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Almeida RAL, Takeuchi KA. Phase-ordering kinetics in the Allen-Cahn (Model A) class: Universal aspects elucidated by electrically induced transition in liquid crystals. Phys Rev E 2021; 104:054103. [PMID: 34942720 DOI: 10.1103/physreve.104.054103] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/20/2021] [Accepted: 09/27/2021] [Indexed: 11/07/2022]
Abstract
The two-dimensional (2D) Ising model is the statistical physics textbook example for phase transitions and their kinetics. Quenched through the Curie point with Glauber rates, the late-time description of the ferromagnetic domain coarsening finds its place at the scalar sector of the Allen-Cahn (or Model A) class, which encompasses phase-ordering kinetics endowed with a nonconserved order parameter. Resisting exact results sought for theoreticians since Lifshitz's first account in 1962, the central quantities of 2D Model A-most scaling exponents and correlation functions-remain known up to approximate theories whose disparate outcomes urge experimental assessment. Here we perform such assessment based on a comprehensive study of the coarsening of 2D twisted nematic liquid crystals whose kinetics is induced by a superfast electrical switching from a spatiotemporally chaotic (disordered) state to a two-phase concurrent, equilibrium one. Tracking the dynamics via optical microscopy, we first show the sharp evidence of well-established Model A aspects, such as the dynamic exponent z=2 and the dynamic scaling hypothesis, to then move forward. We confirm the Bray-Humayun theory for Porod's regime describing intradomain length scales of the two-point spatial correlators and show that their nontrivial decay beyond the Porod's scale can be captured in a free-from-parameter fashion by Gaussian theories, namely the Ohta-Jasnow-Kawasaki (OJK) and Mazenko theories. Regarding time-related statistics, we corroborate the aging hypothesis in Model A systems, which includes the collapse of two-time correlators into a master curve whose format is, actually, best accounted for by a solution of the local scaling invariance theory: the same solution that fits the 2D nonconserved Ising model correlator along with the Fisher-Huse conjecture. We also suggest the true value for the local persistence exponent in Model A class, in disfavor of the exact outcome for the diffusion and OJK equations. Finally, we observe a fractal morphology for persistence clusters and extract their universal dimension. Given its accuracy and possibilities, this experimental setup may work as a prototype to address further universality issues in the realm of nonequilibrium systems.
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Affiliation(s)
- Renan A L Almeida
- Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan.,Departmento de Física, Universidade Federal de Viçosa, 36570-900 Viçosa, MG, Brazil.,Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Kazumasa A Takeuchi
- Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
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2
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Corberi F, Lippiello E, Politi P. Quasideterministic dynamics, memory effects, and lack of self-averaging in the relaxation of quenched ferromagnets. Phys Rev E 2020; 102:020102. [PMID: 32942398 DOI: 10.1103/physreve.102.020102] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/13/2020] [Accepted: 08/06/2020] [Indexed: 11/07/2022]
Abstract
We discuss the interplay between the degree of dynamical stochasticity, memory persistence, and violation of the self-averaging property in the aging kinetics of quenched ferromagnets. We show that, in general, the longest possible memory effects, which correspond to the slowest possible temporal decay of the correlation function, are accompanied by the largest possible violation of self-averaging and a quasideterministic descent into the ergodic components. This phenomenon is observed in different systems, such as the Ising model with long-range interactions, including the mean-field, and the short-range random-field Ising model.
