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de Freitas Pimenta PH, Stariolo DA. Finite-Size Relaxational Dynamics of a Spike Random Matrix Spherical Model. ENTROPY (BASEL, SWITZERLAND) 2023; 25:957. [PMID: 37372301 DOI: 10.3390/e25060957] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/19/2023] [Revised: 06/15/2023] [Accepted: 06/17/2023] [Indexed: 06/29/2023]
Abstract
We present a thorough numerical analysis of the relaxational dynamics of the Sherrington-Kirkpatrick spherical model with an additive non-disordered perturbation for large but finite sizes N. In the thermodynamic limit and at low temperatures, the perturbation is responsible for a phase transition from a spin glass to a ferromagnetic phase. We show that finite-size effects induce the appearance of a distinctive slow regime in the relaxation dynamics, the extension of which depends on the size of the system and also on the strength of the non-disordered perturbation. The long time dynamics are characterized by the two largest eigenvalues of a spike random matrix which defines the model, and particularly by the statistics concerning the gap between them. We characterize the finite-size statistics of the two largest eigenvalues of the spike random matrices in the different regimes, sub-critical, critical, and super-critical, confirming some known results and anticipating others, even in the less studied critical regime. We also numerically characterize the finite-size statistics of the gap, which we hope may encourage analytical work which is lacking. Finally, we compute the finite-size scaling of the long time relaxation of the energy, showing the existence of power laws with exponents that depend on the strength of the non-disordered perturbation in a way that is governed by the finite-size statistics of the gap.
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Affiliation(s)
| | - Daniel A Stariolo
- Departamento de Física, National Institute of Science and Technology for Complex Systems, Universidade Federal Fluminense, Campus da Praia Vermelha, Av. Litorânea s/n, Niterói 24210-346, RJ, Brazil
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2
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Elizondo-Aguilera LF, Rizzo T, Voigtmann T. From Subaging to Hyperaging in Structural Glasses. PHYSICAL REVIEW LETTERS 2022; 129:238003. [PMID: 36563193 DOI: 10.1103/physrevlett.129.238003] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2022] [Revised: 07/14/2022] [Accepted: 11/01/2022] [Indexed: 06/17/2023]
Abstract
We demonstrate nonequilibrium scaling laws for the aging and equilibration dynamics in glass formers that emerge from combining a relaxation equation for the static structure with the equilibrium scaling laws of glassy dynamics. Different scaling regimes are predicted for the evolution of the structural relaxation time τ with age (waiting time t_{w}), depending on the depth of the quench from the liquid into the glass: "simple" aging (τ∼t_{w}) applies for quenches close to the critical point of mode-coupling theory (MCT) and implies "subaging" (τ≈t_{w}^{δ} with δ<1) as a broad equilibration crossover for quenches to nearly arrested equilibrium states; "hyperaging" (or superaging, τ∼t_{w}^{δ^{'}} with δ^{'}>1) emerges for quenches deep into the glass. The latter is cut off by non-mean-field fluctuations that we account for within a recent extension of MCT, the stochastic β-relaxation theory (SBR). We exemplify the scaling laws with a schematic model that quantitatively fits simulation data.
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Affiliation(s)
- Luis F Elizondo-Aguilera
- Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, 72520 Puebla, México
| | - Tommaso Rizzo
- Dipartimento di Fisica, Università di Roma I "La Sapienza," Piazzale A. Moro 2, I-00185 Rome, Italy
- ISC-CNR, UOS Roma, Università di Roma I "La Sapienza," Piazzale A. Moro 2, I-00185 Rome, Italy
| | - Thomas Voigtmann
- Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft-und Raumfahrt (DLR), Linder Höhe, 51170 Köln, Germany
- Department of Physics, Heinrich-Heine-Universität, Universitätsstraße 1, 40225 Düsseldorf, Germany
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3
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Diffusions interacting through a random matrix: universality via stochastic Taylor expansion. Probab Theory Relat Fields 2021. [DOI: 10.1007/s00440-021-01027-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
Abstract
AbstractConsider $$(X_{i}(t))$$
(
X
i
(
t
)
)
solving a system of N stochastic differential equations interacting through a random matrix $${\mathbf {J}} = (J_{ij})$$
J
=
(
J
ij
)
with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of $$(X_i(t))$$
(
X
i
(
t
)
)
, initialized from some $$\mu $$
μ
independent of $${\mathbf {J}}$$
J
, are universal, i.e., only depend on the choice of the distribution $$\mathbf {J}$$
J
through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks.
