1
|
Wiese KJ. Theory and experiments for disordered elastic manifolds, depinning, avalanches, and sandpiles. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2022; 85:086502. [PMID: 35943081 DOI: 10.1088/1361-6633/ac4648] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2021] [Accepted: 12/23/2021] [Indexed: 06/15/2023]
Abstract
Domain walls in magnets, vortex lattices in superconductors, contact lines at depinning, and many other systems can be modeled as an elastic system subject to quenched disorder. The ensuing field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group (RG) flow involves a function, the disorder correlator Δ(w), and is therefore termed the functional RG. Δ(w) is a physical observable, the auto-correlation function of the center of mass of the elastic manifold. In this review, we give a pedagogical introduction into its phenomenology and techniques. This allows us to treat both equilibrium (statics), and depinning (dynamics). Building on these techniques, avalanche observables are accessible: distributions of size, duration, and velocity, as well as the spatial and temporal shape. Various equivalences between disordered elastic manifolds, and sandpile models exist: an elastic string driven at a point and the Oslo model; disordered elastic manifolds and Manna sandpiles; charge density waves and Abelian sandpiles or loop-erased random walks. Each of the mappings between these systems requires specific techniques, which we develop, including modeling of discrete stochastic systems via coarse-grained stochastic equations of motion, super-symmetry techniques, and cellular automata. Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and non-linear surface growth with additional Kardar-Parisi-Zhang (KPZ) terms. On the other hand, KPZ without disorder can be mapped back to disordered elastic manifolds, either on the directed polymer for its steady state, or a single particle for its decay. Other topics covered are the relation between functional RG and replica symmetry breaking, and random-field magnets. Emphasis is given to numerical and experimental tests of the theory.
Collapse
Affiliation(s)
- Kay Jörg Wiese
- Laboratoire de physique, Département de physique de l'ENS, École normale supérieure, UPMC Univ. Paris 06, CNRS, PSL Research University, 75005 Paris, France
| |
Collapse
|
2
|
Tückmantel P, Gaponenko I, Caballero N, Agar JC, Martin LW, Giamarchi T, Paruch P. Local Probe Comparison of Ferroelectric Switching Event Statistics in the Creep and Depinning Regimes in Pb(Zr_{0.2}Ti_{0.8})O_{3} Thin Films. PHYSICAL REVIEW LETTERS 2021; 126:117601. [PMID: 33798378 DOI: 10.1103/physrevlett.126.117601] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/25/2020] [Revised: 10/28/2020] [Accepted: 01/26/2021] [Indexed: 06/12/2023]
Abstract
Ferroelectric materials provide a useful model system to explore the jerky, highly nonlinear dynamics of elastic interfaces in disordered media. The distribution of nanoscale switching event sizes is studied in two Pb(Zr_{0.2}Ti_{0.8})O_{3} thin films with different disorder landscapes using piezoresponse force microscopy. While the switching event statistics show the expected power-law scaling, significant variations in the value of the scaling exponent τ are seen, possibly as a consequence of the different intrinsic disorder landscapes in the samples and of further alterations under high tip bias applied during domain writing. Importantly, higher exponent values (1.98-2.87) are observed when crackling statistics are acquired only for events occurring in the creep regime. The exponents are systematically lowered when all events across both creep and depinning regimes are considered-the first time such a distinction is made in studies of ferroelectric materials. These results show that distinguishing the two regimes is of crucial importance, significantly affecting the exponent value and potentially leading to incorrect assignment of universality class.
Collapse
Affiliation(s)
- Philippe Tückmantel
- Department of Quantum Matter Physics, University of Geneva, CH-1211, Geneva, Switzerland
| | - Iaroslav Gaponenko
- Department of Quantum Matter Physics, University of Geneva, CH-1211, Geneva, Switzerland
| | - Nirvana Caballero
- Department of Quantum Matter Physics, University of Geneva, CH-1211, Geneva, Switzerland
| | - Joshua C Agar
- Department of Material Science and Engineering, University of California, Berkeley, California 94720, USA
| | - Lane W Martin
- Department of Material Science and Engineering, University of California, Berkeley, California 94720, USA
| | - Thierry Giamarchi
- Department of Quantum Matter Physics, University of Geneva, CH-1211, Geneva, Switzerland
| | - Patrycja Paruch
- Department of Quantum Matter Physics, University of Geneva, CH-1211, Geneva, Switzerland
| |
Collapse
|
3
|
Le Priol C, Le Doussal P, Rosso A. Spatial Clustering of Depinning Avalanches in Presence of Long-Range Interactions. PHYSICAL REVIEW LETTERS 2021; 126:025702. [PMID: 33512216 DOI: 10.1103/physrevlett.126.025702] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/25/2020] [Revised: 10/14/2020] [Accepted: 12/11/2020] [Indexed: 06/12/2023]
Abstract
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.
