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Tsubota T, Liu C, Foster B, Knobloch E. Bifurcation delay and front propagation in the real Ginzburg-Landau equation on a time-dependent domain. Phys Rev E 2024; 109:044210. [PMID: 38755931 DOI: 10.1103/physreve.109.044210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2023] [Accepted: 03/12/2024] [Indexed: 05/18/2024]
Abstract
This work analyzes bifurcation delay and front propagation in the one-dimensional real Ginzburg-Landau equation with periodic boundary conditions on isotropically growing or shrinking domains. First, we obtain closed-form expressions for the delay of primary bifurcations on a growing domain and show that the additional domain growth before the appearance of a pattern is independent of the growth time scale. We also quantify primary bifurcation delay on a shrinking domain; in contrast with a growing domain, the time scale of domain compression is reflected in the additional compression before the pattern decays. For secondary bifurcations such as the Eckhaus instability, we obtain a lower bound on the delay of phase slips due to a time-dependent domain. We also construct a heuristic model to classify regimes with arrested phase slips, i.e., phase slips that fail to develop. Then, we study how propagating fronts are influenced by a time-dependent domain. We identify three types of pulled fronts: homogeneous, pattern spreading, and Eckhaus fronts. By following the linear dynamics, we derive expressions for the velocity and profile of homogeneous fronts on a time-dependent domain. We also derive the natural "asymptotic" velocity and front profile and show that these deviate from predictions based on the marginal stability criterion familiar from fixed domain theory. This difference arises because the time dependence of the domain lifts the degeneracy of the spatial eigenvalues associated with speed selection and represents a fundamental distinction from the fixed domain theory that we verify using direct numerical simulations. The effect of a growing domain on pattern spreading and Eckhaus front velocities is inspected qualitatively and found to be similar to that of homogeneous fronts. These more complex fronts can also experience delayed onset. Lastly, we show that dilution-an effect present when the order parameter is conserved-increases bifurcation delay and amplifies changes in the homogeneous front velocity on time-dependent domains. The study provides general insight into the effects of domain growth on pattern onset, pattern transitions, and front propagation in systems across different scientific fields.
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Affiliation(s)
- Troy Tsubota
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
| | - Chang Liu
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
- School of Mechanical, Aerospace, and Manufacturing Engineering, University of Connecticut, Storrs, Connecticut 06269, USA
| | - Benjamin Foster
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
| | - Edgar Knobloch
- Department of Physics, University of California at Berkeley, Berkeley, California 94720, USA
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Tello-Fraile M, Cano A, Donaire M. Topological thermalization via vortex formation in ultrafast quenches. Phys Rev E 2020; 101:052113. [PMID: 32575337 DOI: 10.1103/physreve.101.052113] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/13/2020] [Accepted: 04/20/2020] [Indexed: 06/11/2023]
Abstract
We investigate the thermalization of a two-component scalar field across a second-order phase transition under extremely fast quenches. We find that vortices start developing once the thermal bath sets the control parameter to its final value in the nonsymmetric phase. Specifically, we find that vortices emerge as the fluctuating field relaxes and departs macroscopically from its symmetric configuration. The density of primordial vortices at the relaxation time is a decreasing function of the final temperature of the quench. Subsequently, vortices and antivortices annihilate at a rate that eventually determines the total thermalization time. This rate decreases if the theory contains a discrete anisotropy term, which otherwise leaves the primordial vortex density unaffected. Our results thus establish a link between the topological processes involved in the vortex dynamics and the delay in the thermalization of the system.
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Affiliation(s)
- M Tello-Fraile
- Departamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, Paseo Belén 7, 47011 Valladolid, Spain
| | - A Cano
- Institut Néel, CNRS and Université Grenoble Alpes, 38042 Grenoble, France
| | - M Donaire
- Departamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, Paseo Belén 7, 47011 Valladolid, Spain
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Ashcroft P, Galla T. Pattern formation in individual-based systems with time-varying parameters. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:062104. [PMID: 24483383 DOI: 10.1103/physreve.88.062104] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/28/2013] [Indexed: 06/03/2023]
Abstract
We study the patterns generated in finite-time sweeps across symmetry-breaking bifurcations in individual-based models. Similar to the well-known Kibble-Zurek scenario of defect formation, large-scale patterns are generated when model parameters are varied slowly, whereas fast sweeps produce a large number of small domains. The symmetry breaking is triggered by intrinsic noise, originating from the discrete dynamics at the microlevel. Based on a linear-noise approximation, we calculate the characteristic length scale of these patterns. We demonstrate the applicability of this approach in a simple model of opinion dynamics, a model in evolutionary game theory with a time-dependent fitness structure, and a model of cell differentiation. Our theoretical estimates are confirmed in simulations. In further numerical work, we observe a similar phenomenon when the symmetry-breaking bifurcation is triggered by population growth.