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Affiliation(s)
- Federico Corberi
- Dipartimento di Fisica "E. R. Caianiello," and INFN, Gruppo Collegato di Salerno, and CNISM, Unità di Salerno, Università di Salerno, via Giovanni Paolo II 132, 84084 Fisciano (SA), Italy
| | - Eugenio Lippiello
- Dipartimento di Matematica e Fisica, Università della Campania, Viale Lincoln 5, 81100 Caserta, Italy
| | - Paolo Politi
- Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy.,INFN Sezione di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Italy
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3
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Fierro A, Coniglio A, Zannetti M. Relation between statics and dynamics in the quench of the Ising model to below the critical point. Phys Rev E 2020; 102:012144. [PMID: 32794899 DOI: 10.1103/physreve.102.012144] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/11/2020] [Accepted: 06/16/2020] [Indexed: 11/07/2022]
Abstract
The standard phase-ordering process is obtained by quenching a system, like the Ising model, to below the critical point. This is usually done with periodic boundary conditions to ensure ergodicity breaking in the low-temperature phase. With this arrangement the infinite system is known to remain permanently out of equilibrium, i.e., there exists a well-defined asymptotic state which is time invariant but different from the ordered ferromagnetic state. In this paper we establish the critical nature of this invariant state by demonstrating numerically that the quench dynamics with periodic and antiperiodic boundary conditions are indistinguishable from each other. However, while the asymptotic state does not coincide with the equilibrium state for the periodic case, it coincides instead with the equilibrium state of the antiperiodic case, which in fact is critical. The specific example of the Ising model is shown to be one instance of a more general phenomenon, since an analogous picture emerges in the spherical model, where boundary conditions are kept fixed to periodic, while the breaking or preserving of ergodicity is managed by imposing the spherical constraint either sharply or smoothly.
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Affiliation(s)
- Annalisa Fierro
- CNR-SPIN, c/o Complesso di Monte S. Angelo, I-80126 Napoli, Italy
| | - Antonio Coniglio
- CNR-SPIN, c/o Complesso di Monte S. Angelo, I-80126 Napoli, Italy.,Physics Department, Università degli Studi di Napoli "Federico II", I-80126 Napoli, Italy
| | - Marco Zannetti
- Dipartimento di Fisica "E. R. Caianiello," Università di Salerno, I-84084 Fisciano (SA), Italy
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Albarracín FAG, Rosales HD, Grynberg MD. Phase ordering dynamics of reconstituting particles. Phys Rev E 2017; 95:062130. [PMID: 28709284 DOI: 10.1103/physreve.95.062130] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/30/2017] [Indexed: 11/07/2022]
Abstract
We consider the large-time dynamics of one-dimensional processes involving adsorption and desorption of extended hard-core particles (dimers, trimers, ..., k-mers), while interacting through their constituent monomers. Desorption can occur whether or not these latter adsorbed together, which leads to reconstitution of k-mers and the appearance of sectors of motion with nonlocal conservation laws for k≥3. Dynamic exponents of the sector including the empty chain are evaluated by finite-size scaling analyses of the relaxation times embodied in the spectral gaps of evolution operators. For attractive interactions it is found that in the low-temperature limit such time scales converge to those of the Glauber dynamics, thus suggesting a diffusive universality class for k≥2. This is also tested by simulated quenches down to T=0, where a common scaling function emerges. By contrast, under repulsive interactions the low-temperature dynamics is characterized by metastable states which decay subdiffusively to a highly degenerate and partially jammed phase.
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Affiliation(s)
- F A Gómez Albarracín
- IFLP-CONICET, Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina
| | - H D Rosales
- IFLP-CONICET, Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina
| | - M D Grynberg
- IFLP-CONICET, Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina
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Lee CH, Lucas A. Simple model for multiple-choice collective decision making. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:052804. [PMID: 25493831 DOI: 10.1103/physreve.90.052804] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/09/2014] [Indexed: 06/04/2023]
Abstract
We describe a simple model of heterogeneous, interacting agents making decisions between n≥2 discrete choices. For a special class of interactions, our model is the mean field description of random field Potts-like models and is effectively solved by finding the extrema of the average energy E per agent. In these cases, by studying the propagation of decision changes via avalanches, we argue that macroscopic dynamics is well captured by a gradient flow along E. We focus on the permutation symmetric case, where all n choices are (on average) the same, and spontaneous symmetry breaking (SSB) arises purely from cooperative social interactions. As examples, we show that bimodal heterogeneity naturally provides a mechanism for the spontaneous formation of hierarchies between decisions and that SSB is a preferred instability to discontinuous phase transitions between two symmetric points. Beyond the mean field limit, exponentially many stable equilibria emerge when we place this model on a graph of finite mean degree. We conclude with speculation on decision making with persistent collective oscillations. Throughout the paper, we emphasize analogies between methods of solution to our model and common intuition from diverse areas of physics, including statistical physics and electromagnetism.