Collapse
Affiliation(s)
- Clément Le Priol
- Laboratoire de Physique de l'Ecole Normale Supérieure, Université PSL, CNRS, Sorbonne Université, Université de Paris, 24 rue Lhomond, 75231 Paris Cedex, France
| | - Pierre Le Doussal
- Laboratoire de Physique de l'Ecole Normale Supérieure, Université PSL, CNRS, Sorbonne Université, Université de Paris, 24 rue Lhomond, 75231 Paris Cedex, France
| | - Alberto Rosso
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
4
|
Balog I, Tarjus G, Tissier M. Dimensional reduction breakdown and correction to scaling in the random-field Ising model. Phys Rev E 2020; 102:062154. [PMID: 33466013 DOI: 10.1103/physreve.102.062154] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2020] [Accepted: 11/30/2020] [Indexed: 11/07/2022]
Abstract
We provide a theoretical analysis by means of the nonperturbative functional renormalization group (NP-FRG) of the corrections to scaling in the critical behavior of the random-field Ising model (RFIM) near the dimension d_{DR}≈5.1 that separates a region where the renormalized theory at the fixed point is supersymmetric and critical scaling satisfies the d→d-2 dimensional reduction property (d>d_{DR}) from a region where both supersymmetry and dimensional reduction break down at criticality (d<d_{DR}). We show that the NP-FRG results are in very good agreement with recent large-scale lattice simulations of the RFIM in d=5 and we detail the consequences for the leading correction-to-scaling exponent of the peculiar boundary-layer mechanism by which the dimensional-reduction fixed point disappears and the dimensional-reduction-broken fixed point emerges in d_{DR}.
Collapse
Affiliation(s)
- Ivan Balog
- Institute of Physics, P.O. Box 304, Bijenička cesta 46, HR-10001 Zagreb, Croatia
| | - Gilles Tarjus
- LPTMC, CNRS-UMR 7600, Sorbonne Université, Boîte 121, 4 Pl. Jussieu, 75252 Paris cedex 05, France
| | - Matthieu Tissier
- LPTMC, CNRS-UMR 7600, Sorbonne Université, Boîte 121, 4 Pl. Jussieu, 75252 Paris cedex 05, France
| |
Collapse
|
5
|
Clemmer JT, Robbins MO. Anisotropic avalanches and critical depinning of three-dimensional magnetic domain walls. Phys Rev E 2019; 100:042121. [PMID: 31770980 DOI: 10.1103/physreve.100.042121] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/25/2019] [Indexed: 11/07/2022]
Abstract
Simulations with more than 10^{12} spins are used to study the motion of a domain wall driven through a three-dimensional random-field Ising magnet (RFIM) by an external field H. The interface advances in a series of avalanches whose size diverges at a critical external field H_{c}. Finite-size scaling is applied to determine critical exponents and test scaling relations. Growth is intrinsically anisotropic with the height of an avalanche normal to the interface ℓ_{⊥} scaling as the width along the interface ℓ_{∥} to a power χ=0.85±0.01. The total interface roughness is consistent with self-affine scaling with a roughness exponent ζ≈χ that is much larger than values found previously for the RFIM and related models that explicitly break orientational symmetry by requiring the interface to be single-valued. Because the RFIM maintains orientational symmetry, the interface develops overhangs that may surround unfavorable regions to create uninvaded bubbles. Overhangs complicate measures of the roughness exponent but decrease in importance with increasing system size.
Collapse
Affiliation(s)
- Joel T Clemmer
- Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA
| | - Mark O Robbins
- Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA
| |
Collapse
|
6
|
Dupuis N. Glassy properties of the Bose-glass phase of a one-dimensional disordered Bose fluid. Phys Rev E 2019; 100:030102. [PMID: 31640005 DOI: 10.1103/physreve.100.030102] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/10/2019] [Indexed: 06/10/2023]
Abstract
We study a one-dimensional disordered Bose fluid using bosonization, the replica method, and a nonperturbative functional renormalization-group approach. The Bose-glass phase is described by a fully attractive strong-disorder fixed point characterized by a singular disorder correlator whose functional dependence assumes a cuspy form that is related to the existence of metastable states. At nonzero momentum scale, quantum tunneling between these metastable states leads to a rounding of the nonanalyticity in a quantum boundary layer that encodes the existence of rare superfluid regions responsible for the ω^{2} behavior of the (dissipative) conductivity in the low-frequency limit. These results can be understood within the "droplet" picture put forward for the description of glassy (classical) systems.
