The thermally activated creep motion of an elastic interface weakly driven on a disordered landscape is one of the best examples of glassy universal dynamics. Its understanding has evolved over the last 30 years thanks to a fruitful interplay between elegant scaling arguments, sophisticated analytical calculations, efficient optimization algorithms and creative experiments. In this article, starting from the pioneer arguments, we review the main theoretical and experimental results that lead to the current physical picture of the creep regime. In particular, we discuss recent works unveiling the collective nature of such ultra-slow motion in terms of elementary activated events. We show that these events control the mean velocity of the interface and cluster into "creep avalanches" statistically similar to the deterministic avalanches observed at the depinning critical threshold. The associated spatio-temporal patterns of activated events have been recently observed in experiments with magnetic domain walls. The emergent physical picture is expected to be relevant for a large family of disordered systems presenting thermally activated dynamics.
Elastic systems driven in a disordered medium exhibit a depinning transition at zero temperature and a creep regime at finite temperature and slow drive $f$. We derive functional renormalization group equations which allow to describe in details the properties of the slowly moving states in both cases. Since they hold at finite velocity $v$, they allow to remedy some shortcomings of the previous approaches to zero temperature depinning. In particular, they enable us to derive the depinning law directly from the equation of motion, with no artificial prescription or additional physical assumpti…
When materials are loaded below their short-term strength over extended periods, a slow time-dependent process known as creep deformation takes place. During creep deformation, the structural properties of a material evolve as a function of time. By means of a generic coarse-grained mesoscopic elastoplastic model which envisages deformation as a sequence of stochastically activated discrete events, we study the creep deformation of disordered materials. We find that the structural evolution of the material during creep modifies not only the average material properties but also changes the stat…
We present a mesoscale elastoplastic model of creep in disordered materials which considers temperature-dependent stochastic activation of localized deformation events which are mutually coupled by internal stresses, leading to collective avalanche dynamics. We generalize this stochastic plasticity model by introducing damage in terms of a local strength that decreases, on statistical average, with increasing local plastic strain. As a consequence the model captures failure in terms of strain localization in a catastrophic shear band concomitant with a finite-time singularity of the creep rate…
Viscoelasticity has been described since the time of Maxwell as an interpolation of purely viscous and purely elastic response, but its microscopic atomic-level mechanism in solids has remained elusive. We studied three model disordered solids: a random lattice, the bond-depleted fcc lattice, and the fcc lattice with vacancies. Within the harmonic approximation for central-force lattices, we applied sum-rules for viscoelastic response derived on the basis of non-affine atomic motions. The latter motions are a direct result of local structural disorder, and in particular, of the lack of inversi…
We study the creep motion of an elastic string in a two dimensional pinning landscape by Langevin dynamics simulations. We find that the Velocity-Force characteristics are well described by the creep formula predicted from phenomenological scaling arguments. We analyze the creep exponent $μ$, and the roughness exponent $ζ$. Two regimes are identified: when the temperature is larger than the strength of the disorder we find $μ\approx 1/4$ and $ζ\approx 2/3$, in agreement with the quasi-equilibrium-nucleation picture of creep motion; on the contrary, lowering enough the temperature, the values o…
We present a subcritical fracture growth model, coupled with the elastic redistribution of the acting mechanical stress along rugous rupture fronts. We show the ability of this model to quantitatively reproduce the intermittent dynamics of cracks propagating along weak disordered interfaces. To this end, we assume that the fracture energy of such interfaces (in the sense of a critical energy release rate) follows a spatially correlated normal distribution. We compare various statistical features from the hence obtained fracture dynamics to that from cracks propagating in sintered polymethylmet…
Using a dynamic functional renormalization group treatment of driven elastic interfaces in a disordered medium, we investigate several aspects of the creep-type motion induced by external forces below the depinning threshold $f_c$: i) We show that in the experimentally important regime of forces slightly below $f_c$ the velocity obeys an Arrhenius-type law $v\sim\exp[-U(f)/T]$ with an effective energy barrier $U(f)\propto (f_{c}-f)$ vanishing linearly when f approaches the threshold $f_c$. ii) Thermal fluctuations soften the pinning landscape at high temperatures. Determining the corresponding…
We study the steady state of driven elastic strings in disordered media below the depinning threshold. In the low-temperature limit, for a fixed sample, the steady state is dominated by a single configuration, which we determine exactly from the transition pathways between metastable states. We obtain the dynamical phase diagram in this limit. At variance with a thermodynamic phase transition, the depinning transition is not associated with a divergent length scale of the steady state below threshold, but only of the transient dynamics. We discuss the distribution of barrier heights, and check…
We review in these notes some dynamical properties of interfaces in random media submitted to an external force. We focuss in particular to the response to a very small force (so called creep motion) and discuss various theoretical aspects of this problem. We consider in details in particular the case of a one dimensional interface (domain wall).