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Affiliation(s)
- Ching Hua Lee
- Department of Physics, Stanford University, Stanford, California 94305, USA
| | - Andrew Lucas
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
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Marcoux C, Byington TW, Qian Z, Charbonneau P, Socolar JES. Emergence of limit-periodic order in tiling models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:012136. [PMID: 25122280 DOI: 10.1103/physreve.90.012136] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/02/2014] [Indexed: 06/03/2023]
Abstract
A two-dimensional (2D) lattice model defined on a triangular lattice with nearest- and next-nearest-neighbor interactions based on the Taylor-Socolar monotile is known to have a limit-periodic ground state. The system reaches that state during a slow quench through an infinite sequence of phase transitions. We study the model as a function of the strength of the next-nearest-neighbor interactions and introduce closely related 3D models with only nearest-neighbor interactions that exhibit limit-periodic phases. For models with no next-nearest-neighbor interactions of the Taylor-Socolar type, there is a large degenerate class of ground states, including crystalline patterns and limit-periodic ones, but a slow quench still yields the limit-periodic state. For the Taylor-Socolar lattic model, we present calculations of the diffraction pattern for a particular decoration of the tile that permits exact expressions for the amplitudes and identify domain walls that slow the relaxation times in the ordered phases. For one of the 3D models, we show that the phase transitions are first order, with equilibrium structures that can be more complex than in the 2D case, and we include a proof of aperiodicity for a geometrically simple tile with only nearest-neighbor matching rules.
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Affiliation(s)
- Catherine Marcoux
- Physics Department, Duke University, Durham, North Carolina 27708, USA
| | | | - Zongjin Qian
- University of Chicago Booth School of Business, 5807 S. Woodlawn Avenue, Chicago, Illinois 60637, USA
| | - Patrick Charbonneau
- Chemistry Department, Duke University, Durham, North Carolina 27708, USA and Physics Department, Duke University, Durham, North Carolina 27708, USA
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7
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Sicilia A, Arenzon JJ, Bray AJ, Cugliandolo LF. Domain growth morphology in curvature-driven two-dimensional coarsening. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:061116. [PMID: 18233823 DOI: 10.1103/physreve.76.061116] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/14/2007] [Indexed: 05/25/2023]
Abstract
We study the distribution of domain areas, areas enclosed by domain boundaries ("hulls"), and perimeters for curvature-driven two-dimensional coarsening, employing a combination of exact analysis and numerical studies, for various initial conditions. We show that the number of hulls per unit area, n_{h}(A,t)dA , with enclosed area in the interval (A,A+dA) , is described, for a disordered initial condition, by the scaling function n_{h}(A,t)=2c_{h}(A+lambda_{h}t);{2} , where c_{h}=18pi sqrt[3] approximately 0.023 is a universal constant and lambda_{h} is a material parameter. For a critical initial condition, the same form is obtained, with the same lambda_{h} but with c_{h} replaced by c_{h}2 . For the distribution of domain areas, we argue that the corresponding scaling function has, for random initial conditions, the form n_{d}(A,t)=2c_{d}(lambda_{d}t);{tau'-2}(A+lambda_{d}t);{tau'} , where c_{d} and lambda_{d} are numerically very close to c_{h} and lambda_{h} , respectively, and tau'=18791 approximately 2.055 . For critical initial conditions, one replaces c_{d} by c_{d}2 and the exponent is tau=379187 approximately 2.027 . These results are extended to describe the number density of the length of hulls and domain walls surrounding connected clusters of aligned spins. These predictions are supported by extensive numerical simulations. We also study numerically the geometric properties of the boundaries and areas.
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Affiliation(s)
- Alberto Sicilia
- Université Pierre et Marie Curie-Paris VI, LPTHE UMR 7589, 4 Place Jussieu, Paris Cedex 05, France
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8
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Arenzon JJ, Bray AJ, Cugliandolo LF, Sicilia A. Exact results for curvature-driven coarsening in two dimensions. PHYSICAL REVIEW LETTERS 2007; 98:145701. [PMID: 17501288 DOI: 10.1103/physrevlett.98.145701] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/2006] [Indexed: 05/15/2023]
Abstract
We consider the statistics of the areas enclosed by domain boundaries ("hulls") during the curvature-driven coarsening dynamics of a two-dimensional nonconserved scalar field from a disordered initial state. We show that the number of hulls per unit area that enclose an area greater than A has, for large time t, the scaling form Nh(A,t)=2c/(A+lambdat), demonstrating the validity of dynamical scaling in this system, where c=1/8pisquare root 3 is a universal constant. Domain areas (regions of aligned spins) have a similar distribution up to very large values of A/lambdat. Identical forms are obtained for coarsening from a critical initial state, but with c replaced by c/2.