Collapse
Affiliation(s)
- Nicolas Dupuis
- Laboratoire de Physique Théorique de la Matière Condensée, LPTMC, Sorbonne Université, CNRS, F-75005 Paris, France
| |
Collapse
|
7
|
Baity PG, Sasagawa T, Popović D. Collective Dynamics and Strong Pinning near the Onset of Charge Order in La_{1.48}Nd_{0.4}Sr_{0.12}CuO_{4}. PHYSICAL REVIEW LETTERS 2018; 120:156602. [PMID: 29756879 DOI: 10.1103/physrevlett.120.156602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2016] [Revised: 11/22/2017] [Indexed: 06/08/2023]
Abstract
The dynamics of charge-ordered states is one of the key issues in underdoped cuprate high-temperature superconductors, but static short-range charge-order (CO) domains have been detected in almost all cuprates. We probe the dynamics across the CO (and structural) transition in La_{1.48}Nd_{0.4}Sr_{0.12}CuO_{4} by measuring nonequilibrium charge transport, or resistance R as the system responds to a change in temperature and to an applied magnetic field. We find evidence for metastable states, collective behavior, and criticality. The collective dynamics in the critical regime indicates strong pinning by disorder. Surprisingly, nonequilibrium effects, such as avalanches in R, are revealed only when the critical region is approached from the charge-ordered phase. Our results on La_{1.48}Nd_{0.4}Sr_{0.12}CuO_{4} provide the long-sought evidence for the fluctuating order across the CO transition, and also set important constraints on theories of dynamic stripes.
Collapse
Affiliation(s)
- P G Baity
- National High Magnetic Field Laboratory and Department of Physics, Florida State University, Tallahassee, Florida 32306, USA
| | - T Sasagawa
- Materials and Structures Laboratory, Tokyo Institute of Technology, Kanagawa 226-8503, Japan
| | - Dragana Popović
- National High Magnetic Field Laboratory and Department of Physics, Florida State University, Tallahassee, Florida 32306, USA
| |
Collapse
|
8
|
Abstract
In disordered elastic systems, driven by displacing a parabolic confining potential adiabatically slowly, all advance of the system is in bursts, termed avalanches. Avalanches have a finite extension in time, which is much smaller than the waiting time between them. Avalanches also have a finite extension ℓ in space, i.e., only a part of the interface of size ℓ moves during an avalanche. Here we study their spatial shape 〈S(x)〉_{ℓ} given ℓ, as well as its fluctuations encoded in the second cumulant 〈S^{2}(x)〉_{ℓ}^{c}. We establish scaling relations governing the behavior close to the boundary. We then give analytic results for the Brownian force model, in which the microscopic disorder for each degree of freedom is a random walk. Finally, we confirm these results with numerical simulations. To do this properly we elucidate the influence of discretization effects, which also confirms the assumptions entering into the scaling ansatz. This allows us to reach the scaling limit already for avalanches of moderate size. We find excellent agreement for the universal shape and its fluctuations, including all amplitudes.
Collapse
Affiliation(s)
- Zhaoxuan Zhu
- CNRS, Laboratoire de Physique Théorique, École Normale Supérieure, PSL Research University, Sorbonne Universités, UPMC, 24 Rue Lhomond, 75005 Paris, France
| | - Kay Jörg Wiese
- CNRS, Laboratoire de Physique Théorique, École Normale Supérieure, PSL Research University, Sorbonne Universités, UPMC, 24 Rue Lhomond, 75005 Paris, France
| |
Collapse
|
9
|
Ferrero EE, Foini L, Giamarchi T, Kolton AB, Rosso A. Spatiotemporal Patterns in Ultraslow Domain Wall Creep Dynamics. PHYSICAL REVIEW LETTERS 2017; 118:147208. [PMID: 28430477 DOI: 10.1103/physrevlett.118.147208] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/08/2016] [Indexed: 06/07/2023]
Abstract
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.
