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Chatterjee S, Sutradhar S, Puri S, Paul R. Ordering kinetics in a q-state random-bond clock model: Role of vortices and interfaces. Phys Rev E 2020. [PMID: 32290025 DOI: 10.1103/physreve.98.032109] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
In this article, we present a Monte Carlo study of phase transition and coarsening dynamics in the nonconserved two-dimensional random-bond q-state clock model (RBCM) deriving from a pure clock model [Chatterjee et al., Phys. Rev. E 98, 032109 (2018)10.1103/PhysRevE.98.032109]. Akin to the pure clock model, RBCM also passes through two different phases when quenched from a disordered initial configuration representing at infinite temperature. Our investigation of the equilibrium phase transition affirms that both upper (T_{c}^{1}) and lower (T_{c}^{2}) phase transition temperatures decrease with bond randomness strength ε. Effect of ε on the nonequilibrium coarsening dynamics is investigated following independent rapid quenches in the quasi-long-range ordered (QLRO, T_{c}^{2}<T<T_{c}^{1}) regime and long-range ordered (LRO, T<T_{c}^{2}) regime at temperature T. We report that the dynamical scaling of the correlation function and structure factor is independent of ε and the presence of quenched disorder slows down domain coarsening. Coarsening dynamics in both LRO and QLRO regimes are further characterized by power-law growth with disorder-dependent exponents within our simulation timescales. The growth exponents in the LRO regime decrease from 0.5 in the pure case to 0.22 in the maximum disordered case, whereas the corresponding change in the QLRO regime happens from 0.45 to 0.38. We further explored the coarsening dynamics in the bond-diluted clock model and, in both the models, the effect of the disorder is more significant for the quench in the LRO regime compared to the QLRO regime.
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Affiliation(s)
- Swarnajit Chatterjee
- School of Mathematical & Computational Sciences, Indian Association for the Cultivation of Science, Kolkata 700032, India
| | | | - Sanjay Puri
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
| | - Raja Paul
- School of Mathematical & Computational Sciences, Indian Association for the Cultivation of Science, Kolkata 700032, India
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2
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Michaud J, Szilva A. Social influence with recurrent mobility and multiple options. Phys Rev E 2018; 97:062313. [PMID: 30011445 DOI: 10.1103/physreve.97.062313] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2018] [Indexed: 11/07/2022]
Abstract
In this paper, we discuss the possible generalizations of the social influence with recurrent mobility (SIRM) model [Phys. Rev. Lett. 112, 158701 (2014)PRLTAO0031-900710.1103/PhysRevLett.112.158701]. Although the SIRM model worked approximately satisfying when U.S. election was modeled, it has its limits: it has been developed only for two-party systems and can lead to unphysical behavior when one of the parties has extreme vote share close to 0 or 1. We propose here generalizations to the SIRM model by its extension for multiparty systems that are mathematically well-posed in case of extreme vote shares, too, by handling the noise term in a different way. In addition, we show that our method opens alternative applications for the study of elections by using an alternative calibration procedure and makes it possible to analyze the influence of the "free will" (creating a new party) and other local effects for different commuting network topologies.
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Affiliation(s)
- Jérôme Michaud
- Department of Physics and Astronomy, University of Uppsala, 752 37 Uppsala, Sweden and Department of Sociology, University of Uppsala, 751 20 Uppsala, Sweden
| | - Attila Szilva
- Department of Physics and Astronomy, University of Uppsala, 752 37 Uppsala, Sweden
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3
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Zamparo M, Chianale F, Tebaldi C, Cosentino-Lagomarsino M, Nicodemi M, Gamba A. Dynamic membrane patterning, signal localization and polarity in living cells. SOFT MATTER 2015; 11:838-849. [PMID: 25563791 DOI: 10.1039/c4sm02157f] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
We review the molecular and physical aspects of the dynamic localization of signaling molecules on the plasma membranes of living cells. At the nanoscale, clusters of receptors and signaling proteins play an essential role in the processing of extracellular signals. At the microscale, "soft" and highly dynamic signaling domains control the interaction of individual cells with their environment. At the multicellular scale, individual polarity patterns control the forces that shape multicellular aggregates and tissues.
