Clusters in an Epidemic Model with Long-Range Dispersal.

In the presence of long-range dispersal, epidemics spread in spatially disconnected regions known as clusters. Here, we characterize exactly their statistical properties in a solvable model, in both the supercritical (outbreak) and critical regimes. We identify two diverging length scales, corresponding to the bulk and the outskirt of the epidemic. We reveal a nontrivial critical exponent that governs the cluster number and the distribution of their sizes and of the distances between them. We also discuss applications to depinning avalanches with long-range elasticity.

Scaling Description of Creep Flow in Amorphous Solids.

Amorphous solids such as coffee foam, toothpaste, or mayonnaise display a transient creep flow when a stress Σ is suddenly imposed. The associated strain rate is commonly found to decay in time as γ[over ˙]∼t^{-ν}, followed either by arrest or by a sudden fluidization. Various empirical laws have been suggested for the creep exponent ν and fluidization time τ_{f} in experimental and numerical studies. Here, we postulate that plastic flow is governed by the difference between Σ and the transient yield stress Σ_{t}(γ) that characterizes the stability of configurations visited by the system at strain γ. Assuming the analyticity of Σ_{t}(γ) allows us to predict ν and asymptotic behaviors of τ_{f} in terms of properties of stationary flows. We test successfully our predictions using elastoplastic models and published experimental results.

The fate of shear-oscillated amorphous solids.

The behavior of shear-oscillated amorphous materials is studied using a coarse-grained model. Samples are prepared at different degrees of annealing and then subjected to athermal and quasi-static oscillatory deformations at various fixed amplitudes. The steady-state reached after several oscillations is fully determined by the initial preparation and the oscillation amplitude, as seen from stroboscopic stress and energy measurements. Under small oscillations, poorly annealed materials display shear-annealing, while ultra-stabilized materials are insensitive to them. Yet, beyond a critical oscillation amplitude, both kinds of materials display a discontinuous transition to the same mixed state composed of a fluid shear-band embedded in a marginal solid. Quantitative relations between uniform shear and the steady-state reached with this protocol are established. The transient regime characterizing the growth and the motion of the shear band is also studied.

How collective asperity detachments nucleate slip at frictional interfaces.

Sliding at a quasi-statically loaded frictional interface can occur via macroscopic slip events, which nucleate locally before propagating as rupture fronts very similar to fracture. We introduce a microscopic model of a frictional interface that includes asperity-level disorder, elastic interaction between local slip events, and inertia. For a perfectly flat and homogeneously loaded interface, we find that slip is nucleated by avalanches of asperity detachments of extension larger than a critical radius [Formula: see text] governed by a Griffith criterion. We find that after slip, the density of asperities at a local distance to yielding [Formula: see text] presents a pseudogap [Formula: see text], where θ is a nonuniversal exponent that depends on the statistics of the disorder. This result makes a link between friction and the plasticity of amorphous materials where a pseudogap is also present. For friction, we find that a consequence is that stick-slip is an extremely slowly decaying finite-size effect, while the slip nucleation radius [Formula: see text] diverges as a θ-dependent power law of the system size. We discuss how these predictions can be tested experimentally.

Spatial Clustering of Depinning Avalanches in Presence of Long-Range Interactions.

Disordered elastic interfaces display avalanche dynamics at the depinning transition. For short-range interactions, avalanches correspond to compact reorganizations of the interface well described by the depinning theory. For long-range elasticity, an avalanche is a collection of spatially disconnected clusters. In this Letter we determine the scaling properties of the clusters and relate them to the roughness exponent of the interface. The key observation of our analysis is the identification of a Bienaymé-Galton-Watson process describing the statistics of the number of clusters. Our work has concrete importance for experimental applications where the cluster statistics is a key probe of avalanche dynamics.

Bath-Induced Zeno Localization in Driven Many-Body Quantum Systems.