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4
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Metz FL, Castillo IP. Condensation of degrees emerging through a first-order phase transition in classical random graphs. Phys Rev E 2019; 100:012305. [PMID: 31499853 DOI: 10.1103/physreve.100.012305] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/17/2019] [Indexed: 06/10/2023]
Abstract
Due to their conceptual and mathematical simplicity, Erdös-Rényi or classical random graphs remain as a fundamental paradigm to model complex interacting systems in several areas. Although condensation phenomena have been widely considered in complex network theory, the condensation of degrees has hitherto eluded a careful study. Here we show that the degree statistics of the classical random graph model undergoes a first-order phase transition between a Poisson-like distribution and a condensed phase, the latter characterized by a large fraction of nodes having degrees in a limited sector of their configuration space. The mechanism underlying the first-order transition is discussed in light of standard concepts in statistical physics. We uncover the phase diagram characterizing the ensemble space of the model, and we evaluate the rate function governing the probability to observe a condensed state, which shows that condensation of degrees is a rare statistical event akin to similar condensation phenomena recently observed in several other systems. Monte Carlo simulations confirm the exactness of our theoretical results.
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Affiliation(s)
- Fernando L Metz
- Institute of Physics, Federal University of Rio Grande do Sul, 91501-970 Porto Alegre, Brazil; Physics Department, Federal University of Santa Maria, 97105-900 Santa Maria, Brazil; and London Mathematical Laboratory, 8 Margravine Gardens, London W6 8RH, United Kingdom
| | - Isaac Pérez Castillo
- Department of Quantum Physics and Photonics, Institute of Physics, UNAM, P.O. Box 20-364, 01000 Mexico City, Mexico, and London Mathematical Laboratory, 8 Margravine Gardens, London W6 8RH, United Kingdom
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5
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Martí D, Brunel N, Ostojic S. Correlations between synapses in pairs of neurons slow down dynamics in randomly connected neural networks. Phys Rev E 2018; 97:062314. [PMID: 30011528 DOI: 10.1103/physreve.97.062314] [Citation(s) in RCA: 30] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2017] [Indexed: 01/11/2023]
Abstract
Networks of randomly connected neurons are among the most popular models in theoretical neuroscience. The connectivity between neurons in the cortex is however not fully random, the simplest and most prominent deviation from randomness found in experimental data being the overrepresentation of bidirectional connections among pyramidal cells. Using numerical and analytical methods, we investigate the effects of partially symmetric connectivity on the dynamics in networks of rate units. We consider the two dynamical regimes exhibited by random neural networks: the weak-coupling regime, where the firing activity decays to a single fixed point unless the network is stimulated, and the strong-coupling or chaotic regime, characterized by internally generated fluctuating firing rates. In the weak-coupling regime, we compute analytically, for an arbitrary degree of symmetry, the autocorrelation of network activity in the presence of external noise. In the chaotic regime, we perform simulations to determine the timescale of the intrinsic fluctuations. In both cases, symmetry increases the characteristic asymptotic decay time of the autocorrelation function and therefore slows down the dynamics in the network.
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Affiliation(s)
- Daniel Martí
- Laboratoire de Neurosciences Cognitives, Inserm UMR No. 960, Ecole Normale Supérieure, PSL Research University, 75230 Paris, France
| | - Nicolas Brunel
- Department of Statistics and Department of Neurobiology, University of Chicago, Chicago, Illinois 60637, USA.,Department of Neurobiology and Department of Physics, Duke University, Durham, North Carolina 27710, USA
| | - Srdjan Ostojic
- Laboratoire de Neurosciences Cognitives, Inserm UMR No. 960, Ecole Normale Supérieure, PSL Research University, 75230 Paris, France
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Aurell E, Del Ferraro G, Domínguez E, Mulet R. Cavity master equation for the continuous time dynamics of discrete-spin models. Phys Rev E 2017; 95:052119. [PMID: 28618512 DOI: 10.1103/physreve.95.052119] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/25/2016] [Indexed: 06/07/2023]
Abstract
We present an alternate method to close the master equation representing the continuous time dynamics of interacting Ising spins. The method makes use of the theory of random point processes to derive a master equation for local conditional probabilities. We analytically test our solution studying two known cases, the dynamics of the mean-field ferromagnet and the dynamics of the one-dimensional Ising system. We present numerical results comparing our predictions with Monte Carlo simulations in three different models on random graphs with finite connectivity: the Ising ferromagnet, the random field Ising model, and the Viana-Bray spin-glass model.