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Affiliation(s)
- Jeferson J Arenzon
- Instituto de Física, Universidade Federal do Rio Grande do Sul, CP 15051, 91501-970 Porto Alegre RS, Brazil
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9
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Uchida M, Shirayama S. Effect of initial conditions on Glauber dynamics in complex networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:046105. [PMID: 17500959 DOI: 10.1103/physreve.75.046105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2006] [Revised: 11/09/2006] [Indexed: 05/15/2023]
Abstract
The effect of initial spin configurations on zero-temperature Glauber spin dynamics in complex networks is investigated. In a system in which the initial spins are defined by centrality measures at the vertices of a network, a variety of nontrivial diffusive behaviors arise, particularly in relation to functional relationships between the initial and final fractions of positive spins, some of which exhibit a critical point. Notably, the majority spin in the initial state is not always dominant in the final state and the phenomena that occur as a result of the dynamics differ according to the initial condition, even for the same network. It is thus concluded that the initial condition of a complex network exerts an influence on spin dynamics that is equally as strong as that exerted by the network structure.
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Affiliation(s)
- Makoto Uchida
- Research into Artifacts, Center for Engineering (RACE), University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8568, Japan.
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10
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Greil F, Drossel B. Dynamics of critical Kauffman networks under asynchronous stochastic update. PHYSICAL REVIEW LETTERS 2005; 95:048701. [PMID: 16090847 DOI: 10.1103/physrevlett.95.048701] [Citation(s) in RCA: 51] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/05/2005] [Indexed: 05/03/2023]
Abstract
We show that the mean number of attractors in a critical Boolean network under asynchronous stochastic update grows like a power law and that the mean size of the attractors increases as a stretched exponential with the system size. This is in strong contrast to the synchronous case, where the number of attractors grows faster than any power law.
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Affiliation(s)
- Florian Greil
- Institut für Festkörperphysik, Technische Universität Darmstadt, Germany
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11
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Puri S, Kumar D. Autocorrelation functions for phase separation in ternary mixtures. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:051501. [PMID: 15600618 DOI: 10.1103/physreve.70.051501] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/12/2004] [Indexed: 05/24/2023]
Abstract
We present numerical and analytical results for the autocorrelation functions which characterize domain growth in ternary mixtures. The numerical results are obtained from Monte Carlo simulations of the spin-1 Blume-Emery-Griffiths model with spin-exchange kinetics. Further, we model the autocorrelation functions using an approach based on the continuous-time random walk formalism. The aging property of these functions is related to the time dependence of the domain-size distribution. Our analytical results are found to be in good agreement with the numerical data.
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Affiliation(s)
- Sanjay Puri
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
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12
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Nagaya T, Gilli JM. Experimental study of spinodal decomposition in a 1D conserved order parameter system. PHYSICAL REVIEW LETTERS 2004; 92:145504. [PMID: 15089551 DOI: 10.1103/physrevlett.92.145504] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/21/2002] [Indexed: 05/24/2023]
Abstract
We study the zigzag instability coarsening of splay-bend walls formed in a nematic liquid crystal under external fields. The vertexes of zigzag can be considered as kinks in a one-dimensional order parameter system and the geometrical constraints associated with the necessary equal length sum of zig and zag segments impose a conserved quantity in this Cahn-Hilliard-type problem. In the late stage of coarsening, the characteristic length of the system L(t) shows a logarithmic increase in time and the dynamical scaling law holds. We then try to extract the nontrivial asymptotic scaling exponent lambda of the two-time correlation function, defined by lim(<phi(0,t)phi(0,t('))> approximately [L(t)/L(t('))](-lambda). The scaling exponents with respective time references, t(')=32 and 64 s, after quench are found to be lambda approximately 2 which is larger than the value with respective time reference t(')=0, predicted by numerical simulation.
