Phase-ordering and persistence: relative effects of space-discretization, chaos, and anisotropy

doi:10.1016/S0378-4371(00)00430-1

Physica A: Statistical Mechanics and its Applications

Volume 288, Issues 1–4, 15 December 2000, Pages 326-337

https://doi.org/10.1016/S0378-4371(00)00430-1 Get rights and content

Abstract

The peculiar phase-ordering properties of a lattice of coupled chaotic maps studied recently (Lemaı̂tre, Chaté, Phys. Rev. Lett. 82 (1999) 1140) are revisited with the help of detailed investigations of interface motion. It is shown that “normal”, curvature-driven-like domain growth is recovered at larger scales than considered before, and that the persistence exponent seems to be universal. Using generalized persistence spectra, the properties of interface motion in this deterministic, chaotic, lattice system are found to “interpolate” between those of the two canonical reference systems, the (probabilistic) Ising model, and the (deterministic), space-continuous, time-dependent Ginzburg–Landau equation.

Introduction

Following a quench at low temperature, bistable “ferromagnetic” systems usually exhibit domain coarsening dynamics. This phase separation process has been observed in various experimental setups, as well as in numerous model systems [1]. The Ising model and its continuous counterpart, the so-called time-dependent Ginzburg–Landau (TDGL) equation are usually taken as paradigms of non-conservative coarsening dynamics [2]. But many other systems, e.g. the diffusion equation, also exhibit phase-ordering dynamics once suitable “phases” are defined (for the diffusion equation, one can for instance consider the sign of a zero-mean field). Spatially extended chaotic systems such as coupled map lattices (CMLs) can also exhibit coarsening transients, after which one chaotic phase dominates another, leaving the system in a long-range ordered state that is usually accompanied by a non-trivial evolution of spatially averaged quantities [3], [4], [5], [6], [7], [8].

The standard theory of domain coarsening predicts that, in general, the correlation length L(t) grows algebraically in time, L(t)∼t^1/z [1], [2]. Moreover, the exponent usually takes the value z=2 in systems with a non-conserved, scalar order parameter when the domain growth is driven by curvature. This universality of domain growth processes is now well established. Another important quantifier of phase ordering dynamics is persistence (for a recent review, see, e.g., Ref. [9] and references therein), defined, e.g., as the fraction of space that has remained in the same phase since some given initial time t₀. Persistence is seen to decay algebraically p(t)∝t^−θ with exponent θ in systems with algebraic domain growth, reflecting the stationarity of two-time correlations expressed in logarithmic time. As a matter of fact, the correlation length L(t) provides a better, “natural” measure of time, by which persistence decays as $p(t)∝L(t)^{−θ̄}$ (with $θ ̄ =zθ$ ) even in systems with, say, logarithmic growth of domains. The (degree of) universality of persistence exponents is still largely an open question today. Even models as close to each other as the zero-temperature Ising model and the TDGL equation – they share the same exponent z=2, the same structure function, the same Fisher–Huse exponent – seem to possess different persistence exponents (currently available measurements given θ=0.20 for TDGL [10], [11], [12] and 0.22 for Ising [12], [13]).

In fact, since it depends on the whole history of each point in space, persistence is a rather complex, non-local quantity. Exact or approximate values of persistence exponents are usually not simple numbers. Understanding the extent to which persistence properties are universal is one of the challenging problems of modern statistical physics. It is also important from an experimental point of view, since persistence is an easily measurable quantifier of phase ordering. One can hope to bring new light to this problem by studying non-conventional models like CMLs. In a sense, they are not constrained as are traditional models: their local dynamics cannot be rigorously reduced to that of standard models: there is no Hamiltonian, no detailed balance, and they have been shown (numerically) to exhibit Ising-like phase transitions with critical exponents that are significantly different from those of the Ising model [14], [15].

In Ref. [8], domain growth has been studied in simple CMLs. Numerical simulations showed the expected scaling behavior, but with exponents z and θ continuously varying with the strength of the diffusive coupling between chaotic units, although $θ ̄$ was found to be universal. It was also found that “normal”, TDGL, values were recovered in the continuous space limit of CMLs, suggesting that a role is played by lattice effects.

In this article, we come back to these somewhat surprising results and study in some detail the dynamics of the interfaces delimiting domains, in order to unravel the origin of the peculiar behavior observed on more global quantities, such as L(t).