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Affiliation(s)
- E Aurell
- Department of Computational Biology, AlbaNova University Center, SE-106 91 Stockholm, Sweden
- Departments of Applied Physics and Computer Science, Aalto University, FIN-00076 Aalto, Finland
- Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - G Del Ferraro
- Department of Computational Biology, AlbaNova University Center, SE-106, 91 Stockholm, Sweden
| | - E Domínguez
- Group of Complex Systems and Statistical Physics. Department of Theoretical Physics, Physics Faculty, University of Havana, Cuba
| | - R Mulet
- Group of Complex Systems and Statistical Physics. Department of Theoretical Physics, Physics Faculty, University of Havana, Cuba
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8
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Hase MO, Tomé T, de Oliveira MJ. Aging and fluctuation-dissipation ratio in a nonequilibrium q-state lattice model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:011133. [PMID: 20866591 DOI: 10.1103/physreve.82.011133] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/09/2010] [Revised: 06/28/2010] [Indexed: 05/29/2023]
Abstract
A generalized version of the nonequilibrium linear Glauber model with q states in d dimensions is introduced and analyzed. The model is fully symmetric, its dynamics being invariant under all permutations of the q states. Exact expressions for the two-time autocorrelation and response functions on a d-dimensional lattice are obtained. In the stationary regime, the fluctuation-dissipation theorem holds, while in the transient the aging is observed with the fluctuation-dissipation ratio leading to the value predicted for the linear Glauber model.
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Affiliation(s)
- M O Hase
- Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05314-970 São Paulo, Brazil
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9
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van Duijvendijk K, Jack RL, van Wijland F. Second-order dynamic transition in a p=2 spin-glass model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:011110. [PMID: 20365326 DOI: 10.1103/physreve.81.011110] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/02/2009] [Indexed: 05/29/2023]
Abstract
We consider the dynamics of a disordered p -spin model with p=2 , analyzing the dynamics within Ruelle's thermodynamic formalism, We use an indicator of the dynamical activity to construct the relevant dynamical Gibbs ensemble. We prove that the dynamics in the low-temperature (spin-glass) phase of the model take place at a second-order phase transition between dynamically active and inactive trajectories. We also show that the same behavior is found in a related model of a three-dimensional ferromagnet.
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Affiliation(s)
- Kristina van Duijvendijk
- Laboratoire Matière et Systèmes Complexes (CNRS UMR 7057), Université Paris Diderot-Paris, 75205 Paris Cedex 13, France
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10
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Aron C, Biroli G, Cugliandolo LF. Driven quantum coarsening. PHYSICAL REVIEW LETTERS 2009; 102:050404. [PMID: 19257491 DOI: 10.1103/physrevlett.102.050404] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/02/2008] [Indexed: 05/27/2023]
Abstract
We study the driven dynamics of quantum coarsening. We analyze models of M-component rotors coupled to two electronic reservoirs at different chemical potential that generate a current threading through the system. In the large M limit, we derive the dynamical phase diagram as a function of temperature, strength of quantum fluctuations, voltage, and coupling to the leads. We show that the slow relaxation in the ordering phase is universal. On large time and length scales, the dynamics are analogous to stochastic classical ones, even for the quantum system driven out of equilibrium at zero temperature. We argue that our results apply to generic driven quantum coarsening.
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Affiliation(s)
- Camille Aron
- Université Pierre et Marie Curie - Paris VI, LPTHE UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France
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11
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Ben Arous G, Dembo A, Guionnet A. Cugliandolo-Kurchan equations for dynamics of Spin-Glasses. Probab Theory Relat Fields 2006. [DOI: 10.1007/s00440-005-0491-y] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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12
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13
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Abstract
Mycotoxins are natural contaminants whose presence in food- and feedstuffs cannot be completely avoided. Since several mycotoxins have been associated with and implicated in human and animal diseases there is a need to establish maximum levels, guidelines or action levels for them in some kinds of commodities. International and government authorities in many countries have been investing in mycotoxins research and initiating administrative actions for elaboration of legislation and implementing regulatory measures for the control of mycotoxins. Codex Alimentarius Commission is established international legislation on food and feed. In European Union specific limits and regulations for mycotoxins and other contaminants are constructed under the general Codex standards and based on proposal from European Commission. The legal basis for European Commission became available with the framework Council Regulation (EEC) No 315/93. In this paper, legislation regarding maximum levels for certain mycotoxins in food- and feedstuffs in European Community and other countries were reviewed and discussed.
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Affiliation(s)
- J Zmudzki
- Department of Pharmacology and Toxicology, National Veterinary Institute in Puławy, Al Partyzantów 57, 24-100 Puławy, Poland.