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Affiliation(s)
- Tomoyuki Nagaya
- Department of Electrical and Electronic Engineering, Faculty of Engineering, Okayama University, 3-1-1 Tsushimanaka Okayama, 700-8530 Japan
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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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Majumdar SN, Dean DS. Slow relaxation in a constrained Ising spin chain: toy model for granular compaction. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:056114. [PMID: 12513563 DOI: 10.1103/physreve.66.056114] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/22/2002] [Indexed: 05/24/2023]
Abstract
We present detailed analytical studies on the zero-temperature coarsening dynamics in an Ising spin chain in the presence of a dynamically induced field that favors locally the "-" phase compared to the "+" phase. We show that the presence of such a local kinetic bias drives the system into a late time state with average magnetization m equal to -1. However the magnetization relaxes into this final value extremely slowly in an inverse logarithmic fashion. We further map this spin model exactly onto a simple lattice model of granular compaction that includes the minimal microscopic moves needed for compaction. This toy model then predicts analytically an inverse logarithmic law for the growth of density of granular particles, as seen in recent experiments and thereby provides a mechanism for the inverse logarithmic relaxation. Our analysis utilizes an independent interval approximation for the particle and the hole clusters and is argued to be exact at late times (supported also by numerical simulations).
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Affiliation(s)
- Satya N Majumdar
- Laboratoire de Physique Quantique (UMR 5626 du CNRS), Université Paul Sabatier, 31062 Toulouse Cedex, France
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Prados A, Brey JJ. Analytical solution of a one-dimensional Ising model with zero-temperature dynamics. ACTA ACUST UNITED AC 2001. [DOI: 10.1088/0305-4470/34/33/103] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Majumdar SN, Dean DS, Grassberger P. Coarsening in the presence of kinetic disorders: analogy to granular compaction. PHYSICAL REVIEW LETTERS 2001; 86:2301-2304. [PMID: 11289914 DOI: 10.1103/physrevlett.86.2301] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/27/2000] [Indexed: 05/23/2023]
Abstract
We study the zero temperature dynamics in an Ising chain in the presence of a dynamically induced field that favors locally the " -" phase compared to the " +" phase. At late times, while the " +" domains coarsen as t(1/2), the " -" domains coarsen as t(1/2)log(t). Hence, at late times, the magnetization decays slowly as m(t) = -1+const/log(t). We establish this behavior both analytically within an independent interval approximation and numerically. Our model can be viewed as a simple model for granular compaction, where the system decays into a fully compact state (with all spins " -") in a slow logarithmic manner as seen in recent experiments on granular systems.
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Affiliation(s)
- S N Majumdar
- Laboratoire de Physique Quantique, UMR C5626 du CNRS, Université Paul Sabatier, Toulouse, France
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17
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Cornell S, Droz M, Menyhard N. Unconventional scaling theory for domain growth in the alternating bond Glauber-Ising chain. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/24/4/008] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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18
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Derrida B, Bray AJ, Godreche C. Non-trivial exponents in the zero temperature dynamics of the 1D Ising and Potts models. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/27/11/002] [Citation(s) in RCA: 210] [Impact Index Per Article: 8.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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19
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20
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Derrida B. Exponents appearing in the zero-temperature dynamics of the 1D Potts model. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/28/6/006] [Citation(s) in RCA: 54] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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21
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Monthus C. Exponents appearing in heterogeneous reaction-diffusion models in one dimension. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:4844-4859. [PMID: 9965666 DOI: 10.1103/physreve.54.4844] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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22
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Majumdar SN, Bray AJ, Cornell SJ, Sire C. Global Persistence Exponent for Nonequilibrium Critical Dynamics. PHYSICAL REVIEW LETTERS 1996; 77:3704-3707. [PMID: 10062287 DOI: 10.1103/physrevlett.77.3704] [Citation(s) in RCA: 57] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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23
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Derrida B, Zeitak R. Distribution of domain sizes in the zero temperature Glauber dynamics of the one-dimensional Potts model. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:2513-2525. [PMID: 9965362 DOI: 10.1103/physreve.54.2513] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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24