Creep is a time-dependent deformation of solids at relatively low stresses, leading to the breakdown with time. Here we propose a simple model for creep failure of disordered solids, in which temperature and stress are controllable. Despite its simplicity, this model can reproduce most experimental observations. Time dependence of the strain rate is well fitted with power laws resembling the Omori-Utsu and the inverse Omori laws in the primary and the tertiary creep regimes, respectively. Distribution of the creep lifetime obeys the log-normal distribution, and the average creep lifetime decay…
We consider a particle in one dimension submitted to amplitude and phase disorder. It can be mapped onto the complex Burgers equation, and provides a toy model for problems with interplay of interferences and disorder, such as the NSS model of hopping conductivity in disordered insulators and the Chalker-Coddington model for the (spin) quantum Hall effect. The model has three distinct phases: (I) a {\em high-temperature} or weak disorder phase, (II) a {\em pinned} phase for strong amplitude disorder, and (III) a {\em diffusive} phase for strong phase disorder, but weak amplitude disorder. We c…
Rare Trajectories in a Prototypical Mean-field Disordered Model: Insights into Landscape and Instantons
Duke University（杜克大学）；Institute for Cross-disciplinary Physics and Complex Systems IFISC (CSIC-UIB)（交叉物理与复杂系统研究所（CSIC-UIB））；Bocconi University（博科尼大学）；Institute for Data Science and Analytics, Bocconi University（博科尼大学数据科学与分析研究所）；Institute of Complex Systems (ISC) - CNR, Rome unit（意大利国家研究委员会复杂系统研究所罗马分部）；Sapienza Università di Roma（罗马大学）
This thesis is an attempt to provide a new outlook on complex systems, as well as some physical answers for certain models, taking a computational approach. We have focused on disordered systems, addressing two traditional problems in three spatial dimensions: the Edwards-Anderson spin glass and the Diluted Antiferromagnet in a Field (the physical realisation of the random-field Ising model). These systems have been studied by means of large-scale Monte Carlo simulations, exploiting a variety of platforms, which include the Janus special-purpose supercomputer. Two main themes are explored thro…
We provide some analytical tests of the density of states estimation from the "localization landscape" approach of Ref. [Phys. Rev. Lett. 116, 056602 (2016)]. We consider two different solvable models for which we obtain the distribution of the landscape function and argue that the precise spectral singularities are not reproduced by the estimation of the landscape approach.
We show that the dynamics of simple disordered models, like the directed Trap Model and the Random Energy Model, takes place at a coexistence point between active and inactive dynamical phases. We relate the presence of a dynamic phase transition in these models to the extreme value statistics of the associated random energy landscape.