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14
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Corberi F, Lippiello E, Zannetti M. Scaling of the linear response function from zero-field-cooled and thermoremanent magnetization in phase-ordering kinetics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:046131. [PMID: 14683026 DOI: 10.1103/physreve.68.046131] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/13/2003] [Indexed: 05/24/2023]
Abstract
In this paper we investigate the relation between the scaling properties of the linear response function R(t,s), of the thermoremanent magnetization (TRM) and of the zero-field-cooled (ZFC) magnetization in the context of phase-ordering kinetics. We explain why the retrieval of the scaling properties of R(t,s) from those of TRM and ZFC magnetization is not trivial. Preasymptotic contributions generate a long crossover in TRM, while ZFC magnetization is affected by a dangerous irrelevant variable. Lack of understanding of both these points has generated some confusion in the literature. The full picture relating the exponents of all the quantities involved is explicitly illustrated in the framework of the large-N model. Following this scheme, an assessment of the present status of numerical simulations for the Ising model can be made. We reach the conclusion that on the basis of the data available up to now, statements on the scaling properties of R(t,s) can be made from ZFC magnetization but not from TRM. From ZFC data for the Ising model with d=2,3,4 we confirm the previously found linear dependence on dimensionality of the exponent a entering R(t,s) approximately s(-(1+a))f(t/s). We also find evidence that a recently derived form of the scaling function f(x), using local scale invariance arguments [M. Henkel, M. Pleimling, C. Godrèche, and J. M. Luck, Phys. Rev. Lett. 87, 265701 (2001)], does not hold for the Ising model.
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Affiliation(s)
- Federico Corberi
- Istituto Nazionale di Fisica della Materia, Unità di Salerno and Dipartimento di Fisica E. Caianiello, Università di Salerno, 84081 Baronissi (Salerno), Italy.
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15
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Diezemann G. Dynamic heterogeneities in the out-of-equilibrium dynamics of simple spherical spin models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:021105. [PMID: 14524951 DOI: 10.1103/physreve.68.021105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2003] [Indexed: 05/24/2023]
Abstract
The response of spherical two-spin interaction models, the spherical ferromagnet (s-FM) and the spherical Sherrington-Kirkpatrick (s-SK) model, is calculated for the protocol of the so-called nonresonant hole burning (NHB) experiment for temperatures below the respective critical temperatures. It is shown that it is possible to select dynamic features in the out-of-equilibrium dynamics of both models, one of the hallmarks of dynamic heterogeneities. The behavior of the s-SK model and the s-FM model in three dimensions is very similar, showing dynamic heterogeneities in the long-time behavior, i.e., in the aging regime. The appearance of dynamic heterogeneities in the s-SK model explicitly demonstrates that these are not necessarily related to spatial heterogeneities. For the s-FM model, it is shown that the nature of the dynamic heterogeneities changes as a function of dimensionality. With the increasing dimension, the frequency selectivity of the NHB diminishes and the dynamics in the mean-field limit of the s-FM model becomes homogeneous.
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Affiliation(s)
- Gregor Diezemann
- Institut für Physikalische Chemie, Universität Mainz, Welderweg 11, 55099 Mainz, Germany
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16
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Mayer P, Berthier L, Garrahan JP, Sollich P. Fluctuation-dissipation relations in the nonequilibrium critical dynamics of Ising models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:016116. [PMID: 12935209 DOI: 10.1103/physreve.68.016116] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/28/2003] [Indexed: 05/24/2023]
Abstract
We investigate the relation between two-time multispin correlation and response functions in the nonequilibrium critical dynamics of Ising models in d=1 and d=2 spatial dimensions. In these nonequilibrium situations, the fluctuation-dissipation theorem (FDT) is not satisfied. We find FDT "violations" qualitatively similar to those reported in various glassy materials, but quantitatively dependent on the chosen observable, in contrast to the results obtained in infinite-range glass models. Nevertheless, all FDT violations can be understood by considering separately the contributions from large wave vectors, which are at quasiequilibrium and obey the FDT, and from small wave vectors where a generalized FDT holds with a nontrivial fluctuation-dissipation ratio X infinity. In d=1, we get X(infinity)=1/2 for spin observables, which measure the orientation of domains, while X(infinity)=0 for observables that are sensitive to the domain-wall motion. Numerical simulations in d=2 reveal a unique X infinity approximately equal 0.34 for all observables. Measurement protocols for X infinity are discussed in detail. Our results suggest that the definition of an effective temperature T(eff)=T/X(infinity) for large length scales is generically possible in nonequilibrium critical dynamics.
