Importance sampling for counting statistics in one-dimensional systems.

In this paper, we consider the problem of numerical investigation of the counting statistics for a class of one-dimensional systems. Importance sampling, the cornerstone technique usually implemented for such problems, critically hinges on selecting an appropriate biased distribution. While an exponential tilt in the observable stands as the conventional choice for various problems, its efficiency in the context of counting statistics may be significantly hindered by the genuine discreteness of the observable. To address this challenge, we propose an alternative strategy, which we call importance sampling with the local tilt. We demonstrate the efficiency of the proposed approach through the analysis of three prototypical examples: a set of independent Gaussian random variables, Dyson gas, and symmetric simple exclusion process with a steplike initial condition.

Anomalous scaling of heterogeneous elastic lines: A picture from sample-to-sample fluctuations.

We study a discrete model of an heterogeneous elastic line with internal disorder, submitted to thermal fluctuations. The monomers are connected through random springs with independent and identically distributed elastic constants drawn from p(k)∼k^{µ-1} for k→0. When µ>1, the scaling of the standard Edwards-Wilkinson model is recovered. When µ<1, the elastic line exhibits an anomalous scaling of the type observed in many growth models and experiments. Here we derive and use the exact probability distribution of the line shape at equilibrium, as well as the spectral properties of the matrix containing the random couplings, to fully characterize the sample to sample fluctuations. Our results lead to scaling predictions that partially disagree with previous works, but that are corroborated by numerical simulations. We also provide an interpretation of the anomalous scaling in terms of the abrupt jumps in the line's shape that dominate the average value of the observable.

Occupation time of a system of Brownian particles on the line with steplike initial condition.

We consider a system of noninteracting Brownian particles on the line with steplike initial condition and study the statistics of the occupation time on the positive half-line. We demonstrate that even at large times, the behavior of the occupation time exhibits long-lasting memory effects of the initialization. Specifically, we calculate the mean and the variance of the occupation time, demonstrating that the memory effects in the variance are determined by a generalized compressibility (or Fano factor), associated with the initial condition. In the particular case of the uncorrelated uniform initial condition we conduct a detailed study of two probability distributions of the occupation time: annealed (averaged over all possible initial configurations) and quenched (for a typical configuration). We show that at large times both the annealed and the quenched distributions admit large deviation form and we compute analytically the associated rate functions. We verify our analytical predictions via numerical simulations using importance sampling Monte Carlo strategy.

Unified understanding of the breakdown of thermal mixing dynamic nuclear polarization: The role of temperature and radical concentration.

We reveal an interplay between temperature and radical concentration necessary to establish thermal mixing (TM) as an efficient dynamic nuclear polarization (DNP) mechanism. We conducted DNP experiments by hyperpolarizing widely used DNP samples, i.e., sodium pyruvate-1-13C in water/glycerol mixtures at varying nitroxide radical (TEMPOL) concentrations and microwave irradiation frequencies, measuring proton and carbon-13 spin temperatures. Using a cryogen consumption-free prototype-DNP apparatus, we could probe cryogenic temperatures between 1.5 and 6.5 K, i.e., below and above the boiling point of liquid helium. We identify two mechanisms for the breakdown of TM: (i) Anderson type of quantum localization for low radical concentration, or (ii) quantum Zeno localization occurring at high temperature. This observation allowed us to reconcile the recent diverging observations regarding the relevance of TM as a DNP mechanism by proposing a unifying picture and, consequently, to find a trade-off between radical concentration and electron relaxation times, which offers a pathway to improve experimental DNP performance based on TM.

Generalized disorder averages and current fluctuations in run and tumble particles.

We present exact results for the fluctuations in the number of particles crossing the origin up to time t in a collection of noninteracting run and tumble particles in one dimension. In contrast to passive systems, such active particles are endowed with two inherent degrees of freedom, positions and velocities, which can be used to construct density and magnetization fields. We introduce generalized disorder averages associated with both these fields and perform annealed and quenched averages over various initial conditions. We show that the variance σ^{2} of the current in annealed versus quenched magnetization situations exhibits a surprising difference at short times, σ^{2}∼t vs σ^{2}∼t^{2}, respectively, with a sqrt[t] behavior emerging at large times. Our analytical results demonstrate that in the strictly quenched scenario, where both the density and magnetization fields are initially frozen, the fluctuations in the current are strongly suppressed. Importantly, these anomalous fluctuations cannot be obtained solely by freezing the density field.