What is the best way to divide a rugged landscape? Since ancient times, watersheds separating adjacent water systems that flow, for example, toward different seas, have been used to delimit boundaries. Interestingly, serious and even tense border disputes between countries have relied on the subtle geometrical properties of these tortuous lines. For instance, slight and even anthropogenic modifications of landscapes can produce large changes in a watershed, and the effects can be highly nonlocal. Although the watershed concept arises naturally in geomorphology, where it plays a fundamental rol…
Landau's theory of phase transitions is adapted to treat independently relaxing regions in complex systems using nanothermodynamics. The order parameter we use governs the thermal fluctuations, not a specific static structure. We find that the entropy term dominates the thermal behavior, as is reasonable for disordered systems. Consequently, the thermal equilibrium occurs at the internal-energy maximum, so that the minima in a potential-energy landscape have negligible influence on the dynamics. Instead the dynamics involves normal thermal fluctuations about the free-energy minimum, with a tim…
Ground states of three-dimensional EA Ising spin glasses are calculated for sizes up to 14^3 using a combination of a genetic algorithm and cluster-exact approximation. For each realization several independent ground states are obtained. Then, by applying ballistic search and T=0 Monte-Carlo simulations, it is ensured that each ground state appears with the same probability. Consequently, the results represent the true T=0 thermodynamic behavior. The distribution P(|q|) of overlaps is evaluated. For increasing size the width of P(|q|) and the fraction of the distribution below q_0=0.5 converge…
Granular materials such as sand, powders, and grains are omnipresent in daily life, industrial applications, and earth-science [1]. When unperturbed, they form stable structures that resemble the ones of other amorphous solids like metallic and colloidal glasses [2]. It is commonly conjectured that all these amorphous materials show a universal mechanical response when sheared slowly, i.e., to have an elastic regime, followed by yielding [3]. Here we use X-ray tomography to determine the microscopic dynamics of a cyclically sheared granular system in three dimensions. Independent of the shear …
In this thesis I discuss analytical approaches to disordered systems using field theory. Disordered systems are characterized by a random energy landscape due to heterogeneities, which remains fixed on the time scales of the phenomena considered. I focus specifically on elastic interfaces in random media, such as pinned domain walls in ferromagnets containing defects, fluid contact lines on rough surfaces, etc. The goal is to understand static properties (e.g. the roughness) of such systems, and their dynamics in response to an external force. The interface moves in avalanches, triggered at ra…
We address the effect of disorder geometry on the critical force in disordered elastic systems. We focus on the model system of a long-range elastic line driven in a random landscape. In the collective pinning regime, we compute the critical force perturbatively. Not only our expression for the critical force confirms previous results on its scaling with respect to the microscopic disorder parameters, it also provides its precise dependence on the disorder geometry (represented by the disorder two-point correlation function). Our results are successfully compared to the results of numerical si…
The energy of an elastic manifold in a random landscape at T=0 is shown numerically to obey a probability distribution that depends on size of the box it is put into. If the extent of the spatial fluctuations of the manifold is much less than that of the system, a cross-over takes place to the Gumbel-distribution of extreme statistics. If they are comparable, the distributions have non-Gaussian, stretched exponential tails. The low-energy and high-energy stretching exponents are roughly independent of the internal dimension and the fluctuation degrees of freedom.