Collapse
Affiliation(s)
- Ezequiel E Ferrero
- Université Grenoble Alpes, LIPHY, F-38000 Grenoble, France and CNRS, LIPHY, F-38000 Grenoble, France
| | - Laura Foini
- Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva, Switzerland
| | - Thierry Giamarchi
- Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva, Switzerland
| | - Alejandro B Kolton
- Instituto Balseiro-UNCu and CONICET, Centro Atómico Bariloche, 8400 Bariloche, Argentina
| | - Alberto Rosso
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| |
Collapse
|
10
|
Hentschel HGE, Procaccia I, Gupta BS. Anatomy of plastic events in magnetic amorphous solids. Phys Rev E 2016; 93:033004. [PMID: 27078438 DOI: 10.1103/physreve.93.033004] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/25/2015] [Indexed: 11/07/2022]
Abstract
Plastic events in amorphous solids can be much more than just "shear transformation zones" when the positional degrees of freedom are coupled nontrivially to other degrees of freedom. Here we consider magnetic amorphous solids where mechanical and magnetic degrees of freedom interact, leading to rather complex plastic events whose nature must be disentangled. In this paper we uncover the anatomy of the various contributions to some typical plastic events. These plastic events are seen as Barkhausen noise or other "serrated noises." Using theoretical considerations we explain the observed statistics of the various contributions to the considered plastic events. The richness of contributions and their different characteristics imply that in general the statistics of these serrated noises cannot be universal, but rather highly dependent on the state of the system and on its microscopic interactions.
Collapse
Affiliation(s)
- H George E Hentschel
- Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
| | - Itamar Procaccia
- Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
| | - Bhaskar Sen Gupta
- Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
| |
Collapse
|
11
|
Chevalier T, Talon L. Moving line model and avalanche statistics of Bingham fluid flow in porous media. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2015; 38:76. [PMID: 26187726 DOI: 10.1140/epje/i2015-15076-5] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/26/2015] [Revised: 05/27/2015] [Accepted: 06/09/2015] [Indexed: 06/04/2023]
Abstract
In this article, we propose a simple model to understand the critical behavior of path opening during flow of a yield stress fluid in porous media as numerically observed by Chevalier and Talon (2015). This model can be mapped to the problem of a contact line moving in an heterogeneous field. Close to the critical point, this line presents an avalanche dynamic where the front advances by a succession of waiting time and large burst events. These burst events are then related to the non-flowing (i.e. unyielded) areas. Remarkably, the statistics of these areas reproduce the same properties as in the direct numerical simulations. Furthermore, even if our exponents seem to be close to the mean field universal exponents, we report an unusual bump in the distribution which depends on the disorder. Finally, we identify a scaling invariance of the cluster spatial shape that is well fit, to first order, by a self-affine parabola.
Collapse
Affiliation(s)
- Thibaud Chevalier
- CNRS, Laboratoire FAST, UMR 7608, Université Paris-Sud, F-91405, Orsay, France,
| | | |
Collapse
|
12
|
Sharma A, Andreanov A, Müller M. Avalanches and hysteresis in frustrated superconductors and XY spin glasses. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042103. [PMID: 25375434 DOI: 10.1103/physreve.90.042103] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/26/2014] [Indexed: 06/04/2023]
Abstract
We study avalanches along the hysteresis loop of long-range interacting spin glasses with continuous XY symmetry, which serves as a toy model of granular superconductors with long-range and frustrated Josephson couplings. We identify sudden jumps in the T=0 configurations of the XY phases as an external field is increased. They are initiated by the softest mode of the inverse susceptibility matrix becoming unstable, which induces an avalanche of phase updates (or spin alignments). We analyze the statistics of these events and study the correlation between the nonlinear avalanches and the soft mode that initiates them. We find that the avalanches follow the directions of a small fraction of the softest modes of the inverse susceptibility matrix, similarly as was found in avalanches in jammed systems. In contrast to the similar Ising spin glass (Sherrington-Kirkpatrick) studied previously, we find that avalanches are not distributed with a scale-free power law but rather have a typical size which scales with the system size. We also observe that the Hessians of the spin-glass minima are not part of standard random matrix ensembles as the lowest eigenvector has a fractal support.