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Affiliation(s)
- Peter Ashcroft
- Theoretical Physics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom
| | - Tobias Galla
- Theoretical Physics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom
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Ohnogi H, Shiwa Y. Effect of noise on ordering of hexagonal grains in a phase-field-crystal model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:051603. [PMID: 22181426 DOI: 10.1103/physreve.84.051603] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/19/2011] [Indexed: 05/31/2023]
Abstract
We present a quantitative analysis of grain morphology of self-organizing hexagonal patterns based on the phase-field crystal model to examine the effect of stochastic noise on grain coarsening. We show that the grain size increases with increasing noise strength, resulting in enhanced hexagonal orientation due to noise up to some critical noise level above which the system becomes disordered.
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Affiliation(s)
- H Ohnogi
- Statistical Mechanics Laboratory, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto, Japan
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Ohnogi H, Shiwa Y. Nucleation, growth, and coarsening of crystalline domains in order-order transitions between lamellar and hexagonal phases. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:011611. [PMID: 21867186 DOI: 10.1103/physreve.84.011611] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/2011] [Revised: 04/04/2011] [Indexed: 05/31/2023]
Abstract
Using the numerical solution of the time-dependent Ginzburg-Landau equation, we study the entire process of transformation between the lamellar and the hexagonal phases from the early-stage nucleation and growth to the late-stage coarsening regime. The metastable crystalline structure that nucleates first is identified in terms of the mean-field theory under the single-wave-number approximation. This has been borne out by the numerically efficient preparation of single-crystal structure developed via the noise-induced self-organization. We also present results for the scaling of the late-time domain growth, which is quantified by two measures: the structure factor and the orientational correlation function. In particular, the growth exponent is shown to be robust and indifferent to conservation of the order parameter.
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Affiliation(s)
- H Ohnogi
- Statistical Mechanics Laboratory, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan
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Burrage K, Burrage PM, Tian T. Numerical methods for strong solutions of stochastic differential equations: an overview. Proc Math Phys Eng Sci 2004. [DOI: 10.1098/rspa.2003.1247] [Citation(s) in RCA: 98] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Affiliation(s)
- K. Burrage
- Department of Mathematics and Advanced Computational Modelling Centre, University of Queensland, Brisbane 4072, Australia
| | - P. M. Burrage
- Department of Mathematics and Advanced Computational Modelling Centre, University of Queensland, Brisbane 4072, Australia
| | - T. Tian
- Department of Mathematics and Advanced Computational Modelling Centre, University of Queensland, Brisbane 4072, Australia
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Galla T, Moro E. Defect formation in the Swift-Hohenberg equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:035101. [PMID: 12689119 DOI: 10.1103/physreve.67.035101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/26/2002] [Indexed: 05/24/2023]
Abstract
We study numerically and analytically the dynamics of defect formation during a finite-time quench of the two-dimensional Swift-Hohenberg (SH) model of Rayleigh-Bénard convection. We find that the Kibble-Zurek picture of defect formation can be applied to describe the density of defects produced during the quench. Our study reveals the relevance of two factors: the effect of local variations of the striped patterns within defect-free domains and the presence of both pointlike and extended defects. Taking into account these two aspects we are able to identify the characteristic length scale selected during the quench and to relate it to the density of defects. We discuss possible consequences of our study for the analysis of the coarsening process of the SH model.
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Affiliation(s)
- Tobias Galla
- Theoretical Physics, University of Oxford, 1 Keble Road, United Kingdom.
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Dziarmaga J. Stochastic gene expression: density of defects frozen into permanent Turing patterns. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:011909. [PMID: 11304289 DOI: 10.1103/physreve.63.011909] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2000] [Revised: 04/28/2000] [Indexed: 05/23/2023]
Abstract
We estimate density of defects frozen into a biological Turing pattern which was turned on at a finite rate. Self-locking of gene expression in individual cells, which makes the Turing transition discontinuous, stabilizes the pattern together with its defects. Defects frozen into the pattern are a permanent record of the transition-they give an animal its own characteristic lifelong "fingerprints" or, as for vital organ formation, they can be fatal. Density of defects scales like the fourth root of the transition rate. This dependence is so weak that there is not enough time during morphogenesis to get rid of defects simply by slowing down the rate. A defect-free pattern can be obtained by spatially inhomogeneous activation of the genes. If the supercritical density of activator spreads slower than certain threshold velocity, then the Turing pattern is expressed without any defects.
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Affiliation(s)
- J Dziarmaga
- Theory Division T-6, MS-B288, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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