The presence of disorder in semiconductors can dramatically change their physical properties. Yet, models faithfully accounting for it are still scarce and computationally inefficient. We present a mathematical and computational model able to simulate the optoelectronic response of semiconductor alloys of several tens of nanometer sidelength, while at the same time accounting for the quantum localization effects induced by the compositional disorder at the nano-scale. The model is based on a Wigner-Weyl analysis of the structure of electron and hole eigenstates in phase space made possible by …
Biological evolution of a population is governed by the fitness landscape, which is a map from genotype to fitness. However, a fitness landscape depends on the organisms environment, and evolution in changing environments is still poorly understood. We study a particular model of antibiotic resistance evolution in bacteria where the antibiotic concentration is an environmental parameter and the fitness landscapes incorporate tradeoffs between adaptation to low and high antibiotic concentration. With evolutionary dynamics that follow fitness gradients, the evolution of the system under slowly c…
We consider the complexity of random ferromagnetic landscapes on the hypercube $\{\pm 1\}^N$ given by Ising models on the complete graph with i.i.d. non-negative edge-weights. This includes, in particular, the case of Bernoulli disorder corresponding to the Ising model on a dense random graph $\mathcal G(N,p)$. Previous results had shown that, with high probability as $N\to\infty$, the gradient search (energy-lowering) algorithm, initialized uniformly at random, converges to one of the homogeneous global minima (all-plus or all-minus). Here, we devise two modified algorithms tailored to explor…
We argue that the freezing transition scenario, previously conjectured to occur in the statistical mechanics of 1/f-noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials p_N(θ) of large N\times N random unitary (CUE) matrices; i.e. the extreme value statistics of p_N(θ) when N \rightarrow\infty. In addition, we argue that it leads to multifractal-like behaviour in the total length μ_N(x) of the intervals in which |p_N(θ)|>N^x, x>0, in the same limit. We speculate that our results extend to the large v…
This lecture deals with glassy dynamics and aging in disordered systems. Special emphasis is put on dynamic mean field theory. In the first part I present some of the systems of interest, in particular spin-glasses, supercooled liquids and glasses, drift, creep and pinning of a particle in a random potential, neural networks, graph partitioning as an example of combinatorial optimisation, the K-sat problem and the minority game as a model for the behaviour of agents on markets. The second part deals with the dynamics of the spherical p-spin-glass with long ranged interactions. This model is a …
We consider a model of an elastic manifold driven on a disordered energy landscape, with generalized long range elasticity. Varying the form of the elastic kernel by progressively allowing for the existence of zero-modes, the model interpolates smoothly between mean field depinning and finite dimensional yielding. We find that the critical exponents of the model change smoothly in this process. Also, we show that in all cases the Herschel-Buckley exponent of the flowcurve depends on the analytical form of the microscopic pinning potential. This is a compelling indication that within the presen…
We study numerically the depinning transition of driven elastic interfaces in a random-periodic medium with localized periodic-correlation peaks in the direction of motion. The analysis of the moving interface geometry reveals the existence of several characteristic lengths separating different length-scale regimes of roughness. We determine the scaling behavior of these lengths as a function of the velocity, temperature, driving force, and transverse periodicity. A dynamical roughness diagram is thus obtained which contains, at small length scales, the critical and fast-flow regimes typical o…
Learning the structure--dynamics correlation in disordered systems is a long-standing problem. Here, we use unsupervised machine learning employing graph neural networks (GNN) to investigate the local structures in disordered systems. We test our approach on 2D binary A65B35 LJ glasses and extract structures corresponding to liquid, supercooled and glassy states at different cooling rates. The neighborhood representation of atoms learned by a GNN in an unsupervised fashion, when clustered, reveal local structures with varying potential energies. These clusters exhibit dynamical heterogeneity i…
We report a comparative study of magnetic field driven domain wall motion in thin films made of different magnetic materials for a wide range of field and temperature. The full thermally activated creep motion, observed below the depinning threshold, is shown to be described by a unique universal energy barrier function. Our findings should be relevant for other systems whose dynamics can be modeled by elastic interfaces moving on disordered energy landscapes.
Effects of Disorder on the Energy Landscape and Motional Mechanisms Involved in Lithium Ion Dynamics and Transport in Solid Electrolytes: Li5.5PS4.5Cl1.5 Argyrodite as a Case Study
The Ma-Dasgupta real-space renormalization methods allow to study disordered systems which are governed by strong disorder fixed points. After a general introduction to the qualitative ideas and to the quantitative renormalization rules, we describe the explicit exact results that can be obtained in various one-dimensional models, either classical or quantum, either for dynamics or statics. The main part of this dissertation is devoted to statistical physics models, with special attention to (i) the off-equilibrium dynamics of a particle diffusing in a Brownian potential or in a trap landscape…