Collapse
Affiliation(s)
- Auditya Sharma
- International Institute of Physics, Federal University of Rio Grande do Norte, Natal, RN, Brazil and Tel Aviv University, Tel Aviv, Israel
| | - Alexei Andreanov
- The Abdus Salam ICTP, Strada Costiera 11, I-34151 Trieste, Italy and Max Planck Institute for Physics of Complex Systems Nöthnitzer Str. 38, 01187 Dresden, Germany
| | - Markus Müller
- The Abdus Salam ICTP, Strada Costiera 11, I-34151 Trieste, Italy
| |
Collapse
|
13
|
Dobrinevski A, Le Doussal P, Wiese KJ. Statistics of avalanches with relaxation and Barkhausen noise: a solvable model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:032106. [PMID: 24125213 DOI: 10.1103/physreve.88.032106] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2013] [Indexed: 06/02/2023]
Abstract
We study a generalization of the Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model of a particle in a Brownian force landscape, including retardation effects. We show that under monotonous driving the particle moves forward at all times, as it does in absence of retardation (Middleton's theorem). This remarkable property allows us to develop an analytical treatment. The model with an exponentially decaying memory kernel is realized in Barkhausen experiments with eddy-current relaxation and has previously been shown numerically to account for the experimentally observed asymmetry of Barkhausen pulse shapes. We elucidate another qualitatively new feature: the breakup of each avalanche of the standard ABBM model into a cluster of subavalanches, sharply delimited for slow relaxation under quasistatic driving. These conditions are typical for earthquake dynamics. With relaxation and aftershock clustering, the present model includes important ingredients for an effective description of earthquakes. We analyze quantitatively the limits of slow and fast relaxation for stationary driving with velocity v>0. The v-dependent power-law exponent for small velocities, and the critical driving velocity at which the particle velocity never vanishes, are modified. We also analyze nonstationary avalanches following a step in the driving magnetic field. Analytically, we obtain the mean avalanche shape at fixed size, the duration distribution of the first subavalanche, and the time dependence of the mean velocity. We propose to study these observables in experiments, allowing a direct measurement of the shape of the memory kernel and tracing eddy current relaxation in Barkhausen noise.
Collapse
Affiliation(s)
- Alexander Dobrinevski
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75005 Paris, France
| | | | | |
Collapse
|
14
|
Le Doussal P, Wiese KJ. Avalanche dynamics of elastic interfaces. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:022106. [PMID: 24032774 DOI: 10.1103/physreve.88.022106] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/27/2013] [Indexed: 06/02/2023]
Abstract
Slowly driven elastic interfaces, such as domain walls in dirty magnets, contact lines wetting a nonhomogeneous substrate, or cracks in brittle disordered material proceed via intermittent motion, called avalanches. Here we develop a field-theoretic treatment to calculate, from first principles, the space-time statistics of instantaneous velocities within an avalanche. For elastic interfaces at (or above) their (internal) upper critical dimension d≥d(uc) (d(uc)=2,4 respectively for long-ranged and short-ranged elasticity) we show that the field theory for the center of mass reduces to the motion of a point particle in a random-force landscape, which is itself a random walk [Alessandro, Beatrice, Bertotti, and Montorsi (ABBM) model]. Furthermore, the full spatial dependence of the velocity correlations is described by the Brownian-force model (BFM) where each point of the interface sees an independent Brownian-force landscape. Both ABBM and BFM can be solved exactly in any dimension d (for monotonous driving) by summing tree graphs, equivalent to solving a (nonlinear) instanton equation. We focus on the limit of slow uniform driving. This tree approximation is the mean-field theory (MFT) for realistic interfaces in short-ranged disorder, up to the renormalization of two parameters at d=d(uc). We calculate a number of observables of direct experimental interest: Both for the center of mass, and for a given Fourier mode q, we obtain various correlations and probability distribution functions (PDF's) of the velocity inside an avalanche, as well as the avalanche shape and its fluctuations (second shape). Within MFT we find that velocity correlations at nonzero q are asymmetric under time reversal. Next we calculate, beyond MFT, i.e., including loop corrections, the one-time PDF of the center-of-mass velocity u[over ·] for dimension d<d(uc). The singularity at small velocity P(u[over ·])~1/u[over ·](a) is substantially reduced from a=1 (MFT) to a=1-2/9(4-d)+... (short-ranged elasticity) and a=1-4/9(2-d)+... (long-ranged elasticity). We show how the dynamical theory recovers the avalanche-size distribution, and how the instanton relates to the response to an infinitesimal step in the force.