Darcy's law of yield stress fluids on a treelike network.

Understanding the flow of yield stress fluids in porous media is a major challenge. In particular, experiments and extensive numerical simulations report a nonlinear Darcy law as a function of the pressure gradient. In this letter we consider a treelike porous structure for which the problem of the flow can be resolved exactly due to a mapping with the directed polymer (DP) with disordered bond energies on the Cayley tree. Our results confirm the nonlinear behavior of the flow and expresses its full pressure dependence via the density of low-energy paths of DP restricted to vanishing overlap. These universal predictions are confirmed by extensive numerical simulations.

Current fluctuations in stochastically resetting particle systems.

We consider a system of noninteracting particles on a line with initial positions distributed uniformly with density ρ on the negative half-line. We consider two different models: (i) Each particle performs independent Brownian motion with stochastic resetting to its initial position with rate r and (ii) each particle performs run-and-tumble motion, and with rate r its position gets reset to its initial value and simultaneously its velocity gets randomized. We study the effects of resetting on the distribution P(Q,t) of the integrated particle current Q up to time t through the origin (from left to right). We study both the annealed and the quenched current distributions and in both cases, we find that resetting induces a stationary limiting distribution of the current at long times. However, we show that the approach to the stationary state of the current distribution in the annealed and the quenched cases are drastically different for both models. In the annealed case, the whole distribution P_{an}(Q,t) approaches its stationary limit uniformly for all Q. In contrast, the quenched distribution P_{qu}(Q,t) attains its stationary form for QQ_{crit}(t). We show that Q_{crit}(t) increases linearly with t for large t. On the scale where Q∼Q_{crit}(t), we show that P_{qu}(Q,t) has an unusual large deviation form with a rate function that has a third-order phase transition at the critical point. We have computed the associated rate functions analytically for both models. Using an importance sampling method that allows to probe probabilities as tiny as 10^{-14000}, we were able to compute numerically this nonanalytic rate function for the resetting Brownian dynamics and found excellent agreement with our analytical prediction.

Local time of a system of Brownian particles on the line with steplike initial condition.

We consider a system of noninteracting Brownian particles on a line with a steplike initial condition, and we investigate the behavior of the local time at the origin at large times. We compute the mean and the variance of the local time, and we show that the memory effects are governed by the Fano factor associated with the initial condition. For the uniform initial condition, we show that the probability distribution of the local time admits a large deviation form, and we compute the corresponding large deviation functions for the annealed and quenched averaging schemes. The two resulting large deviation functions are very different. Our analytical results are supported by extensive numerical simulations.

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.

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.

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.

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.

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.

Statistical properties of avalanches via the c-record process.

We study the statistics of avalanches, as a response to an applied force, undergone by a particle hopping on a one-dimensional lattice where the pinning forces at each site are independent and identically distributed (i.i.d.), each drawn from a continuous f(x). The avalanches in this model correspond to the interrecord intervals in a modified record process of i.i.d. variables, defined by a single parameter c>0. This parameter characterizes the record formation via the recursive process R_{k}>R_{k-1}-c, where R_{k} denotes the value of the kth record. We show that for c>0, if f(x) decays slower than an exponential for large x, the record process is nonstationary as in the standard c=0 case. In contrast, if f(x) has a faster than exponential tail, the record process becomes stationary and the avalanche size distribution π(n) has a decay faster than 1/n^{2} for large n. The marginal case where f(x) decays exponentially for large x exhibits a phase transition from a nonstationary phase to a stationary phase as c increases through a critical value c_{crit}. Focusing on f(x)=e^{-x} (with x≥0), we show that c_{crit}=1 and for c<1, the record statistics is nonstationary. However, for c>1, the record statistics is stationary with avalanche size distribution π(n)∼n^{-1-λ(c)} for large n. Consequently, for c>1, the mean number of records up to N steps grows algebraically ∼N^{λ(c)} for large N. Remarkably, the exponent λ(c) depends continuously on c for c>1 and is given by the unique positive root of c=-ln(1-λ)/λ. We also unveil the presence of nontrivial correlations between avalanches in the stationary phase that resemble earthquake sequences.

