application to granular gases, the Chapman-Enskog approach is preferable for deriving kinetic coefficients.

Diffusion of impurities in a granular gas

Diffusion of impurities in a granular gas undergoing homogeneous cooling state is studied. The results are obtained by solving the Boltzmann-Lorentz equation by means of the Chapman-Enskog method. In the first order in the density gradient of impurities, the diffusion coefficient D is determined as the solution of a linear integral equation which is approximately solved by making an expansion in Sonine polynomials. In this paper, we evaluate D up to the second order in the Sonine expansion and get explicit expressions for D in terms of the coefficients of restitution for the impurity-gas and gas-gas collisions as well as the ratios of mass and particle sizes. To check the reliability of the Sonine polynomial solution, analytical results are compared with those obtained from numerical solutions of the Boltzmann equation by means of the direct simulation Monte Carlo method. In the simulations, the diffusion coefficient is measured via the mean-square displacement of impurities. The comparison between theory and simulation shows in general an excellent agreement, except for the cases in which the gas particles are much heavier and/or much larger than impurities. In these cases, the second Sonine approximation to D improves significantly the qualitative predictions made from the first Sonine approximation. A discussion on the convergence of the Sonine polynomial expansion is also carried out.

Transport theory of granular swarms

The transport of trace granular gas (swarm) in a carrier granular fluid is studied by means of the Boltzmann-Lorentz kinetic equation. Time-dependent perturbation theory is used to follow the evolution of the granular swarm from an arbitrary initial distribution. A nonhydrodynamic extension of the diffusion equation is derived, with transport coefficients that are time dependent and implicitly depend on the wave vector. Transport coefficients of any order are obtained as velocity moments of the solutions of the corresponding kinetic equations derived from the Boltzmann-Lorentz equation. For the special case of the initial distribution of swarm particles, transport coefficients are identified as time derivatives of the moments of the number density. Finally the granular particle transport theory is extended by the introduction of the concept of non-particle-conserving collisions.

Scaling, multiscaling, and nontrivial exponents in inelastic collision processes

We investigate velocity statistics of homogeneous inelastic gases using the Boltzmann equation. Employing an approximate uniform collision rate, we obtain analytic results valid in arbitrary dimension. In the freely evolving case, the velocity distribution is characterized by an algebraic large-velocity tail, P(v,t) approximately v(-sigma). The exponent sigma(d,epsilon), a nontrivial root of an integral equation, varies continuously with the spatial dimension d and the dissipation coefficient epsilon. Although the velocity distribution follows a scaling form, its moments exhibit multiscaling asymptotic behavior. Furthermore, the velocity autocorrelation function decays algebraically with time, A(t)=<v(0).v(t)> approximately t(-alpha), with a nonuniversal dissipation-dependent exponent alpha=1/epsilon. In the forced case, the steady state Fourier transform is obtained via a cumulant expansion. Even in this case, velocity correlations develop and the velocity distribution is non-Maxwellian.

Diffusion in a granular fluid. I. Theory

Many important properties of granular fluids can be represented by a system of hard spheres with inelastic collisions. Traditional methods of nonequilibrium statistical mechanics are effective for analysis and description of the inelastic case as well. This is illustrated here for diffusion of an impurity particle in a fluid undergoing homogeneous cooling. An appropriate scaling of the Liouville equation is described such that the homogeneous cooling ensemble and associated time correlation functions map to those of a stationary state. In this form the familiar methods of linear response can be applied, leading to Green-Kubo and Einstein representations of diffusion in terms of the velocity and mean-square displacement correlation functions. These correlation functions are evaluated approximately using a cumulant expansion and from kinetic theory, providing the diffusion coefficient as a function of the density and the restitution coefficients. Comparisons with results from molecular-dynamics simulation are given in the following companion paper.

Hydrodynamics of granular gases of viscoelastic particles

Our study examines the long-time behaviour of a force-free granular gas of viscoelastic particles, for which the coefficient of restitution depends on the impact velocity, as it follows from the solution of the impact problem for viscoelastic spheres. Starting from the Boltzmann equation, we derived the hydrodynamic equations and obtained microscopic expressions for the transport coefficients in terms of the elastic and dissipative parameters of the particle material. We performed the stability analysis of the linearized set of equations and found that any inhomogeneities and vortices vanish after a long time and the system approaches the flow-free stage of homogeneous density. This behaviour is in contrast to that of a gas consisting of particles which interact via a (non-realistic) constant coefficient of restitution, for which inhomogeneities (clusters) and vortex patterns have been proven to arise and to continuously develop.

Extremal collision sequences of particles on a line: optimal transmission of kinetic energy

The transmission of kinetic energy through chains of inelastically colliding spheres is investigated for the case of constant coefficient of restitution epsilon=const and impact-velocity-dependent coefficient epsilon(v) for viscoelastic particles. We derive a theory for the optimal distribution of particle masses which maximize the energy transfer along the chain and check it numerically. We found that for epsilon=const, the mass distribution is a monotonous function which does not depend on the value of epsilon. In contrast, for epsilon(v) the mass distribution reveals a pronounced maximum, depending on the particle properties and on the chain length. The system investigated demonstrates that even for small and simple systems, the velocity dependence of the coefficient of restitution may lead to new effects with respect to the same systems under the simplifying approximation epsilon=const.

Velocity distribution in granular gases of viscoelastic particles

The velocity distribution in a homogeneously cooling granular gas has been studied in the viscoelastic regime, when the restitution coefficient of colliding particles depends on the impact velocity. We show that for viscoelastic particles a simple scaling hypothesis is violated, i.e., that the time dependence of the velocity distribution does not scale with the mean square velocity as in the case of particles interacting via a constant restitution coefficient. The deviation from the Maxwellian distribution does not depend on time monotonically. For the case of small dissipation we detected two regimes of evolution of the velocity distribution function: Starting from the initial Maxwellian distribution, the deviation first increases with time on a collision time scale saturating at some maximal value; then it decays to zero on a much larger time scale which corresponds to the temperature relaxation. For larger values of the dissipation parameter there appears an additional intermediate relaxation regime. Analytical calculations for small dissipation agree well with the results of a numerical analysis.