Collapse
Affiliation(s)
- Pierre Le Doussal
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75005 Paris, France
| | | |
Collapse
|
15
|
Tarjus G, Baczyk M, Tissier M. Avalanches and dimensional reduction breakdown in the critical behavior of disordered systems. PHYSICAL REVIEW LETTERS 2013; 110:135703. [PMID: 23581342 DOI: 10.1103/physrevlett.110.135703] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/23/2012] [Indexed: 06/02/2023]
Abstract
We investigate the connection between a formal property of the critical behavior of several disordered systems, known as "dimensional reduction," and the presence in these systems at zero temperature of collective events known as "avalanches." Avalanches generically produce nonanalyticities in the functional dependence of the cumulants of the renormalized disorder. We show that this leads to a breakdown of the dimensional reduction predictions if and only if the fractal dimension characterizing the scaling properties of the avalanches is exactly equal to the difference between the dimension of space and the scaling dimension of the primary field. This is proven by combining scaling theory and the functional renormalization group. We therefore clarify the puzzle of why dimensional reduction remains valid in random field systems above a nontrivial dimension (but fails below), always applies to the statistics of branched polymer, and is always wrong in elastic models of interfaces in a random environment.
Collapse
Affiliation(s)
- Gilles Tarjus
- LPTMC, CNRS-UMR 7600, Université Pierre et Marie Curie, Boîte 121, 4 Place Jussieu, 75252 Paris Cédex 05, France.
| | | | | |
Collapse
|
16
|
Agoritsas E, Bustingorry S, Lecomte V, Schehr G, Giamarchi T. Finite-temperature and finite-time scaling of the directed polymer free energy with respect to its geometrical fluctuations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:031144. [PMID: 23030903 DOI: 10.1103/physreve.86.031144] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/28/2012] [Indexed: 06/01/2023]
Abstract
We study the fluctuations of the directed polymer in 1+1 dimensions in a Gaussian random environment with a finite correlation length ξ and at finite temperature. We address the correspondence between the geometrical transverse fluctuations of the directed polymer, described by its roughness, and the fluctuations of its free energy, characterized by its two-point correlator. Analytical arguments are provided in favor of a generic scaling law between those quantities, at finite time, nonvanishing ξ, and explicit temperature dependence. Numerical results are in good agreement both for simulations on the discrete directed polymer and on a continuous directed polymer (with short-range correlated disorder). Applications to recent experiments on liquid crystals are discussed.
Collapse
Affiliation(s)
- Elisabeth Agoritsas
- DPMC-MaNEP, Université de Genève, 24 quai Ernest Ansermet, 1211 Genève, Switzerland
| | | | | | | | | |
Collapse
|
17
|
Le Doussal P, Wiese KJ. First-principles derivation of static avalanche-size distributions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:061102. [PMID: 23005046 DOI: 10.1103/physreve.85.061102] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/11/2011] [Indexed: 06/01/2023]
Abstract
We study the energy minimization problem for an elastic interface in a random potential plus a quadratic well. As the position of the well is varied, the ground state undergoes jumps, called shocks or static avalanches. We introduce an efficient and systematic method to compute the statistics of avalanche sizes and manifold displacements. The tree-level calculation, i.e., mean-field limit, is obtained by solving a saddle-point equation. Graphically, it can be interpreted as the sum of all tree graphs. The 1-loop corrections are computed using results from the functional renormalization group. At the upper critical dimension the shock statistics is described by the Brownian force model (BFM), the static version of the so-called Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model in the nonequilibrium context of depinning. This model can itself be treated exactly in any dimension and its shock statistics is that of a Lévy process. Contact is made with classical results in probability theory on the Burgers equation with Brownian initial conditions. In particular we obtain a functional extension of an evolution equation introduced by Carraro and Duchon, which recursively constructs the tree diagrams in the field theory.