We study a quantum interacting spin system subject to an external drive and coupled to a thermal bath of vibrational modes, uncorrelated for different spins, serving as a model for dynamic nuclear polarization protocols. We show that even when the many-body eigenstates of the system are ergodic, a sufficiently strong coupling to the bath may effectively localize the spins due to many-body quantum Zeno effect. Our results provide an explanation of the breakdown of the thermal mixing regime experimentally observed above 4-5 K in these protocols.

Random critical point separates brittle and ductile yielding transitions in amorphous materials.

We combine an analytically solvable mean-field elasto-plastic model with molecular dynamics simulations of a generic glass former to demonstrate that, depending on their preparation protocol, amorphous materials can yield in two qualitatively distinct ways. We show that well-annealed systems yield in a discontinuous brittle way, as metallic and molecular glasses do. Yielding corresponds in this case to a first-order nonequilibrium phase transition. As the degree of annealing decreases, the first-order character becomes weaker and the transition terminates in a second-order critical point in the universality class of an Ising model in a random field. For even more poorly annealed systems, yielding becomes a smooth crossover, representative of the ductile rheological behavior generically observed in foams, emulsions, and colloidal glasses. Our results show that the variety of yielding behaviors found in amorphous materials does not necessarily result from the diversity of particle interactions or microscopic dynamics but is instead unified by carefully considering the role of the initial stability of the system.

Universal Scaling of the Velocity Field in Crack Front Propagation.

The propagation of a crack front in disordered materials is jerky and characterized by bursts of activity, called avalanches. These phenomena are the manifestation of an out-of-equilibrium phase transition originated by the disorder. As a result avalanches display universal scalings which are, however, difficult to characterize in experiments at a finite drive. Here, we show that the correlation functions of the velocity field along the front allow us to extract the critical exponents of the transition and to identify the universality class of the system. We employ these correlations to characterize the universal behavior of the transition in simulations and in an experiment of crack propagation. This analysis is robust, efficient, and can be extended to all systems displaying avalanche dynamics.

Darcy's Law for Yield Stress Fluids.

Predicting the flow of non-Newtonian fluids in a porous structure is still a challenging issue due to the interplay between the microscopic disorder and the nonlinear rheology. In this Letter, we study the case of a yield stress fluid in a two-dimensional structure. Thanks to an efficient optimization algorithm, we show that the system undergoes a continuous phase transition in the behavior of the flow, controlled by the applied pressure difference. In analogy with studies of plastic depinning of vortex lattices in high-T_{c} superconductors, we characterize the nonlinearity of the flow curve and relate it to the change in the geometry of the open channels. In particular, close to the transition, a universal scale-free distribution of the channel length is observed and explained theoretically via a mapping to the Kardar-Parisi-Zhang equation.

Correction: Soft modes and strain redistribution in continuous models of amorphous plasticity: the Eshelby paradigm, and beyond?

Correction for 'Soft modes and strain redistribution in continuous models of amorphous plasticity: the Eshelby paradigm, and beyond?' by Xiangyu Cao et al., Soft Matter, 2018, DOI: 10.1039/c7sm02510f.

Soft modes and strain redistribution in continuous models of amorphous plasticity: the Eshelby paradigm, and beyond?

The deformation of disordered solids relies on swift and localised rearrangements of particles. The inspection of soft vibrational modes can help predict the locations of these rearrangements, while the strain that they actually redistribute mediates collective effects. Here, we study soft modes and strain redistribution in a two-dimensional continuous mesoscopic model based on a Ginzburg-Landau free energy for perfect solids, supplemented with a plastic disorder potential that accounts for shear softening and rearrangements. Regardless of the disorder strength, our numerical simulations show soft modes that are always sharply peaked at the softest point of the material (unlike what happens for the depinning of an elastic interface). Contrary to widespread views, the deformation halo around this peak does not always have a quadrupolar (Eshelby-like) shape. Instead, for finite and narrowly-distributed disorder, it looks like a fracture, with a strain field that concentrates along some easy directions. These findings are rationalised with analytical calculations in the case where the plastic disorder is confined to a point-like 'impurity'. In this case, we unveil a continuous family of elastic propagators, which are identical for the soft modes and for the equilibrium configurations. This family interpolates between the standard quadrupolar propagator and the fracture-like one as the anisotropy of the elastic medium is increased. Therefore, we expect to see a fracture-like propagator when extended regions on the brink of failure have already softened along the shear direction and thus rendered the material anisotropic, but not failed yet. We speculate that this might be the case in carefully aged glasses just before macroscopic failure.