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Affiliation(s)
- M Zamparo
- Human Genetics Foundation - Torino, Italy.
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4
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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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5
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Kim M, Park SC, Noh JD. Coarsening dynamics of nonequilibrium chiral Ising models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:012129. [PMID: 23410305 DOI: 10.1103/physreve.87.012129] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/05/2012] [Indexed: 06/01/2023]
Abstract
We investigate a nonequilibrium coarsening dynamics of a one-dimensional Ising spin system with chirality. Only spins at domain boundaries are updated so that the model undergoes a coarsening to either of equivalent absorbing states with all spins + or -. Chirality is imposed by assigning different transition rates to events at down (+-) kinks and up (-+) kinks. The coarsening is characterized by power-law scalings of the kink density ρ~t(-δ) and the characteristic length scale ξ~t(1/z) with time t. Surprisingly the scaling exponents vary continuously with model parameters, which is not the case for systems without chirality. These results are obtained from extensive Monte Carlo simulations and spectral analyses of the time evolution operator. Our study uncovers the novel universality class of the coarsening dynamics with chirality.
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Affiliation(s)
- Mina Kim
- Department of Physics, University of Seoul, Seoul 130-743, Korea
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6
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Lambiotte R, Saramäki J, Blondel VD. Dynamics of latent voters. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:046107. [PMID: 19518301 DOI: 10.1103/physreve.79.046107] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/11/2008] [Indexed: 05/27/2023]
Abstract
We study the effect of latency on binary-choice opinion formation models. Latency is introduced into the models as an additional dynamic rule: after a voter changes its opinion, it enters a waiting period of stochastic length where no further changes take place. We first focus on the voter model and show that as a result of introducing latency, the average magnetization is not conserved, and the system is driven toward zero magnetization, independently of initial conditions. The model is studied analytically in the mean-field case and by simulations in one dimension. We also address the behavior of the majority-rule model with added latency, and show that the competition between imitation and latency leads to a rich phenomenology.
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Affiliation(s)
- Renaud Lambiotte
- Department of Mathematical Engineering, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium
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7
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Sire C. Crossing intervals of non-Markovian Gaussian processes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:011121. [PMID: 18763933 DOI: 10.1103/physreve.78.011121] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/13/2008] [Indexed: 05/26/2023]
Abstract
We review the properties of time intervals between the crossings at a level M of a smooth stationary Gaussian temporal signal. The distribution of these intervals and the persistence are derived within the independent interval approximation (IIA). These results grant access to the distribution of extrema of a general Gaussian process. Exact results are obtained for the persistence exponents and the crossing interval distributions, in the limit of large |M|. In addition, the small-time behavior of the interval distributions and the persistence is calculated analytically, for any M. The IIA is found to reproduce most of these exact results, and its accuracy is also illustrated by extensive numerical simulations applied to non-Markovian Gaussian processes appearing in various physical contexts.
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Affiliation(s)
- Clément Sire
- Laboratoire de Physique Théorique--IRSAMC, CNRS, Université Paul Sabatier, 31062 Toulouse, France.
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8
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Lu Q, Korniss G, Szymanski BK. Naming games in two-dimensional and small-world-connected random geometric networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:016111. [PMID: 18351919 DOI: 10.1103/physreve.77.016111] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/27/2007] [Indexed: 05/26/2023]
Abstract
We investigate a prototypical agent-based model, the naming game, on two-dimensional random geometric networks. The naming game [Baronchelli, J. Stat. Mech.: Theory Exp. (2006) P06014] is a minimal model, employing local communications that captures the emergence of shared communication schemes (languages) in a population of autonomous semiotic agents. Implementing the naming games with local broadcasts on random geometric graphs, serves as a model for agreement dynamics in large-scale, autonomously operating wireless sensor networks. Further, it captures essential features of the scaling properties of the agreement process for spatially embedded autonomous agents. Among the relevant observables capturing the temporal properties of the agreement process, we investigate the cluster-size distribution and the distribution of the agreement times, both exhibiting dynamic scaling. We also present results for the case when a small density of long-range communication links are added on top of the random geometric graph, resulting in a "small-world"-like network and yielding a significantly reduced time to reach global agreement. We construct a finite-size scaling analysis for the agreement times in this case.