Collapse
Affiliation(s)
- Pierre Le Doussal
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75005 Paris, France
| | | |
Collapse
|
18
|
Dobrinevski A, Le Doussal P, Wiese KJ. Nonstationary dynamics of the Alessandro-Beatrice-Bertotti-Montorsi model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:031105. [PMID: 22587036 DOI: 10.1103/physreve.85.031105] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/29/2011] [Indexed: 05/31/2023]
Abstract
We obtain an exact solution for the motion of a particle driven by a spring in a Brownian random-force landscape, the Alessandro-Beatrice-Bertotti-Montorsi (ABBM) model. Many experiments on quasistatic driving of elastic interfaces (Barkhausen noise in magnets, earthquake statistics, shear dynamics of granular matter) exhibit the same universal behavior as this model. It also appears as a limit in the field theory of elastic manifolds. Here we discuss predictions of the ABBM model for monotonous, but otherwise arbitrary, time-dependent driving. Our main result is an explicit formula for the generating functional of particle velocities and positions. We apply this to derive the particle-velocity distribution following a quench in the driving velocity. We also obtain the joint avalanche size and duration distribution and the mean avalanche shape following a jump in the position of the confining spring. Such nonstationary driving is easy to realize in experiments, and provides a way to test the ABBM model beyond the stationary, quasistatic regime. We study extensions to two elastically coupled layers, and to an elastic interface of internal dimension d, in the Brownian force landscape. The effective action of the field theory is equal to the action, up to one-loop corrections obtained exactly from a functional determinant. This provides a connection to renormalization-group methods.
Collapse
Affiliation(s)
- Alexander Dobrinevski
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, Paris, France.
| | | | | |
Collapse
|
19
|
Wiese KJ, Majumdar SN, Rosso A. Perturbation theory for fractional Brownian motion in presence of absorbing boundaries. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:061141. [PMID: 21797336 DOI: 10.1103/physreve.83.061141] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/01/2010] [Indexed: 05/31/2023]
Abstract
Fractional Brownian motion is a Gaussian process x(t) with zero mean and two-time correlations (x(t(1))x(t(2)))=D(t(1)(2H)+t(2)(2H)-|t(1)-t(2)|(2H)), where H, with 0<H<1, is called the Hurst exponent. For H=1/2, x(t) is a Brownian motion, while for H≠1/2, x(t) is a non-Markovian process. Here we study x(t) in presence of an absorbing boundary at the origin and focus on the probability density P(+)(x,t) for the process to arrive at x at time t, starting near the origin at time 0, given that it has never crossed the origin. It has a scaling form P(+)(x,t)~t(-H)R(+)(x/t(H)). Our objective is to compute the scaling function R(+)(y), which up to now was only known for the Markov case H=1/2. We develop a systematic perturbation theory around this limit, setting H=1/2+ε, to calculate the scaling function R(+)(y) to first order in ε. We find that R(+)(y) behaves as R(+)(y)~y(ϕ) as y→0 (near the absorbing boundary), while R(+)(y)~y(γ)exp(-y(2)/2) as y→∞, with ϕ=1-4ε+O(ε(2)) and γ=1-2ε+O(ε(2)). Our ε-expansion result confirms the scaling relation ϕ=(1-H)/H proposed in Zoia, Rosso, and Majumdar [Phys. Rev. Lett. 102, 120602 (2009)]. We verify our findings via numerical simulations for H=2/3. The tools developed here are versatile, powerful, and adaptable to different situations.
Collapse
Affiliation(s)
- Kay Jörg Wiese
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, Paris, France
| | | | | |
Collapse
|
20
|
Dobrinevski A, Le Doussal P, Wiese KJ. Interference in disordered systems: a particle in a complex random landscape. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:061116. [PMID: 21797311 DOI: 10.1103/physreve.83.061116] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2011] [Indexed: 05/31/2023]
Abstract
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 Nguyen-Spivak-Shklovskii model of hopping conductivity in disordered insulators and the Chalker-Coddington model for the (spin) quantum Hall effect. We also propose a direct realization in an experiment with cold atoms. The model has three distinct phases: (I) a high-temperature or weak disorder phase, (II) a pinned phase for strong amplitude disorder, and (III) a diffusive phase for strong phase disorder, but weak amplitude disorder. We compute analytically the renormalized disorder correlator, equivalent to the Burgers velocity-velocity correlator at long times. In phase III, it assumes a universal form. For strong phase disorder, interference leads to a logarithmic singularity, related to zeros of the partition sum, or poles of the complex Burgers velocity field. These results are valuable in the search for the adequate field theory for higher-dimensional systems.
Collapse
Affiliation(s)
- Alexander Dobrinevski
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, Paris, France.
| | | | | |
Collapse
|
21
|
Le Doussal P, Wiese KJ. Elasticity of a contact-line and avalanche-size distribution at depinning. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:011108. [PMID: 20866566 DOI: 10.1103/physreve.82.011108] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/14/2010] [Indexed: 05/29/2023]
Abstract
Motivated by recent experiments, we extend the Joanny-deGennes calculation of the elasticity of a contact line to an arbitrary contact angle and an arbitrary plate inclination in presence of gravity. This requires a diagonalization of the elastic modes around the nonlinear equilibrium profile, which is carried out exactly. We then make detailed predictions for the avalanche-size distribution at quasistatic depinning: we study how the universal (i.e., short-scale independent) rescaled size distribution and the ratio of moments of local to global avalanches depend on the precise form of the elastic kernel.