Eigenstate versus Zeeman-based approaches to the solid effect.

The solid effect is one of the simplest and most effective mechanisms for dynamic nuclear polarization. It involves the exchange of polarization between one electron and one nuclear spin coupled via the hyperfine interaction. Even for such a small spin system, the theoretical understanding is complicated by the contact with the lattice and the microwave irradiation. Both being weak, they can be treated within perturbation theory. In this work, we analyze the two most popular perturbation schemes: the Zeeman and the eigenstate-based approaches, which differ in the way the hyperfine interaction is treated. For both schemes, we derive from first principles an effective Liouville equation that describes the density matrix of the spin system; we then study numerically the behavior of the nuclear polarization for several values of the hyperfine coupling. In general, we obtain that the Zeeman-based approach underestimates the value of the nuclear polarization. By performing a projection onto the diagonal part of the spin-system density matrix, we are able to understand the origin of the discrepancy, which is due to the presence of parasite leakage transitions appearing whenever the Zeeman basis is employed.

Liouville Field Theory and Log-Correlated Random Energy Models.

An exact mapping is established between the c≥25 Liouville field theory (LFT) and the Gibbs measure statistics of a thermal particle in a 2D Gaussian free field plus a logarithmic confining potential. The probability distribution of the position of the minimum of the energy landscape is obtained exactly by combining the conformal bootstrap and one-step replica symmetry-breaking methods. Operator product expansions in the LFT allow us to unveil novel universal behaviors of the log-correlated random energy class. High-precision numerical tests are given.

Spatiotemporal Patterns in Ultraslow Domain Wall Creep Dynamics.

In the presence of impurities, ferromagnetic and ferroelectric domain walls slide only above a finite external field. Close to this depinning threshold, they proceed by large and abrupt jumps called avalanches, while, at much smaller fields, these interfaces creep by thermal activation. In this Letter, we develop a novel numerical technique that captures the ultraslow creep regime over huge time scales. We point out the existence of activated events that involve collective reorganizations similar to avalanches, but, at variance with them, display correlated spatiotemporal patterns that resemble the complex sequence of aftershocks observed after a large earthquake. Remarkably, we show that events assemble in independent clusters that display at large scales the same statistics as critical depinning avalanches. We foresee these correlated dynamics being experimentally accessible by magnetooptical imaging of ferromagnetic films.

Scaling description of non-local rheology.

The plastic flow of amorphous materials displays non-local effects, characterized by a cooperativity length scale ξ. We argue that these effects enter in the more general description of surface phenomena near critical points. Using this approach, we obtain a scaling relation between exponents that describe the strain rate profiles in shear driven and pressure driven flow, which we confirm both in numerical models and experimental data. We find empirically that the cooperative length follows closely the characteristic length previously extracted in homogenous bulk flows. This analysis shows that the often used mean field exponents fail to capture quantitatively the non-local effects. Our analysis also explains the unusually large finite size effects previously observed in pressure driven flows.

Scaling description of the yielding transition in soft amorphous solids at zero temperature.