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Affiliation(s)
- Qiming Lu
- Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute,Troy, New York 12180-3590, USA.
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9
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Sire C. Probability distribution of the maximum of a smooth temporal signal. PHYSICAL REVIEW LETTERS 2007; 98:020601. [PMID: 17358591 DOI: 10.1103/physrevlett.98.020601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/06/2006] [Indexed: 05/14/2023]
Abstract
We present an approximate calculation for the distribution of the maximum of a smooth stationary temporal signal X(t). As an application, we compute the persistence exponent associated with the probability that the process remains below a nonzero level M. When X(t) is a Gaussian process, our results are expressed explicitly in terms of the two-time correlation function, f(t)=X(0)X(t).
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Affiliation(s)
- Clément Sire
- Laboratoire de Physique Théorique, UMR 5152 du CNRS, Université Paul Sabatier, Toulouse Cedex 4, France
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10
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Andrenacci N, Corberi F, Lippiello E. Crossover between Ising and XY -like behavior in the off-equilibrium kinetics of the one-dimensional clock model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:031111. [PMID: 17025598 DOI: 10.1103/physreve.74.031111] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/17/2006] [Revised: 07/20/2006] [Indexed: 05/12/2023]
Abstract
We study the phase-ordering kinetics following a quench to a final temperature Tf of the one-dimensional p-state clock model. We show the existence of a critical value pc=4, where the properties of the dynamics change. At Tf=0, for p<or=pc the dynamics is analogous to that of the kinetic Ising model, characterized by Brownian motion and annihilation of interfaces. Dynamical scaling is obeyed with the same dynamical exponents and scaling functions of the Ising model. For p>pc, instead, the dynamics is dominated by a texture mechanism analogous to the one-dimensional XY model and dynamical scaling is violated. During the phase-ordering process at Tf>0, before equilibration occurs, a crossover between an early XY-like regime and a late Ising-like dynamics is observed for p>pc.
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Affiliation(s)
- Natascia Andrenacci
- Dipartimento di Fisica E. Caianiello and CNR-INFM Istituto Nazionale di Fisica della Materia, Università di Salerno, 84081 Baronissi (Salerno), Italy.
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11
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Sopik J, Sire C, Chavanis PH. Dynamics of the Bose-Einstein condensation: analogy with the collapse dynamics of a classical self-gravitating Brownian gas. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:011112. [PMID: 16907065 DOI: 10.1103/physreve.74.011112] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/21/2005] [Indexed: 05/11/2023]
Abstract
We consider the dynamics of a gas of free bosons within a semiclassical Fokker-Planck equation for which we give a physical justification. In this context, we find a striking similarity between the Bose-Einstein condensation in the canonical ensemble, and the gravitational collapse of a gas of classical self-gravitating Brownian particles. The paper is mainly devoted to the complete study of the Bose-Einstein "collapse" within this model. We find that at the Bose-Einstein condensation temperature Tc, the chemical potential mu(t) vanishes exponentially with a universal rate that we compute exactly. Below Tc, we show analytically that square root mu(t) vanishes linearly in a finite time t coll. After t coll, the mass of the condensate grows linearly with time and saturates exponentially to its equilibrium value for large time. We also give analytical results for the density scaling functions, for the corrections to scaling, and for the exponential relaxation time. Finally, we find that the equilibration time (above Tc) and the collapse time T coll(below Tc) both behave like -T -3 c ln|T-Tc|, near Tc.
