Resolving a paradox of anomalous scalings in the diffusion of granular materials

Self-diffusion of spherocylindrical particles flowing under non-uniform shear rate

This work is devoted to study numerically the self-diffusion of spherocylindrical particles flowing down an inclined plane, using the discrete element method (DEM). This system is challenging due to particles being non-spherical and because they are subjected to a non-uniform shear rate. We performed simulations for several aspect ratios and inclination angles, tracking individual particle trajectories. Using the simulation data, we computed the diffusion coefficients D, and a coarse-graining methodology allowed accessing the shear rate spatial profiles (z). This data enabled us to identify the spatial regions where the diffusivity strongly correlates with the local shear rate. Introducing an effective particle size d_⊥, we proposed a well-rationalized scaling law between D and . Our findings also identified specific locations where the diffusivity does not correlate with the shear rate. This observation corresponds to zones where  has non-linear spatial variation, and the velocity probability density distributions exhibit asymmetric shapes.

Time averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes

How different are the results of constant-rate resetting of anomalous-diffusion processes in terms of their ensemble-averaged versus time-averaged mean-squared displacements (MSDs versus TAMSDs) and how does stochastic resetting impact nonergodicity? We examine, both analytically and by simulations, the implications of resetting on the MSD- and TAMSD-based spreading dynamics of particles executing fractional Brownian motion (FBM) with a long-time memory, heterogeneous diffusion processes (HDPs) with a power-law space-dependent diffusivity D(x)=D_{0}|x|^{γ} and their "combined" process of HDP-FBM. We find, inter alia, that the resetting dynamics of originally ergodic FBM for superdiffusive Hurst exponents develops disparities in scaling and magnitudes of the MSDs and mean TAMSDs indicating weak ergodicity breaking. For subdiffusive HDPs we also quantify the nonequivalence of the MSD and TAMSD and observe a new trimodal form of the probability density function. For reset FBM, HDPs and HDP-FBM we compute analytically and verify by simulations the short-time MSD and TAMSD asymptotes and long-time plateaus reminiscent of those for processes under confinement. We show that certain characteristics of these reset processes are functionally similar despite a different stochastic nature of their nonreset variants. Importantly, we discover nonmonotonicity of the ergodicity-breaking parameter EB as a function of the resetting rate r. For all reset processes studied we unveil a pronounced resetting-induced nonergodicity with a maximum of EB at intermediate r and EB∼(1/r)-decay at large r. Alongside the emerging MSD-versus-TAMSD disparity, this r-dependence of EB can be an experimentally testable prediction. We conclude by discussing some implications to experimental systems featuring resetting dynamics.

Dynamics of a vibration-driven single disk

Granular particles exhibit rich collective behaviors on vibration beds, but the motion of an isolated particle is not well understood even for uniform particles with a simple shape such as disks or spheres. Here we measured the motion of a single disk confined to a quasi-two-dimensional horizontal box on a vertically vibrating stage. The translational displacements obey compressed exponential distributions whose exponent [Formula: see text] increases with the frequency, while the rotational displacements exhibit unimodal distributions at low frequencies and bimodal distributions at high frequencies. During short time intervals, the translational displacements are subdiffusive and negatively correlated, while the rotational displacements are superdiffusive and positively correlated. After prolonged periods, the rotational displacements become diffusive and their correlations decay to zero. Both the rotational and the translational displacements exhibit white noise at low frequencies, and blue noise for translational motions and Brownian noise for rotational motions at high frequencies. The translational kinetic energy obeys Boltzmann distribution while the rotational kinetic energy deviates from it. Most energy is distributed in translational motions at low frequencies and in rotational motions at high frequencies, which violates the equipartition theorem. Translational and rotational motions are not correlated. These experimental results show that the random diffusion of such driven particles is distinct from thermal motion in both the translational and rotational degrees of freedom, which poses new challenges to theory. The results cast new light on the motion of individual particles and the collective motion of driven granular particles.

Symmetry in Self-Similarity in Space and Time—Short Time Transients and Power-Law Spatial Asymptotes

The self-similarity in space and time (hereafter self-similarity), either deterministic or statistical, is characterized by similarity exponents and a function of scaled variable, called the scaling function. In the present paper, we address mainly the self-similarity in the limit of early stage, as opposed to the latter one, and also consider the scaling functions that decay or grow algebraically, as opposed to the rapidly decaying functions such as Gaussian or error function. In particular, in the case of simple diffusion, our symmetry analysis shows a mathematical mechanism by which the rapidly decaying scaling functions are generated by other polynomial scaling functions. While the former is adapted to the self-similarity in the late-stage processes, the latter is adapted to the early stages. This paper sheds some light on the internal structure of the family of self-similarities generated by a simple diffusion equation. Then, we present an example of self-similarity for the late stage whose scaling function has power-law tail, and also several cases of self-similarity for the early stages. These examples show the utility of self-similarity to a wider range of phenomena other than the late stage behaviors with rapidly decaying scaling functions.

Intermediate asymptotics of the capillary-driven thin-film equation

We present an analytical and numerical study of the two-dimensional capillary-driven thin-film equation. In particular, we focus on the intermediate asymptotics of its solutions. Linearising the equation enables us to derive the associated Green's function and therefore obtain a complete set of solutions. Moreover, we show that the rescaled solution for any summable initial profile uniformly converges in time towards a universal self-similar attractor that is precisely the rescaled Green's function. Finally, a numerical study on compact-support initial profiles enables us to conjecture the extension of our results to the nonlinear equation.

Numerical solutions of thin-film equations for polymer flows

We report on the numerical implementation of thin-film equations that describe the capillary-driven evolution of viscous films, in two-dimensional configurations. After recalling the general forms and features of these equations, we focus on two particular cases inspired by experiments: the leveling of a step at the free surface of a polymer film, and the leveling of a polymer droplet over an identical film. In each case, we first discuss the long-term self-similar regime reached by the numerical solution before comparing it to the experimental profile. The agreement between theory and experiment is excellent, thus providing a versatile probe for nanorheology of viscous liquids in thin-film geometries.