Universal Poisson statistics of a passive tracer diffusing in dilute active suspensions.

The statistics of a passive tracer immersed in a suspension of active particles (swimmers) is derived from first principles by considering a perturbative expansion of the tracer interaction with the microscopic swimmer field. To first order in the swimmer density, the tracer statistics is shown to be exactly represented by a spatial Poisson process combined with independent tracer-swimmer scattering events, rigorously reducing the multiparticle dynamics to two-body interactions. The Poisson representation is valid in any dimension, for arbitrary interaction forces and for a large class of swimmer dynamics. The framework not only allows for the systematic calculation of the tracer statistics in various dynamical regimes but highlights in particular surprising universal features that are independent of the swimmer dynamics such as a time-independent velocity distribution.

Machine learning approaches for the optimization of packing densities in granular matter.

The fundamental question of how densely granular matter can pack and how this density depends on the shape of the constituent particles has been a longstanding scientific problem. Previous work has mainly focused on empirical approaches based on simulations or mean-field theory to investigate the effect of shape variation on the resulting packing densities, focusing on a small set of pre-defined shapes like dimers, ellipsoids, and spherocylinders. Here we discuss how machine learning methods can support the search for optimally dense packing shapes in a high-dimensional shape space. We apply dimensional reduction and regression techniques based on random forests and neural networks to find novel dense packing shapes by numerical optimization. Moreover, an investigation of the regression function in the dimensionally reduced shape representation allows us to identify directions in the packing density landscape that lead to a strongly non-monotonic variation of the packing density. The predictions obtained by machine learning are compared with packing simulations. Our approach can be more widely applied to optimize the properties of granular matter by varying the shape of its constituent particles.

Exponential increase of transition rates in metastable systems driven by non-Gaussian noise.

Noise-induced escape from metastable states governs a plethora of transition phenomena in physics, chemistry, and biology. While the escape problem in the presence of thermal Gaussian noise has been well understood since the seminal works of Arrhenius and Kramers, many systems, in particular living ones, are effectively driven by non-Gaussian noise for which the conventional theory does not apply. Here we present a theoretical framework based on path integrals that allows the calculation of both escape rates and optimal escape paths for a generic class of non-Gaussian noises. We find that non-Gaussian noise always leads to more efficient escape and can enhance escape rates by many orders of magnitude compared with thermal noise, highlighting that away from equilibrium escape rates cannot be reliably modelled based on the traditional Arrhenius-Kramers result. Our analysis also identifies a new universality class of non-Gaussian noises, for which escape paths are dominated by large jumps.

Structural analysis of disordered dimer packings.

Jammed disordered packings of non-spherical particles show significant variation in the packing density as a function of particle shape for a given packing protocol. Rotationally symmetric elongated shapes such as ellipsoids, spherocylinders, and dimers, e.g., pack significantly denser than spheres over a narrow range of aspect ratios, exhibiting a characteristic peak at aspect ratios of αmax ≈ 1.4-1.5. However, the structural features that underlie this non-monotonic behaviour in the packing density are unknown. Here, we study disordered packings of frictionless dimers in three dimensions generated by a gravitational pouring protocol in LAMMPS. Focusing on the characteristics of contacts as well as orientational and translational order metrics, we identify a number of structural features that accompany the formation of maximally dense packings as the dimer aspect ratio α is varied from the spherical limit. Our results highlight that dimer packings undergo significant structural changes as α increases up to αmax manifest in the reorganisation of the contact configurations between neighbouring dimers, increasing nematic order, and decreasing local translational order. Remarkably, for α > αmax our metrics remain largely unchanged, indicating that the peak in the packing density is related to the interplay of structural rearrangements for α < αmax and subsequent excluded volume effects with unchanged structure for α > αmax.

Kinetics of random sequential adsorption of two-dimensional shapes on a one-dimensional line.

Saturated random sequential adsorption packings built of two-dimensional ellipses, spherocylinders, rectangles, and dimers placed on a one-dimensional line are studied to check analytical prediction concerning packing growth kinetics [A. Baule, Phys. Rev. Lett. 119, 028003 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.028003]. The results show that the kinetics is governed by the power law with the exponent d=1.5 and 2.0 for packings built of ellipses and rectangles, respectively, which is consistent with analytical predictions. However, for spherocylinders and dimers of moderate width-to-height ratio, a transition between these two values is observed. We argue that this transition is a finite-size effect that arises for spherocylinders due to the properties of the contact function. In general, it appears that the kinetics of packing growth can depend on packing size even for very large packings.

Loopy Lévy flights enhance tracer diffusion in active suspensions.