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Affiliation(s)
- Julien Sopik
- Laboratoire de Physique Théorique (UMR 5152 du CNRS), Université Paul Sabatier 118, Route de Narbonne, 31062 Toulouse Cedex 4, France
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12
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Golovin AA, Pismen LM. Dynamic phase separation: from coarsening to turbulence via structure formation. CHAOS (WOODBURY, N.Y.) 2004; 14:845-854. [PMID: 15446995 DOI: 10.1063/1.1784751] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We investigate some new two-dimensional evolution models belonging to the class of convective Cahn-Hilliard models: (i) a local model with a scalar order parameter, (ii) a nonlocal model with a scalar order parameter, and (iii) a model with a vector order parameter. These models are applicable to phase-separating system where concentration gradients cause hydrodynamic motion due to buoyancy or Marangoni effect. The numerical study of the models shows transition from coarsening, typical of Cahn-Hilliard systems, to spatiotemporally irregular behavior (turbulence), typical of the Kuramoto-Sivashinsky equation, which is obtained in the limit of very strong driving. The transition occurs not in a straightforward way, but through the formation of spatial patterns that emerge for intermediate values of the driving intensity. As in driven one-dimensional models studied before, the mere presence of the driving force, however small, breaks the symmetry between the two separating phases, as well as increases the coarsening rate. With increasing driving, coarsening stops. The dynamics is generally irregular at strong driving, but exhibits specific structural features.
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Affiliation(s)
- A A Golovin
- Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA
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13
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Krishnamurthy S, Rajesh R, Zaboronski O. Persistence properties of a system of coagulating and annihilating random walkers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:046103. [PMID: 14682998 DOI: 10.1103/physreve.68.046103] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/29/2003] [Indexed: 05/24/2023]
Abstract
We study a d-dimensional system of diffusing particles that on contact either annihilate with probability 1/(q-1) or coagulate with probability (q-2)/(q-1). In one dimension, the system models the zero-temperature Glauber dynamics of domain walls in the q-state Potts model. We calculate P(m,t), the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly (m-1) coagulations. Using perturbative renormalization group analysis for d<2, we show that, if the number of coagulations m is much less than the typical number M(t), then P(m,t) approximately m(zeta/d)t(-theta), with theta=dQ+Q(Q-1/2)epsilon+O(epsilon(2)), zeta=(2Q-1)epsilon+(2Q-1)(Q-1)(1/2+AQ)epsilon(2)+O(epsilon(3)), where Q=(q-1)/q, epsilon=2-d and A=-0.006....M(t) is shown to scale as M(t) approximately t(d/2-delta), where delta=d(1-Q)+(Q-1)(Q-1/2)epsilon+O(epsilon(2)). In two dimensions, we show that P(m,t) approximately ln(t)(Q(3-2Q))ln(m)((2Q-1)(2))t(-2Q) for m<<t(2Q-1). We also derive an exact nonperturbative relation between the exponents: namely delta(Q)=theta(1-Q). The one-dimensional results corresponding to epsilon=1 are compared with results from Monte Carlo simulations.
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14
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Ben-Naim E, Krapivsky PL. Exchange-driven growth. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:031104. [PMID: 14524747 DOI: 10.1103/physreve.68.031104] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/08/2003] [Indexed: 05/24/2023]
Abstract
We study a class of growth processes in which clusters evolve via exchange of particles. We show that depending on the rate of exchange there are three possibilities: (I) Growth-clusters grow indefinitely, (II) gelation-all mass is transformed into an infinite gel in a finite time, and (III) instant gelation. In regimes I and II, the cluster size distribution attains a self-similar form. The large size tail of the scaling distribution is Phi(x) approximately exp(-x(2-nu)), where nu is a homogeneity degree of the rate of exchange. At the borderline case nu=2, the distribution exhibits a generic algebraic tail, Phi(x) approximately x(-5). In regime III, the gel nucleates immediately and consumes the entire system. For finite systems, the gelation time vanishes logarithmically, T approximately [lnN](-(nu-2)), in the large system size limit N--> infinity. The theory is applied to coarsening in the infinite range Ising-Kawasaki model and in electrostatically driven granular layers.