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Affiliation(s)
- Peter Mayer
- Department of Mathematics, King's College, Strand, London WC2R 2LS, United Kingdom
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17
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Crisanti A, Ritort F. Violation of the fluctuation–dissipation theorem in glassy systems: basic notions and the numerical evidence. ACTA ACUST UNITED AC 2003. [DOI: 10.1088/0305-4470/36/21/201] [Citation(s) in RCA: 291] [Impact Index Per Article: 13.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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18
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Picone A, Henkel M, Richert J. Competition between dynamic and thermal relaxation in non-equilibrium critical spin systems above the critical point. ACTA ACUST UNITED AC 2003. [DOI: 10.1088/0305-4470/36/5/305] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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19
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Calabrese P, Gambassi A. Two-loop critical fluctuation-dissipation ratio for the relaxational dynamics of the O(N) Landau-Ginzburg Hamiltonian. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:066101. [PMID: 12513341 DOI: 10.1103/physreve.66.066101] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/22/2002] [Revised: 09/12/2002] [Indexed: 05/24/2023]
Abstract
The off-equilibrium purely dissipative dynamics (model A) of the O(N) vector model is considered at criticality in an epsilon=4-d>0 expansion up to O(epsilon(2)). The scaling behavior of two-time response and correlation functions at zero momentum, the associated universal scaling functions and the nontrivial limit of the fluctuation-dissipation ratio are determined in the aging regime.
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Affiliation(s)
- Pasquale Calabrese
- Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, I-56126 Pisa, Italy.
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20
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Fusco N, Zannetti M. Relaxation and overlap-probability function in the spherical and mean-spherical models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:066113. [PMID: 12513353 DOI: 10.1103/physreve.66.066113] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2002] [Indexed: 05/24/2023]
Abstract
The problem of the equivalence of the spherical and mean-spherical models, which has been thoroughly studied and is understood in equilibrium, is considered anew from the dynamical point of view during the time evolution following a quench from above to below the critical temperature. It is found that there exists a crossover time t(*) approximately V(2/d) such that for t<t(*) the two models are equivalent, while for t>t(*) macroscopic discrepancies arise. The relation between the off equilibrium response function and the structure of the equilibrium state that usually holds for phase ordering systems is found to hold for the spherical model but not for the mean-spherical one. The latter model offers an explicit example of a system which is not stochastically stable.
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Affiliation(s)
- Nicola Fusco
- Istituto Nazionale per la Fisica della Materia, Unità di Salerno, Università di Salerno, 84081 Baronissi (Salerno), Italy.
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21
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Caiazzo A, Coniglio A, Nicodemi M. Dynamics and thermodynamics of the spherical frustrated Blume-Emery-Griffiths model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:046101. [PMID: 12443253 DOI: 10.1103/physreve.66.046101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/05/2002] [Indexed: 05/24/2023]
Abstract
We introduce a spherical version of the frustrated Blume-Emery-Griffiths model and solve exactly the statics and the Langevin dynamics for zero particle-particle interaction (K=0). In this case the model exhibits an equilibrium transition from a disordered to a spin glass phase, which is always continuous for nonzero temperature. The same phase diagram results from the study of the dynamics. Furthermore, we note the existence of a nonequilibrium time regime in a region of the disordered phase, characterized by aging, as occurs in the glassy phase. Due to a finite equilibration time, the system displays in this region the pattern of interrupted aging.
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Affiliation(s)
- A Caiazzo
- Dipartimento di Scienze Fisiche, INFM, Unitá di Napoli, Monte Sant'Angelo, I-80126 Napoli, Italy
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22
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Picone A, Henkel M. Response of non-equilibrium systems with long-range initial correlations. ACTA ACUST UNITED AC 2002. [DOI: 10.1088/0305-4470/35/27/304] [Citation(s) in RCA: 57] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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23
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Calabrese P, Gambassi A. Aging in ferromagnetic systems at criticality near four dimensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:066120. [PMID: 12188796 DOI: 10.1103/physreve.65.066120] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/19/2002] [Indexed: 05/23/2023]
Abstract
We study the off-equilibrium response and correlation functions and the corresponding fluctuation-dissipation ratio for a purely dissipative relaxation of an O(N) symmetric vector model (model A) below its upper critical dimension. The scaling behavior of these quantities is analyzed and the associated universal functions are determined at first order in epsilon=4-d in the high-temperature phase and at criticality. A nontrivial limit of the fluctuation-dissipation ratio is found in the aging regime X( infinity )=1/2(1-(epsilon/4)(N+2)/(N+8))+O(epsilon(2)).
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Affiliation(s)
- Pasquale Calabrese
- Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, I-56126 Pisa, Italy.
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24
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Corberi F, de Candia A, Lippiello E, Zannetti M. Off-equilibrium response function in the one-dimensional random-field Ising model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:046114. [PMID: 12005933 DOI: 10.1103/physreve.65.046114] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/20/2001] [Indexed: 05/23/2023]
Abstract
A thorough numerical investigation of the slow dynamics in the d=1 random-field Ising model in the limit of an infinite ferromagnetic coupling is presented in this paper. Crossovers from the preasymptotic pure regime to the asymptotic Sinai regime are investigated for the average domain size, the autocorrelation function, and staggered magnetization. By switching on an additional small random field at the time t(w) the linear off-equilibrium response function is obtained, which displays as well the crossover from the nontrivial behavior of the d=1 pure Ising model to the asymptotic behavior where it vanishes identically.