The influence of the brittle-ductile transition zone on aftershock and foreshock occurrence.

Aftershock occurrence is characterized by scaling behaviors with quite universal exponents. At the same time, deviations from universality have been proposed as a tool to discriminate aftershocks from foreshocks. Here we show that the change in rheological behavior of the crust, from velocity weakening to velocity strengthening, represents a viable mechanism to explain statistical features of both aftershocks and foreshocks. More precisely, we present a model of the seismic fault described as a velocity weakening elastic layer coupled to a velocity strengthening visco-elastic layer. We show that the statistical properties of aftershocks in instrumental catalogs are recovered at a quantitative level, quite independently of the value of model parameters. We also find that large earthquakes are often anticipated by a preparatory phase characterized by the occurrence of foreshocks. Their magnitude distribution is significantly flatter than the aftershock one, in agreement with recent results for forecasting tools based on foreshocks.

Current fluctuations in noninteracting run-and-tumble particles in one dimension.

We present a general framework to study the distribution of the flux through the origin up to time t, in a noninteracting one-dimensional system of particles with a step initial condition with a fixed density ρ of particles to the left of the origin. We focus principally on two cases: (i) particles undergoing diffusive dynamics (passive case) and (ii) run-and-tumble dynamics for each particle (active case). In analogy with disordered systems, we consider the flux distribution for both the annealed and the quenched initial conditions, for passive and active particles. In the annealed case, we show that, for arbitrary particle dynamics, the flux distribution is a Poissonian with a mean µ(t) that we compute exactly in terms of the Green's function of the single-particle dynamics. For the quenched case, we show that, for the run-and-tumble dynamics, the quenched flux distribution takes an anomalous large-deviation form at large times, P_{qu}(Q,t)∼exp[-ρv_{0}γt^{2}ψ_{RTP}(Q/ρv_{0}t)], where γ is the rate of tumbling and v_{0} is the ballistic speed between two successive tumblings. In this paper, we compute the rate function ψ_{RTP}(q) and show that it is nontrivial. Our method also gives access to the probability of the rare event that, at time t, there is no particle to the right of the origin. For diffusive and run-and-tumble dynamics, we find that this probability decays with time as a stretched exponential, ∼exp(-csqrt[t]), where the constant c can be computed exactly. We verify our results for these large deviations by using an importance sampling Monte Carlo method.

Stochastic growth in time-dependent environments.

We study the Kardar-Parisi-Zhang (KPZ) growth equation in one dimension with a noise variance c(t) depending on time. We find that for c(t)∝t^{-α} there is a transition at α=1/2. When α>1/2, the solution saturates at large times towards a nonuniversal limiting distribution. When α<1/2 the fluctuation field is governed by scaling exponents depending on α and the limiting statistics are similar to the case when c(t) is constant. We investigate this problem using different methods: (1) Elementary changes of variables mapping the time-dependent case to variants of the KPZ equation with constant variance of the noise but in a deformed potential. (2) An exactly solvable discretization, the log-gamma polymer model. (3) Numerical simulations.

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.

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.

Seismiclike organization of avalanches in a driven long-range elastic string as a paradigm of brittle cracks.

Crack growth in heterogeneous materials sometimes exhibits crackling dynamics, made of successive impulselike events with specific scale-invariant time and size organization reminiscent of earthquakes. Here, we examine this dynamics in a model which identifies the crack front with a long-range elastic line driven in a random potential. We demonstrate that, under some circumstances, fracture grows intermittently, via scale-free impulse organized into aftershock sequences obeying the fundamental laws of statistical seismology. We examine the effects of the driving rate and system overall stiffness (unloading factor) onto the scaling exponents and cutoffs associated with the time and size organization. We unravel the specific conditions required to observe a seismiclike organization in the crack propagation problem. Beyond failure problems, implications of these results to other crackling systems are finally discussed.