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Affiliation(s)
- E Ben-Naim
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
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15
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Das SK, Puri S. Kinetics of inhomogeneous cooling in granular fluids. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:011302. [PMID: 12935132 DOI: 10.1103/physreve.68.011302] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/06/2002] [Indexed: 05/24/2023]
Abstract
We study the dynamical behavior of a freely evolving granular gas, where the particles undergo inelastic collisions. The velocity and density fields exhibit complex pattern dynamics, which is reminiscent of phase ordering systems. For example, in the initial time regime, the density field stays (approximately) uniform, and the system is said to be in a homogeneous cooling state (HCS). At later times, the density field undergoes nonlinear clustering, and the system continues to lose energy in an inhomogeneous cooling state (ICS). We quantitatively characterize the HCS-->ICS crossover as a function of system parameters. Furthermore, we study nonlinear growth processes in the ICS by invoking analogies from studies of phase ordering dynamics.
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Affiliation(s)
- Subir K Das
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
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16
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Vazquez F, Krapivsky PL, Redner S. Constrained opinion dynamics: freezing and slow evolution. ACTA ACUST UNITED AC 2003. [DOI: 10.1088/0305-4470/36/3/103] [Citation(s) in RCA: 103] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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17
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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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18
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Hellén EKO, Alava MJ. Persistence in cluster-cluster aggregation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:026120. [PMID: 12241250 DOI: 10.1103/physreve.66.026120] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/20/2001] [Revised: 04/25/2002] [Indexed: 11/07/2022]
Abstract
Persistence is considered in one-dimensional diffusion-limited cluster-cluster aggregation when the diffusion coefficient of a cluster depends on its size s as D(s) approximately s(gamma). The probabilities that a site has been either empty or covered by a cluster all the time define the empty and filled site persistences. The cluster persistence gives the probability of a cluster remaining intact. The empty site and cluster persistences are universal whereas the filled site depends on the initial concentration. For gamma>0 the universal persistences decay algebraically with the exponent 2/(2-gamma). For the empty site case the exponent remains the same for gamma<0 but the cluster persistence shows a stretched exponential behavior as it is related to the small s behavior of the cluster size distribution. The scaling of the intervals between persistent regions demonstrates the presence of two length scales: the one related to the distances between clusters and that between the persistent regions.
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Affiliation(s)
- E K O Hellén
- Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 HUT, Finland
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19
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O'Donoghue SJ, Bray AJ. Fraction of uninfected walkers in the one-dimensional Potts model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:051114. [PMID: 12059536 DOI: 10.1103/physreve.65.051114] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/26/2001] [Indexed: 05/23/2023]
Abstract
The dynamics of the one-dimensional q-state Potts model, in the zero-temperature limit, can be formulated through the motion of random walkers which either annihilate (A+A-->Phi) or coalesce (A+A-->A) with a q-dependent probability. We consider all of the walkers in this model to be mutually infectious. Whenever two walkers meet, they experience mutual contamination. Walkers which avoid an encounter with another random walker up to time t remain uninfected. The fraction of uninfected walkers is known to obey a power-law decay U(t) approximately t(-phi(q)), with a nontrivial exponent phi(q) [C. Monthus, Phys. Rev. E 54, 4844 (1996); S. N. Majumdar and S. J. Cornell, ibid. 57, 3757 (1998)]. We probe the numerical values of phi(q) to a higher degree of accuracy than previous simulations and relate the exponent phi(q) to the persistence exponent theta(q) [B. Derrida, V. Hakim, and V. Pasquier, Phys. Rev. Lett. 75, 751 (1995)], through the relation phi(q)=gamma(q)theta(q) where gamma is an exponent introduced in [S. J. O'Donoghue and A. J. Bray, preceding paper, Phys. Rev. E 65, 051113 (2002)]. Our study is extended to include the coupled diffusion-limited reaction A+A-->B, B+B-->A in one dimension with equal initial densities of A and B particles. We find that the density of walkers decays in this model as rho(t) approximately t(-1/2). The fraction of sites unvisited by either an A or a B particle is found to obey a power law, P(t) approximately t(-theta) with theta approximately 1.33. We discuss these exponents within the context of the q-state Potts model and present numerical evidence that the fraction of walkers which remain uninfected decays as U(t) approximately t(-phi), where phi approximately 1.13 when infection occurs between like particles only, and phi approximately 1.93 when we also include cross-species contamination. We find that the relation between phi and theta in this model can also be characterized by an exponent gamma, where similarly, phi=gamma(theta).