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Yeung C, Rao M, Desai RC. Bounds on the decay of the autocorrelation in phase ordering dynamics. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:3073-3077. [PMID: 9964613 DOI: 10.1103/physreve.53.3073] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Majumdar SN, Sengupta AM. Nonequilibrium dynamics following a quench to the critical point in a semi-infinite system. PHYSICAL REVIEW LETTERS 1996; 76:2394-2397. [PMID: 10060686 DOI: 10.1103/physrevlett.76.2394] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Castellano C, Zannetti M. Multiscaling to standard-scaling crossover in the Bray-Humayun model for phase-ordering kinetics. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:1430-1440. [PMID: 9964403 DOI: 10.1103/physreve.53.1430] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Derrida B, Hakim V, Pasquier V. Exact first-passage exponents of 1D domain growth: Relation to a reaction-diffusion model. PHYSICAL REVIEW LETTERS 1995; 75:751-754. [PMID: 10060105 DOI: 10.1103/physrevlett.75.751] [Citation(s) in RCA: 112] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Sire C, Majumdar SN. Coarsening in the q-state Potts model and the Ising model with globally conserved magnetization. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:244-254. [PMID: 9963428 DOI: 10.1103/physreve.52.244] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Majumdar SN, Huse DA. Growth of long-range correlations after a quench in phase-ordering systems. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:270-284. [PMID: 9963430 DOI: 10.1103/physreve.52.270] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Corberi F, Coniglio A, Zannetti M. Early stage scaling in phase ordering kinetics. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:5469-5475. [PMID: 9963280 DOI: 10.1103/physreve.51.5469] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Bray AJ, Derrida B. Exact exponent lambda of the autocorrelation function for a soluble model of coarsening. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:R1633-R1636. [PMID: 9962943 DOI: 10.1103/physreve.51.r1633] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Rutenberg AD, Bray AJ. Phase-ordering kinetics of one-dimensional nonconserved scalar systems. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:1900-1911. [PMID: 9962192 DOI: 10.1103/physreve.50.1900] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Majumdar SN, Huse DA, Lubachevsky BD. Growth of long-range correlations after a quench in conserved-order-parameter systems. PHYSICAL REVIEW LETTERS 1994; 73:182-185. [PMID: 10056750 DOI: 10.1103/physrevlett.73.182] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Privman V. Exact results for diffusion-limited reactions with synchronous dynamics. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:50-53. [PMID: 9961943 DOI: 10.1103/physreve.50.50] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Blundell RE, Bray AJ, Sattler S. Absolute test for theories of phase-ordering dynamics. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:2476-2480. [PMID: 9960880 DOI: 10.1103/physreve.48.2476] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Majumdar SN, Sire C. Phase separation model with conserved order parameter on the Bethe lattice. PHYSICAL REVIEW LETTERS 1993; 70:4022-4025. [PMID: 10054025 DOI: 10.1103/physrevlett.70.4022] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Bray AJ, Humayun K. Universal amplitudes of power-law tails in the asymptotic structure factor of systems with topological defects. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 47:R9-R12. [PMID: 9960066 DOI: 10.1103/physreve.47.r9] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Privman V. Exact solution of a phase separation model with conserved order parameter dynamics. PHYSICAL REVIEW LETTERS 1992; 69:3686-3688. [PMID: 10046887 DOI: 10.1103/physrevlett.69.3686] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Cornell SJ, Droz M. Comment on "Universal scaling function for domain growth in the Glauber-Ising chain". PHYSICAL REVIEW. B, CONDENSED MATTER 1992; 46:14261-14262. [PMID: 10003512 DOI: 10.1103/physrevb.46.14261] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Privman V. Exact results for a three-body reaction-diffusion system. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1992; 46:R6140-R6142. [PMID: 9907997 DOI: 10.1103/physreva.46.r6140] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Lin Z, Wang X, Tao R. Universal scaling function for domain growth in the Glauber-Ising chain. PHYSICAL REVIEW. B, CONDENSED MATTER 1992; 45:8131-8133. [PMID: 10000630 DOI: 10.1103/physrevb.45.8131] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Cornell SJ, Kaski K, Stinchcombe RB. Domain scaling and glassy dynamics in a one-dimensional Kawasaki Ising model. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 44:12263-12274. [PMID: 9999381 DOI: 10.1103/physrevb.44.12263] [Citation(s) in RCA: 49] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Derrida B, Godrèche C, Yekutieli I. Scale-invariant regimes in one-dimensional models of growing and coalescing droplets. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 44:6241-6251. [PMID: 9905756 DOI: 10.1103/physreva.44.6241] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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