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Affiliation(s)
- F Corberi
- Istituto Nazionale di Fisica della Materia, Unità di Salerno and Dipartimento di Fisica E. Caianiello, Università di Salerno, 84081 Baronissi, Salerno, Italy.
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Corberi F, Lippiello E, Zannetti M. Slow relaxation in the large-N model for phase ordering. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:046136. [PMID: 12005955 DOI: 10.1103/physreve.65.046136] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2001] [Indexed: 05/23/2023]
Abstract
The basic features of the slow relaxation phenomenology arising in phase ordering processes are obtained analytically in the large-N model through the exact separation of the order parameter into the sum of thermal and condensation components. The aging contribution in the response function chi(ag)(t,t(w)) is found to obey a pattern of behavior, under variation of dimensionality, qualitatively similar to the one observed in Ising systems. There exists a critical dimensionality (d=4) above which chi(ag)(t,t(w)) is proportional to the defect density rho(D)(t), while for d<4 it vanishes more slowly than rho(D)(t) and at d=2 does not vanish. As in the Ising case, this behavior can be understood in terms of the dependence on dimensionality of the interplay between the defect density and the effective response associated to a single defect.
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Affiliation(s)
- Federico Corberi
- Istituto Nazionale per la Fisica della Materia, Unità di Salerno and Dipartimento di Fisica, E. Caianiello, Università di Salerno, 84081 Baronissi, Salerno, Italy.
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26
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Kawasaki M. Out-of-equilibrium thermodynamic relations in systems with aging and slow relaxation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:046145. [PMID: 12005964 DOI: 10.1103/physreve.65.046145] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/05/2000] [Revised: 12/27/2001] [Indexed: 05/23/2023]
Abstract
The experimental time scale dependence of thermodynamic relations in out-of-equilibrium systems with aging phenomena is investigated theoretically by using only aging properties of the two-time correlation functions and the generalized fluctuation-dissipation theorem. We show that there are two experimental time regimes characterized by different thermal properties. In the first regime where the waiting time is much longer than the measurement time, the principle of minimum work holds even though a system is out of equilibrium. In the second regime where both the measurement time and the waiting time are long, the thermal properties are completely different from properties in equilibrium. For the single-correlation-scale systems such as p-spin spherical spin glasses, contrary to a fundamental assumption of thermodynamics, the work done in an infinitely slow operation depends on the path of change of the external field even when the waiting time is infinite. On the other hand, for the multi-correlation-scale systems such as Sherrington-Kirkpatrick model, the work done in an infinitely slow operation is independent of the path. Our results imply that in order to describe thermodynamic properties of systems with aging it is essential to consider the experimental time scales and history of a system as a state variable is necessary.
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Arous GB, Bovier A, Gayrard V. Aging in the random energy model. PHYSICAL REVIEW LETTERS 2002; 88:087201. [PMID: 11863971 DOI: 10.1103/physrevlett.88.087201] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/08/2001] [Indexed: 05/23/2023]
Abstract
The random energy model (REM) has become a key reference model for glassy systems. In particular, it is expected to provide a prime example of a system whose dynamics shows aging, a universal phenomenon characterizing the dynamics of complex systems. The analysis of its activated dynamics is based on so-called trap models, introduced by Bouchaud, that are also used to mimic the dynamics of more complex disordered systems. In this Letter we report the first results that justify rigorously the trap model predictions in the REM.
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Fielding S, Sollich P. Observable dependence of fluctuation-dissipation relations and effective temperatures. PHYSICAL REVIEW LETTERS 2002; 88:050603. [PMID: 11863711 DOI: 10.1103/physrevlett.88.050603] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/01/2001] [Indexed: 05/23/2023]
Abstract
We study the nonequilibrium fluctuation-dissipation theorem (FDT) in the glass phase of Bouchaud's trap model. We incorporate an arbitrary observable m and obtain its correlation and response functions in closed form. A limiting nonequilibrium FDT plot is approached at long times for most choices of m. In contrast to standard mean field models, however, the shape of the plot depends nontrivially on the observable, and its slope varies continuously even though there is a single scaling of relaxation times with age. Nonequilibrium FDT plots can therefore not be used to define a meaningful effective temperature T(eff) in this model. Consequences for the wider applicability of an FDT-derived T(eff) are discussed.