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Affiliation(s)
- S J O'Donoghue
- Department of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom
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Conti M, Meerson B, Peleg A, Sasorov PV. Phase ordering with a global conservation law: Ostwald ripening and coalescence. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:046117. [PMID: 12005936 DOI: 10.1103/physreve.65.046117] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/11/2001] [Indexed: 05/23/2023]
Abstract
Globally conserved phase ordering dynamics is investigated in systems with short range correlations at t=0. A Ginzburg-Landau equation with a global conservation law is employed as the phase field model. The conditions are found under which the sharp-interface limit of this equation is reducible to the area-preserving motion by curvature. Numerical simulations show that, for both critical and off-critical quench, the equal-time pair correlation function exhibits dynamic scaling, and the characteristic coarsening length obeys l(t) approximately t(1/2). For the critical quench, our results are in excellent agreement with earlier results. For off-critical quench (Ostwald ripening) we investigate the dynamics of the size distribution function of the minority phase domains. The simulations show that, at large times, this distribution function has a self-similar form with growth exponent 1/2. The scaled distribution, however, strongly differs from the classical Wagner distribution. We attribute this difference to coalescence of domains. A theory of Ostwald ripening is developed that takes into account binary coalescence events. The theoretical scaled distribution function agrees well with that obtained in the simulations.
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Affiliation(s)
- Massimo Conti
- Dipartimento di Matematica e Fisica, Università di Camerino, and Istituto Nazionale di Fisica della Materia, 62032 Camerino, Italy
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Sen P, Dasgupta S. Short-time scaling in the critical dynamics of an antiferromagnetic Ising system with conserved magnetization. ACTA ACUST UNITED AC 2002. [DOI: 10.1088/0305-4470/35/12/305] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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22
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Das SK, Puri S. Dynamics of phase separation in multicomponent mixtures. PHYSICAL REVIEW E 2002; 65:026141. [PMID: 11863621 DOI: 10.1103/physreve.65.026141] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2001] [Indexed: 11/07/2022]
Abstract
We study the dynamics of phase separation in multicomponent mixtures through Monte Carlo simulations of the q-state Potts model with conserved kinetics. We use the Monte Carlo renormalization-group method to investigate the asymptotic regime. The domain growth law is found to be consistent with the Lifshitz-Slyozov law, L(t) equivalent to t(1/3) (where t is time), regardless of the value of q. We also present results for the scaled correlation functions and domain-size distribution functions for a range of q values.
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Affiliation(s)
- Subir K Das
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
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Fontes LR, Isopi M, Newman CM, Stein DL. Aging in 1D discrete spin models and equivalent systems. PHYSICAL REVIEW LETTERS 2001; 87:110201. [PMID: 11531509 DOI: 10.1103/physrevlett.87.110201] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2001] [Indexed: 05/23/2023]
Abstract
We derive exact expressions for a number of aging functions that are scaling limits of nonequilibrium correlations, R(t(w),t(w)+t) as t(w)-->infinity, t/t(w)-->theta, in the 1D homogenous q-state Potts model for all q with T = 0 dynamics following a quench from T = infinity. One such quantity is <sigma-->(0)(t(w));sigma-->(n)(t(w)+t)> when n/square root of ([t(w))-->z. Exact, closed-form expressions are also obtained when an interlude of T = infinity dynamics occurs. Our derivations express the scaling limit via coalescing Brownian paths and a "Brownian space-time spanning tree," which also yields other aging functions, such as the persistence probability of no spin flip at 0 between t(w) and t(w)+t.