Yield stress materials flow if a sufficiently large shear stress is applied. Although such materials are ubiquitous and relevant for industry, there is no accepted microscopic description of how they yield, even in the simplest situations in which temperature is negligible and in which flow inhomogeneities such as shear bands or fractures are absent. Here we propose a scaling description of the yielding transition in amorphous solids made of soft particles at zero temperature. Our description makes a connection between the Herschel-Bulkley exponent characterizing the singularity of the flow curve near the yield stress Σc, the extension and duration of the avalanches of plasticity observed at threshold, and the density P(x) of soft spots, or shear transformation zones, as a function of the stress increment x beyond which they yield. We argue that the critical exponents of the yielding transition may be expressed in terms of three independent exponents, θ, df, and z, characterizing, respectively, the density of soft spots, the fractal dimension of the avalanches, and their duration. Our description shares some similarity with the depinning transition that occurs when an elastic manifold is driven through a random potential, but also presents some striking differences. We test our arguments in an elasto-plastic model, an automaton model similar to those used in depinning, but with a different interaction kernel, and find satisfying agreement with our predictions in both two and three dimensions.

Exact Short-Time Height Distribution in the One-Dimensional Kardar-Parisi-Zhang Equation and Edge Fermions at High Temperature.

We consider the early time regime of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions in curved (or droplet) geometry. We show that for short time t, the probability distribution P(H,t) of the height H at a given point x takes the scaling form P(H,t)∼exp[-Φ_{drop}(H)/sqrt[t]] where the rate function Φ_{drop}(H) is computed exactly for all H. While it is Gaussian in the center, i.e., for small H, the probability distribution function has highly asymmetric non-Gaussian tails that we characterize in detail. This function Φ_{drop}(H) is surprisingly reminiscent of the large deviation function describing the stationary fluctuations of finite-size models belonging to the KPZ universality class. Thanks to a recently discovered connection between the KPZ equation and free fermions, our results have interesting implications for the fluctuations of the rightmost fermion in a harmonic trap at high temperature and the full counting statistics at the edge.

Evidence of spin-temperature in dynamic nuclear polarization: an exact computation of the EPR spectrum.

In dynamic nuclear polarization (DNP) experiments, the compound is driven out-of-equilibrium by the microwave (MW) irradiation of the radical electron spins. Their stationary state has been recently probed via electron double resonance (ELDOR) techniques showing, at low temperature, a broad depolarization of the electron paramagnetic resonance (EPR) spectrum under microwave irradiation. In this theoretical manuscript, we develop a numerical method to compute exactly the EPR spectrum in the presence of dipolar interactions. Our results reproduce the observed broad depolarisation and provide a microscopic justification for the spectral diffusion mechanism. We show the validity of the spin-temperature approach for typical radical concentration used in dissolution DNP protocols. In particular once the interactions are properly taken into account, the spin-temperature is consistent with the non-monotonic behavior of the EPR spectrum with a wide minimum around the irradiated frequency.

Spatial extent of an outbreak in animal epidemics.

Characterizing the spatial extent of epidemics at the outbreak stage is key to controlling the evolution of the disease. At the outbreak, the number of infected individuals is typically small, and therefore, fluctuations around their average are important: then, it is commonly assumed that the susceptible-infected-recovered mechanism can be described by a stochastic birth-death process of Galton-Watson type. The displacements of the infected individuals can be modeled by resorting to brownian motion, which is applicable when long-range movements and complex network interactions can be safely neglected, like in the case of animal epidemics. In this context, the spatial extent of an epidemic can be assessed by computing the convex hull enclosing the infected individuals at a given time. We derive the exact evolution equations for the mean perimeter and the mean area of the convex hull, and we compare them with Monte Carlo simulations.

Dynamic Nuclear Polarization and the Paradox of Quantum Thermalization.

Dynamic nuclear polarization (DNP) is to date the most effective technique to increase the nuclear polarization opening disruptive perspectives for medical applications. In a DNP setting, the interacting spin system is quasi-isolated and brought out of equilibrium by microwave irradiation. Here we show that the resulting stationary state strongly depends on the ergodicity properties of the spin many-body eigenstates. In particular, the dipolar interactions compete with the disorder induced by local magnetic fields resulting in two distinct dynamical phases: while for weak interaction, only a small enhancement of polarization is observed, for strong interactions the spins collectively equilibrate to an extremely low effective temperature that boosts DNP efficiency. We argue that these two phases are intimately related to the problem of thermalization in closed quantum systems where a many-body localization transition can occur varying the strength of the interactions.