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Affiliation(s)
- Suzanne Fielding
- Department of Physics and Astronomy, University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom
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Henkel M, Pleimling M, Godrèche C, Luck JM. Aging, phase ordering, and conformal invariance. PHYSICAL REVIEW LETTERS 2001; 87:265701. [PMID: 11800840 DOI: 10.1103/physrevlett.87.265701] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2001] [Indexed: 05/23/2023]
Abstract
In a variety of systems which exhibit aging, the two-time response function scales as R(t,s) approximately s(-1-a)f(t/s). We argue that dynamical scaling can be extended towards conformal invariance, thus obtaining the explicit form of the scaling function f. This quantitative prediction is confirmed in several spin systems, both for T<T(c) (phase ordering) and T = T(c) (nonequilibrium critical dynamics). The 2D and 3D Ising models with Glauber dynamics are studied numerically, while exact results are available for the spherical model with a nonconserved order parameter, both for short-ranged and long-ranged interactions, as well as for the mean-field spherical spin glass.
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Affiliation(s)
- M Henkel
- Laboratoire de Physique des Matériaux, Université Henri Poincaré Nancy I, BP 239, F-54506 Vandoeuvre lès Nancy Cedex, France
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Berthier L, Cugliandolo LF, Iguain JL. Glassy systems under time-dependent driving forces: application to slow granular rheology. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:051302. [PMID: 11414897 DOI: 10.1103/physreve.63.051302] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/19/2000] [Indexed: 05/23/2023]
Abstract
We study the dynamics of a glassy model with infinite range interactions externally driven by an oscillatory force. We find a well-defined transition in the (temperature-amplitude-frequency) phase diagram between (i) a "glassy" state characterized by the slow relaxation of one-time quantities, aging in two-time quantities and a modification of the equilibrium fluctuation-dissipation relation; and (ii) a "liquid" state with a finite relaxation time. In the glassy phase, the degrees of freedom governing the slow relaxation are thermalized to an effective temperature. Using Monte Carlo simulations, we investigate the effect of trapping regions in phase space on the driven dynamics. We find that it alternates between periods of rapid motion and periods of trapping. These results confirm the strong analogies between the slow granular rheology and the dynamics of glasses. They also provide a theoretical underpinning to earlier attempts to present a thermodynamic description of moderately driven granular materials.
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Affiliation(s)
- L Berthier
- Laboratoire de Physique, Ecole Normale Supérieure de Lyon, 46 Allée d'Italie, F-69007 Lyon, France
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Barrat A, Kurchan J, Loreto V, Sellitto M. Edwards' measures: a thermodynamic construction for dense granular media and glasses. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:051301. [PMID: 11414896 DOI: 10.1103/physreve.63.051301] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/29/2000] [Indexed: 05/23/2023]
Abstract
We present numerical support for the hypothesis that macroscopic observables of dense granular media and glasses can be evaluated from averages over typical blocked configurations: we construct the corresponding measure for a class of finite-dimensional systems and compare its predictions for various observables with the outcome of the out of equilibrium dynamics at large times. We discuss in detail the connection with the effective temperatures that appear in out of equilibrium glass theories, as well as the relation between our computation and those based on "inherent structure" arguments. A short version of this work has appeared in Phys. Rev. Lett. 85, 5034 (2000).
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Affiliation(s)
- A Barrat
- Laboratoire de Physique Théorique, CNRS UMR 8627, Université de Paris-Sud, Bâtiment 210, 91405 Orsay, France
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32
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Berthier L, Holdsworth PCW, Sellitto M. Nonequilibrium critical dynamics of the two-dimensionalXYmodel. ACTA ACUST UNITED AC 2001. [DOI: 10.1088/0305-4470/34/9/301] [Citation(s) in RCA: 105] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Barrat A, Kurchan J, Loreto V, Sellitto M. Edwards' measures for powders and glasses. PHYSICAL REVIEW LETTERS 2000; 85:5034-5037. [PMID: 11102180 DOI: 10.1103/physrevlett.85.5034] [Citation(s) in RCA: 29] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2000] [Indexed: 05/23/2023]
Abstract
Can one construct a thermodynamics for compact, slowly moving powders and grains? A few years ago, Edwards proposed a possible step in this direction, raising the fascinating perspective that such systems have a statistical mechanics of their own, different from that of Maxwell, Boltzmann, and Gibbs, allowing us to have some information while still ignoring dynamic details. Recent developments in the theory of glasses have come to confirm these ideas within mean field. In order to go beyond, we explicitly generate Edwards' measure in a 3D model. Comparison of the results with the irreversible compaction data shows very good agreement. The present framework immediately suggests new experimental checks.