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Affiliation(s)
- L R Fontes
- Instituto de Matemática e Estatística, Universidade de São Paulo, 05315-970 São Paulo, Brasil
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Peleg A, Conti M, Meerson B. Normal scaling in globally conserved interface-controlled coarsening of fractal clusters. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:036127. [PMID: 11580414 DOI: 10.1103/physreve.64.036127] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2000] [Revised: 04/25/2001] [Indexed: 05/23/2023]
Abstract
We find that globally conserved interface-controlled coarsening of diffusion-limited aggregates exhibits dynamic scale invariance (DSI) and normal scaling. This is demonstrated by a numerical solution of the Ginzburg-Landau equation with a global conservation law. The general sharp-interface limit of this equation is introduced and reduced to volume preserving motion by mean curvature. A simple example of globally conserved interface-controlled coarsening system: the sublimation/deposition dynamics of a solid and its vapor in a small closed vessel, is presented in detail. The results of the numerical simulations show that the scaled form of the correlation function has a power-law tail accommodating the fractal initial condition. The coarsening length exhibits normal dynamic scaling. A decrease of the cluster radius with time, predicted by DSI, is observed. The difference between global and local conservation is discussed.
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Affiliation(s)
- A Peleg
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
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25
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Peleg A, Meerson B, Vilenkin A, Conti M. Area-preserving dynamics of a long slender finger by curvature: a test case for globally conserved phase ordering. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:066101. [PMID: 11415167 DOI: 10.1103/physreve.63.066101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/25/2001] [Indexed: 05/23/2023]
Abstract
A long and slender finger can serve as a simple "test bed" for different phase-ordering models. In this work, the globally conserved, interface-controlled dynamics of a long finger is investigated, analytically and numerically, in two dimensions. An important limit is considered when the finger dynamics is reducible to area-preserving motion by curvature. A free boundary problem for the finger shape is formulated. An asymptotic perturbation theory is developed that uses the finger aspect ratio as a small parameter. The leading-order approximation is a modification of the Mullins finger (a well-known analytic solution) whose width is allowed to slowly vary with time. This time dependence is described, in the leading order, by an exponential law with the characteristic time proportional to the (constant) finger area. The subleading terms of the asymptotic theory are also calculated. Finally, the finger dynamics is investigated numerically, employing the Ginzburg-Landau equation with a global conservation law. The theory is in very good agreement with the numerical solution.
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Affiliation(s)
- A Peleg
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
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26
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Sire C, Majumdar SN, Rudinger A. Analytical results for random walk persistence. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:1258-69. [PMID: 11046403 DOI: 10.1103/physreve.61.1258] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/13/1998] [Indexed: 04/15/2023]
Abstract
In this paper, we present a detailed calculation of the persistence exponent straight theta for a nearly Markovian Gaussian process X(t), a problem initially introduced elsewhere in [Phys. Rev. Lett. 77, 1420 (1996)], describing the probability that the walker never crosses the origin. Resummed perturbative and nonperturbative expressions for straight theta are derived, which suggest a connection with the result of the alternative independent interval approximation. The perturbation theory is extended to the calculation of straight theta for non-Gaussian processes, by making a strong connection between the problem of persistence and the calculation of the energy eigenfunctions of a quantum mechanical problem. Finally, we give perturbative and nonperturbative expressions for the persistence exponent straight theta(X0), describing the probability that the process remains larger than X(0)sqrt[<X2(t)>].
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Affiliation(s)
- C Sire
- Laboratoire de Physique Quantique (UMR C5626 du CNRS), Universite Paul Sabatier, 31062, Toulouse Cedex, France
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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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Majumdar SN, Sire C. Survival Probability of a Gaussian Non-Markovian Process: Application to the T=0 Dynamics of the Ising Model. PHYSICAL REVIEW LETTERS 1996; 77:1420-1423. [PMID: 10063074 DOI: 10.1103/physrevlett.77.1420] [Citation(s) in RCA: 31] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Ispolatov I, Krapivsky PL, Redner S. War: The dynamics of vicious civilizations. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:1274-1289. [PMID: 9965196 DOI: 10.1103/physreve.54.1274] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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31
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Rutenberg AD. Nonequilibrium phase ordering with a global conservation law. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:972-973. [PMID: 9965150 DOI: 10.1103/physreve.54.972] [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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