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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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Forrester A, Chu HC, Williams GA. Exact solution for vortex dynamics in temperature quenches of two-dimensional superfluids. PHYSICAL REVIEW LETTERS 2013; 110:165303. [PMID: 23679613 DOI: 10.1103/physrevlett.110.165303] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/27/2011] [Indexed: 06/02/2023]
Abstract
An exact analytic solution for the dynamics of vortex pairs is obtained for rapid temperature quenches of a superfluid film starting from the line of critical points below the critical temperature T(KT). An approximate solution for quenches at and above above T(KT) provides insights into the origin of logarithmic transients in the vortex decay, and is in general agreement with recent simulations of the quenched XY model. These results confirm that there is no "creation" of vortices whose density increases with the quench rate as predicted by the Kibble-Zurek theory but only monotonic decay of the thermal vortices already present at the initial temperature.
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Affiliation(s)
- Andrew Forrester
- Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
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Burioni R, Corberi F, Vezzani A. Complex phase ordering of the one-dimensional Heisenberg model with conserved order parameter. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:041119. [PMID: 19518185 DOI: 10.1103/physreve.79.041119] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/21/2008] [Revised: 02/05/2009] [Indexed: 05/27/2023]
Abstract
We study the phase-ordering kinetics of the one-dimensional Heisenberg model with conserved order parameter by means of scaling arguments and numerical simulations. We find a rich dynamical pattern with a regime characterized by two distinct growing lengths. Spins are found to be coplanar over regions of a typical size LV(t), while inside these regions smooth rotations associated to a smaller length LC(t) are observed. Two different and coexisting ordering mechanisms are associated to these lengths, leading to different growth laws LV(t) approximately t1/3 and LC(t) approximately t1/4 violating dynamical scaling.
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Affiliation(s)
- R Burioni
- Dipartimento di Fisica and INFN, Università di Parma, Parco Area delle Scienze 7/A, I-423100 Parma, Italy
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Dutta S, Roy SK. Dynamical scaling in two-dimensional quenched uniaxial nematic liquid crystals. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:026119. [PMID: 15783390 DOI: 10.1103/physreve.71.026119] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/15/2004] [Indexed: 05/24/2023]
Abstract
The phase-ordering kinetics of the two-dimensional uniaxial nematic has been studied using a cell dynamic scheme. The system after quench from T=infinity was found to scale dynamically with an asymptotic growth law similar to that of the two-dimensional O(2) model (quenched from above the Kosterlitz-Thouless transition temperature), i.e., L (t) approximately [t/ln (t/ t(0) ) ](1/2) (with nonuniversal time scale t(0) ). We obtained the true asymptotic limit of the growth law by performing our simulation for a sufficiently long time. The presence of topologically stable 1/2 -disclination points is reflected in the observed large-momentum dependence k(-4) of the structure factor. The correlation function was also found to tally with the theoretical prediction of the correlation function for the two-dimensional O(2) system.
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Affiliation(s)
- Subhrajit Dutta
- Department of Physics, Jadavpur University, Calcutta 700 032, India.
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Schehr G, Le Doussal P. Exact multilocal renormalization of the effective action: application to the random sine Gordon model statics and nonequilibrium dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:046101. [PMID: 14682996 DOI: 10.1103/physreve.68.046101] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/23/2003] [Indexed: 05/24/2023]
Abstract
We extend the exact multilocal renormalization group (RG) method to study the flow of the effective action functional. This important physical quantity satisfies an exact RG equation which is then expanded in multilocal components. Integrating the nonlocal parts yields a closed exact RG equation for the local part, to a given order in the local part. The method is illustrated on the O(N) model by straightforwardly recovering the eta exponent and scaling functions. Then it is applied to study the glass phase of the Cardy-Ostlund, random phase sine Gordon model near the glass transition temperature. The static correlations and equilibrium dynamical exponent z are recovered and several results are obtained, such as the equilibrium two-point scaling functions. The nonequilibrium, finite momentum, two-time t,t' response and correlations are computed. They are shown to exhibit scaling forms, characterized by exponents lambda(R) not equal lambda(C), as well as universal scaling functions that we compute. The fluctuation dissipation ratio is found to be nontrivial and of the form X[q(z)(t-t'),t/t']. Analogies and differences with pure critical models are discussed.