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Affiliation(s)
- A Barrat
- Laboratoire de Physique Theorique, Batiment 210, Universite de Paris-Sud, 91405 Orsay Cedex, France
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34
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Godrèche C, Luck JM. Response of non-equilibrium systems at criticality: ferromagnetic models in dimension two and above. ACTA ACUST UNITED AC 2000. [DOI: 10.1088/0305-4470/33/50/302] [Citation(s) in RCA: 150] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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35
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Lippiello E, Zannetti M. Fluctuation dissipation ratio in the one-dimensional kinetic ising model. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:3369-3374. [PMID: 11088111 DOI: 10.1103/physreve.61.3369] [Citation(s) in RCA: 21] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/22/1999] [Indexed: 05/23/2023]
Abstract
The exact relation between the response function R(t,t(')) and the two time correlation function C(t,t(')) is derived analytically in the one-dimensional kinetic Ising model subjected to a temperature quench. The fluctuation dissipation ratio X(t,t(')) is found to depend on time through C(t,t(')) in the time region where scaling C(t,t('))=f(t/t(')) holds. The crossover from the nontrivial form X[C(t,t('))] to X(t,t(')) identical with1 takes place as the waiting time t(w) is increased from below to above the equilibration time t(eq).
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Affiliation(s)
- E Lippiello
- Istituto Nazionale di Fisica della Materia, Unita di Salerno and Dipartimento di Fisica, Universita di Salerno, 84081 Baronissi, Salerno, Italy
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36
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Heerema M, Ritort F. Damage spreading transition in glasses: a probe for the ruggedness of the configurational landscape. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:3646-65. [PMID: 11970198 DOI: 10.1103/physreve.60.3646] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/1998] [Revised: 06/10/1999] [Indexed: 04/18/2023]
Abstract
We consider damage spreading transitions in the framework of mode-coupling theory. This theory describes relaxation processes in glasses in the mean-field approximation which are known to be characterized by the presence of an exponentially large number of metastable states. For systems evolving under identical but arbitrarily correlated noises, we demonstrate that there exists a critical temperature T0 which separates two different dynamical regimes depending on whether damage spreads or not in the asymptotic long-time limit. This transition exists for generic noise correlations such that the zero damage solution is stable at high temperatures, being minimal for maximal noise correlations. Although this dynamical transition depends on the type of noise correlations, we show that the asymptotic damage has the good properties of a dynamical order parameter, such as (i) independence of the initial damage; (ii) independence of the class of initial condition; and (iii) stability of the transition in the presence of asymmetric interactions which violate detailed balance. For maximally correlated noises we suggest that damage spreading occurs due to the presence of a divergent number of saddle points (as well as metastable states) in the thermodynamic limit consequence of the ruggedness of the free-energy landscape which characterizes the glassy state. These results are then compared to extensive numerical simulations of a mean-field glass model (the Bernasconi model) with Monte Carlo heat-bath dynamics. The freedom of choosing arbitrary noise correlations for Langevin dynamics makes damage spreading an interesting tool to probe the ruggedness of the configurational landscape.
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Affiliation(s)
- M Heerema
- Institute of Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands.
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39
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Parisi G, Ranieri P, Ricci-Tersenghi F, Ruiz-Lorenzo JJ. Mean field dynamical exponents in finite-dimensional Ising spin glass. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/30/20/015] [Citation(s) in RCA: 22] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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41
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Laughton SN, Coolen ACC, Sherrington D. Order-parameter flow in the SK spin-glass: II. Inclusion of microscopic memory effects. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/29/4/007] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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42
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Campellone M, Parisi G, Ranieri P. Dynamical fluctuations in an exactly solvable model of spin glasses. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/31/8/003] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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43
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Cugliandolo LF, Dean DS. On the dynamics of a spherical spin-glass in a magnetic field. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/28/17/001] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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44
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Cugliandolo LF, Kurchan J, Monasson R, Parisi G. A mean-field hard-spheres model of glass. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/29/7/007] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Thirumalai D, Ashwin V, Bhattacharjee JK. Dynamics of Random Hydrophobic-Hydrophilic Copolymers with Implications for Protein Folding. PHYSICAL REVIEW LETTERS 1996; 77:5385-5388. [PMID: 10062790 DOI: 10.1103/physrevlett.77.5385] [Citation(s) in RCA: 20] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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47
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Bonilla LL, Padilla FG, Parisi G, Ritort F. Closure of the Monte Carlo dynamical equations in the spherical Sherrington-Kirkpatrick model. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 54:4170-4182. [PMID: 9986321 DOI: 10.1103/physrevb.54.4170] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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48
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Coolen AC, Laughton SN, Sherrington D. Dynamical replica theory for disordered spin systems. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 53:8184-8187. [PMID: 9982303 DOI: 10.1103/physrevb.53.8184] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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49
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Cugliandolo LF. Large time nonequilibrium dynamics of a particle in a random potential. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:1525-1552. [PMID: 9964416 DOI: 10.1103/physreve.53.1525] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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