Brownian motion is widely used as a model of diffusion in equilibrium media throughout the physical, chemical and biological sciences. However, many real-world systems are intrinsically out of equilibrium owing to energy-dissipating active processes underlying their mechanical and dynamical features1. The diffusion process followed by a passive tracer in prototypical active media, such as suspensions of active colloids or swimming microorganisms2, differs considerably from Brownian motion, as revealed by a greatly enhanced diffusion coefficient3-10 and non-Gaussian statistics of the tracer displacements6,9,10. Although these characteristic features have been extensively observed experimentally, there is so far no comprehensive theory explaining how they emerge from the microscopic dynamics of the system. Here we develop a theoretical framework to model the hydrodynamic interactions between the tracer and the active swimmers, which shows that the tracer follows a non-Markovian coloured Poisson process that accounts for all empirical observations. The theory predicts a long-lived Lévy flight regime11 of the loopy tracer motion with a non-monotonic crossover between two different power-law exponents. The duration of this regime can be tuned by the swimmer density, suggesting that the optimal foraging strategy of swimming microorganisms might depend crucially on their density in order to exploit the Lévy flights of nutrients12. Our framework can be applied to address important theoretical questions, such as the thermodynamics of active systems13, and practical ones, such as the interaction of swimming microorganisms with nutrients and other small particles14 (for example, degraded plastic) and the design of artificial nanoscale machines15.

Optimal Random Deposition of Interacting Particles.

Irreversible random sequential deposition of interacting particles is widely used to model aggregation phenomena in physical, chemical, and biophysical systems. We show that in one dimension the exact time-dependent solution of such processes can be found for arbitrary interaction potentials with finite range. The exact solution allows us to rigorously prove characteristic features of the deposition kinetics, which have previously only been accessible by simulations. We show in particular that a unique interaction potential exists that leads to a maximally dense line coverage for a given interaction range. Remarkably, this potential is singular and can only be expressed as a mathematical limit. The relevance of these results for models of nucleosome packing on DNA is discussed. The results highlight how the generation of an optimally dense packing requires a highly coordinated packing dynamics, which can be effectively tuned by the interaction potential even in the presence of intrinsic randomness.

Erratum: Anomalous Processes with General Waiting Times: Functionals and Multipoint Structure [Phys. Rev. Lett. 115, 110601 (2015)].

This corrects the article DOI: 10.1103/PhysRevLett.115.110601.

Weak Galilean invariance as a selection principle for coarse-grained diffusive models.

How does the mathematical description of a system change in different reference frames? Galilei first addressed this fundamental question by formulating the famous principle of Galilean invariance. It prescribes that the equations of motion of closed systems remain the same in different inertial frames related by Galilean transformations, thus imposing strong constraints on the dynamical rules. However, real world systems are often described by coarse-grained models integrating complex internal and external interactions indistinguishably as friction and stochastic forces. Since Galilean invariance is then violated, there is seemingly no alternative principle to assess a priori the physical consistency of a given stochastic model in different inertial frames. Here, starting from the Kac-Zwanzig Hamiltonian model generating Brownian motion, we show how Galilean invariance is broken during the coarse-graining procedure when deriving stochastic equations. Our analysis leads to a set of rules characterizing systems in different inertial frames that have to be satisfied by general stochastic models, which we call "weak Galilean invariance." Several well-known stochastic processes are invariant in these terms, except the continuous-time random walk for which we derive the correct invariant description. Our results are particularly relevant for the modeling of biological systems, as they provide a theoretical principle to select physically consistent stochastic models before a validation against experimental data.

Shape Universality Classes in the Random Sequential Adsorption of Nonspherical Particles.

Random sequential adsorption (RSA) of particles of a particular shape is used in a large variety of contexts to model particle aggregation and jamming. A key feature of these models is the observed algebraic time dependence of the asymptotic jamming coverage ∼t^{-ν} as t→∞. However, the exact value of the exponent ν is not known apart from the simplest case of the RSA of monodisperse spheres adsorbed on a line (Renyi's seminal "car parking problem"), where ν=1 can be derived analytically. Empirical simulation studies have conjectured on a case-by-case basis that for general nonspherical particles, ν=1/(d+d[over ˜]), where d denotes the dimension of the domain, and d[over ˜] the number of orientational degrees of freedom of a particle. Here, we solve this long-standing problem analytically for the d=1 case-the "Paris car parking problem." We prove, in particular, that the scaling exponent depends on the particle shape, contrary to the original conjecture and, remarkably, falls into two universality classes: (i) ν=1/(1+d[over ˜]/2) for shapes with a smooth contact distance, e.g., ellipsoids, and (ii) ν=1/(1+d[over ˜]) for shapes with a singular contact distance, e.g., spherocylinders and polyhedra. The exact solution explains, in particular, why many empirically observed scalings fall in between these two limits.

Gaussian white noise as a resource for work extraction.

We show that uncorrelated Gaussian noise can drive a system out of equilibrium and can serve as a resource from which work can be extracted. We consider an overdamped particle in a periodic potential with an internal degree of freedom and a state-dependent friction, coupled to an equilibrium bath. Applying additional Gaussian white noise drives the system into a nonequilibrium steady state and causes a finite current if the potential is spatially asymmetric. The model thus operates as a Brownian ratchet, whose current we calculate explicitly in three complementary limits. Since the particle current is driven solely by additive Gaussian white noise, this shows that the latter can potentially perform work against an external load. By comparing the extracted power to the energy injection due to the noise, we discuss the efficiency of such a ratchet.