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Affiliation(s)
- Gregory Schehr
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, F-75231 Paris, France
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Das SK, Puri S. Nonequilibrium dynamics of the complex Ginzburg-Landau equation: numerical results in two and three dimensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:046123. [PMID: 12005942 DOI: 10.1103/physreve.65.046123] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/14/2001] [Indexed: 05/23/2023]
Abstract
This paper is the second of a two-stage exposition, in which we study the nonequilibrium dynamics of the complex Ginzburg-Landau (CGL) equation. We use spiral defects to characterize the system evolution and morphologies. In the first paper of this exposition [S.K. Das, S. Puri, and M.C. Cross, Phys. Rev E 64, 046206 (2001)], we presented analytical results for the correlation function of a single spiral defect, and its short-distance singular behavior. We had also examined the utility of the Gaussian auxiliary field ansatz for characterizing multispiral morphologies. In this paper, we present results from an extensive numerical study of nonequilibrium dynamics in the CGL equation with dimensionality d=2,3. We discuss the behavior of domain growth laws; real-space correlation functions; and momentum-space structure factors. We also compare numerical results for the correlation functions and structure factors with analytical results presented in our first paper.
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Affiliation(s)
- Subir K Das
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
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Puri S, Das SK, Cross MC. Nonequilibrium dynamics in the complex Ginzburg-Landau equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:056140. [PMID: 11736047 DOI: 10.1103/physreve.64.056140] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/18/2001] [Indexed: 05/23/2023]
Abstract
Results from a comprehensive analytical and numerical study of nonequilibrium dynamics in the two-dimensional complex Ginzburg-Landau equation have been presented. In particular, spiral defects have been used to characterize the domain growth law and the evolution morphology. An asymptotic analysis of the single-spiral correlation function shows a sequence of singularities-analogous to those seen for time-dependent Ginzburg-Landau models with O(n) symmetry, where n is even.
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Affiliation(s)
- S Puri
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi-110067, India
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Rojas F, Bray AJ. Structure factor tail for the ordering kinetics of nonconserved systems without topological defects. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:4686-4695. [PMID: 9964796 DOI: 10.1103/physreve.53.4686] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Puri S, Bray AJ, Rojas F. Ordering kinetics of conserved XY models. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:4699-4703. [PMID: 9963965 DOI: 10.1103/physreve.52.4699] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Lee JR, Lee SJ, Kim B. Ordering kinetics of two-dimensional O(2) models: Scaling and temperature dependence. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:1550-1557. [PMID: 9963576 DOI: 10.1103/physreve.52.1550] [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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Zapotocky M, Zakrzewski W. Kinetics of phase ordering with topological textures. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:R5189-R5192. [PMID: 9963392 DOI: 10.1103/physreve.51.r5189] [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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Rutenberg AD, Bray AJ. Energy-scaling approach to phase-ordering growth laws. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:5499-5514. [PMID: 9963283 DOI: 10.1103/physreve.51.5499] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Zapotocky M, Goldbart PM, Goldenfeld N. Kinetics of phase ordering in uniaxial and biaxial nematic films. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:1216-1235. [PMID: 9962764 DOI: 10.1103/physreve.51.1216] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Rojas F, Bray AJ. Phase-ordering dynamics of systems with a conserved vector order parameter. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:188-197. [PMID: 9962631 DOI: 10.1103/physreve.51.188] [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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Filipe JA, Bray AJ. Gaussian approach for phase ordering in nonconserved scalar systems with long-range interactions. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:204-211. [PMID: 9962633 DOI: 10.1103/physreve.51.204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Blundell RE, Bray AJ. Phase-ordering dynamics of the O(n) model: Exact predictions and numerical results. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:4925-4937. [PMID: 9961813 DOI: 10.1103/physreve.49.4925] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Rao M, Chakrabarti A. Dynamical scaling functions in conserved vector order-parameter systems without topological defects. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:3727-3730. [PMID: 9961658 DOI: 10.1103/physreve.49.3727] [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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Bray AJ, Rutenberg AD. Growth laws for phase ordering. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:R27-R30. [PMID: 9961301 DOI: 10.1103/physreve.49.r27] [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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Bray AJ, Humayun K. Towards a systematic calculation of the scaling functions for the ordering kinetics of nonconserved fields. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:R1609-R1612. [PMID: 9960859 DOI: 10.1103/physreve.48.r1609] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Hayakawa H. Phase ordering in random media. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 47:11696-11702. [PMID: 10005336 DOI: 10.1103/physrevb.47.11696] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Bray AJ, Puri S, Blundell RE, Somoza AM. Structure factor for phase ordering in nematic liquid crystals. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 47:R2261-R2264. [PMID: 9960348 DOI: 10.1103/physreve.47.r2261] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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