Collapse
Affiliation(s)
- Pierre Le Doussal
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 73231 Paris Cedex 05, France
| | | |
Collapse
|
22
|
Jagla EA, Kolton AB. A mechanism for spatial and temporal earthquake clustering. ACTA ACUST UNITED AC 2010. [DOI: 10.1029/2009jb006974] [Citation(s) in RCA: 48] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
|
23
|
Le Doussal P, Wiese KJ. Size distributions of shocks and static avalanches from the functional renormalization group. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:051106. [PMID: 19518415 DOI: 10.1103/physreve.79.051106] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/20/2009] [Indexed: 05/27/2023]
Abstract
Interfaces pinned by quenched disorder are often used to model jerky self-organized critical motion. We study static avalanches, or shocks, defined here as jumps between distinct global minima upon changing an external field. We show how the full statistics of these jumps is encoded in the functional-renormalization-group fixed-point functions. This allows us to obtain the size distribution P(S) of static avalanches in an expansion in the internal dimension d of the interface. Near and above d=4 this yields the mean-field distribution P(S) approximately S;{-3/2}e;{-S4S_{m}} , where S_{m} is a large-scale cutoff, in some cases calculable. Resumming all one-loop contributions, we find P(S) approximately S;{-tau}exp(C(SS_{m});{1/2}-B/4(S/S_{m});{delta}) , where B , C , delta , and tau are obtained to first order in =4-d . Our result is consistent to O() with the relation tau=tau_{zeta}:=2-2/d+zeta , where zeta is the static roughness exponent, often conjectured to hold at depinning. Our calculation applies to all static universality classes, including random-bond, random-field, and random-periodic disorders. Extended to long-range elastic systems, it yields a different size distribution for the case of contact-line elasticity, with an exponent compatible with tau=2-1/d+zeta to O(=2-d) . We discuss consequences for avalanches at depinning and for sandpile models, relations to Burgers turbulence and the possibility that the relation tau=tau_{zeta} be violated to higher loop order. Finally, we show that the avalanche-size distribution on a hyperplane of codimension one is in mean field (valid close to and above d=4 ) given by P(S) approximately K_{13}(S)S , where K is the Bessel- K function, thus tau_{hyperplane}=4/3 .
Collapse
Affiliation(s)
- Pierre Le Doussal
- Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, CNRS, 24 rue Lhomond, 75231 Paris Cedex, France
| | | |
Collapse
|
24
|
Le Doussal P, Wiese KJ. Driven particle in a random landscape: disorder correlator, avalanche distribution, and extreme value statistics of records. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:051105. [PMID: 19518414 DOI: 10.1103/physreve.79.051105] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/30/2008] [Indexed: 05/27/2023]
Abstract
We review how the renormalized force correlator Delta(micro) , the function computed in the functional renormalization-group (RG) field theory, can be measured directly in numerics and experiments on the dynamics of elastic manifolds in the presence of pinning disorder. We show how this function can be computed analytically for a particle dragged through a one-dimensional random-force landscape. The limit of small velocity allows one to access the critical behavior at the depinning transition. For uncorrelated forces one finds three universality classes, corresponding to the three extreme value statistics, Gumbel, Weibull, and Fréchet. For each class we obtain analytically the universal function Delta(micro) , the corrections to the critical force, and the joint probability distribution of avalanche sizes s and waiting times w . We find P(s)=P(w) for all three cases. All results are checked numerically. For a Brownian force landscape, known as the Alessandro, Beatrice, Bertotti, and Montorsi (ABBM) model, avalanche distributions and Delta(micro) can be computed for any velocity. For two-dimensional disorder, we perform large-scale numerical simulations to calculate the renormalized force correlator tensor Delta_{ij}(micro[over ]) , and to extract the anisotropic scaling exponents zeta_{x}>zeta_{y} . We also show how the Middleton theorem is violated. Our results are relevant for the record statistics of random sequences with linear trends, as encountered, e.g., in some models of global warming. We give the joint distribution of the time s between two successive records and their difference in value w .
Collapse
Affiliation(s)
- Pierre Le Doussal
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France
| | | |
Collapse
|