Anomalous processes with general waiting times: functionals and multipoint structure.

Many transport processes in nature exhibit anomalous diffusive properties with nontrivial scaling of the mean square displacement, e.g., diffusion of cells or of biomolecules inside the cell nucleus, where typically a crossover between different scaling regimes appears over time. Here, we investigate a class of anomalous diffusion processes that is able to capture such complex dynamics by virtue of a general waiting time distribution. We obtain a complete characterization of such generalized anomalous processes, including their functionals and multipoint structure, using a representation in terms of a normal diffusive process plus a stochastic time change. In particular, we derive analytical closed form expressions for the two-point correlation functions, which can be readily compared with experimental data.

Langevin formulation of a subdiffusive continuous-time random walk in physical time.

Systems living in complex nonequilibrated environments often exhibit subdiffusion characterized by a sublinear power-law scaling of the mean square displacement. One of the most common models to describe such subdiffusive dynamics is the continuous-time random walk (CTRW). Stochastic trajectories of a CTRW can be described in terms of the subordination of a normal diffusive process by an inverse Lévy-stable process. Here, we propose an equivalent Langevin formulation of a force-free CTRW without subordination. By introducing a different type of non-Gaussian noise, we are able to express the CTRW dynamics in terms of a single Langevin equation in physical time with additive noise. We derive the full multipoint statistics of this noise and compare it with the scaled Brownian motion (SBM), an alternative stochastic model describing subdiffusive dynamics. Interestingly, these two noises are identical up to the second order correlation functions, but different in the higher order statistics. We extend our formalism to general waiting time distributions and force fields and compare our results with those of the SBM. In the presence of external forces, our proposed noise generates a different class of stochastic processes, resembling a CTRW but with forces acting at all times.

Adhesive loose packings of small dry particles.

We explore adhesive loose packings of small dry spherical particles of micrometer size using 3D discrete-element simulations with adhesive contact mechanics and statistical ensemble theory. A dimensionless adhesion parameter (Ad) successfully combines the effects of particle velocities, sizes and the work of adhesion, identifying a universal regime of adhesive packings for Ad > 1. The structural properties of the packings in this regime are well described by an ensemble approach based on a coarse-grained volume function that includes the correlation between bulk and contact spheres. Our theoretical and numerical results predict: (i) an equation of state for adhesive loose packings that appear as a continuation from the frictionless random close packing (RCP) point in the jamming phase diagram and (ii) the existence of an asymptotic adhesive loose packing point at a coordination number Z = 2 and a packing fraction ϕ = 1/2(3). Our results highlight that adhesion leads to a universal packing regime at packing fractions much smaller than the random loose packing (RLP), which can be described within a statistical mechanical framework. We present a general phase diagram of jammed matter comprising frictionless, frictional, adhesive as well as non-spherical particles, providing a classification of packings in terms of their continuation from the spherical frictionless RCP.

Fundamental challenges in packing problems: from spherical to non-spherical particles.

Random packings of objects of a particular shape are ubiquitous in science and engineering. However, such jammed matter states have eluded any systematic theoretical treatment due to the strong positional and orientational correlations involved. In recent years progress on a fundamental description of jammed matter could be made by starting from a constant volume ensemble in the spirit of conventional statistical mechanics. Recent work has shown that this approach, first introduced by S. F. Edwards more than two decades ago, can be cast into a predictive framework to calculate the packing fractions of both spherical and non-spherical particles.

Weak-noise limit of a piecewise-smooth stochastic differential equation.

We investigate the validity and accuracy of weak-noise (saddle-point or instanton) approximations for piecewise-smooth stochastic differential equations (SDEs), taking as an illustrative example a piecewise-constant SDE, which serves as a simple model of Brownian motion with solid friction. For this model, we show that the weak-noise approximation of the path integral correctly reproduces the known propagator of the SDE at lowest order in the noise power, as well as the main features of the exact propagator with higher-order corrections, provided the singularity of the path integral associated with the nonsmooth SDE is treated with some heuristics. We also show that, as in the case of smooth SDEs, the deterministic paths of the noiseless system correctly describe the behavior of the nonsmooth SDE in the low-noise limit. Finally, we consider a smooth regularization of the piecewise-constant SDE and study to what extent this regularization can rectify some of the problems encountered when dealing with discontinuous drifts and singularities in SDEs.

Mean-field theory of random close packings of axisymmetric particles.

Finding the optimal random packing of non-spherical particles is an open problem with great significance in a broad range of scientific and engineering fields. So far, this search has been performed only empirically on a case-by-case basis, in particular, for shapes like dimers, spherocylinders and ellipsoids of revolution. Here we present a mean-field formalism to estimate the packing density of axisymmetric non-spherical particles. We derive an analytic continuation from the sphere that provides a phase diagram predicting that, for the same coordination number, the density of monodisperse random packings follows the sequence of increasing packing fractions: spheres