Section snippets

Domain growth and interface dynamics in chaotic CMLs

We consider d=2 dimensional square lattices $L$ of diffusively coupled identical maps S_μ acting on real variables $(X_{r})_{r∈L}$ : $X_{r}^{t+1} =(1−2dg)S_{μ} (X_{r}^{t})+g ∑ e ∈ V S_{μ} (X_{r+e}^{t}),$ where $V$ is the set of 2d nearest neighbors $e$ of site $0$ . For simplicity, we present results obtained for the piecewise linear, odd, local map S_μ defined by $S_{μ} (X)= μX if X∈[−1/3,1/3], 2μ/3−μX if X∈[1/3,1], −2μ/3−μX if X∈[−1,−1/3] .$ Choosing μ∈[1,2] guarantees that the local map has two symmetric invariant intervals, allowing us to define “spins” as $σ_{r} = sign$

Persistence at late times

We now turn to the other set of results presented in Ref. [8], those related to the decay of p(t), the fraction of persistent sites. Fig. 6a shows the decay of p(t) for the same run as in Fig. 5. The behavior of the local exponent clearly changes at late times, in coincidence with the crossover observed in the growth of L(t). The persistence exponents estimated in Ref. [8] thus reflect only the short-time behavior of the coarsening process.

Given that L(t) reaches its asymptotic behavior rather

Conclusion

The results presented here have elucidated the surprising coarsening behavior of chaotic CMLs previously reported in Ref. [8], which we have shown to be representative only of the long, intermediate scaling region present in such deterministic systems. “Normal”, z=2 phase-ordering is recovered at large time/length scales for all coupling strengths g>g_e.

In fact, both the growth law and the persistence exponent θ seem to be independent of g in this limit. Although our numerical data does not

References (22)

H. Chaté et al.
Physica D
(1997)
A.J. Bray
Adv. Phys.
(1994)
P.C. Hohenberg et al.
Rev. Mod. Phys.
(1977)
H. Chaté et al.
Prog. Theor. Phys.
(1992)
H. Chaté et al.
Europhys. Lett.
(1992)
A. Lemaı̂tre et al.
Phys. Rev. Lett.
(1998)
A. Lemaı̂tre et al.
J. Stat. Phys.
(1999)
A. Lemaı̂tre et al.
Phys. Rev. Lett.
(1999)
S.N. Majumdar
Current Science (India)
(1999)
S. Cueille et al.
J. Phys. A
(1998)

S. Cueille et al.

Eur. Phys. J. B

(1999)

Cited by (12)

Transition to coarse-grained order in coupled logistic maps: Effect of delay and asymmetry
2020, Chaos, Solitons and Fractals
Citation Excerpt :
In this work, we have coarse-grained the variables and studied persistence in the coupled map lattices. Persistence in spatially extended systems has been studied before [7] and can be useful in identifying phases [8–10]. In particular, fully or partially arrested states can be identified using persistence.
We study one-dimensional coupled logistic maps with delayed linear or nonlinear nearest-neighbor coupling. Taking the nonzero fixed point of the map x* as reference, we coarse-grain the system by identifying values above x* with the spin-up state and values below x* with the spin-down state. We define persistent sites at time T as the sites which did not change their spin state even once for all even times till time T. A clear transition from asymptotic zero persistence to non-zero persistence is seen in the parameter space. The transition is accompanied by the emergence of antiferromagnetic, or ferromagnetic order in space. We observe antiferromagnetic order for nonlinear coupling and even delay, or linear coupling and odd delay. We observe ferromagnetic order for linear coupling and even delay, or nonlinear coupling and odd delay. For symmetric coupling, we observe a power-law decay of persistence. The persistence exponent is close to 0.375 for the transition to antiferromagnetic order and close to 0.285 for ferromagnetic order. The number of domain walls decays with an exponent close to 0.5 in all cases as expected. The persistence decays as a stretched exponential and not a power-law at the critical point, in the presence of asymmetry.
Phase ordering in chaotic map lattices with additive noise
2001, Physics Letters, Section A: General, Atomic and Solid State Physics
We present some result about phase separation in coupled map lattices with additive noise. We show that additive noise acts as an ordering agent in this class of systems. In particular, in the weak coupling region, a suitable quantity of noise leads to complete phase separation. Extrapolating our results at small coupling, we deduce that this phenomenon could take place also in the limit of zero coupling.
Transition to coarse-grained order in coupled logistic maps: effect of delay and asymmetry
2020, arXiv
Synchronization and phase ordering in globally coupled chaotic maps
2015, Springer Proceedings in Mathematics and Statistics
Phase growth in bistable systems with impurities
2008, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
Phase ordering induced by defects in chaotic bistable media
2007, European Physical Journal: Special Topics

View all citing articles on Scopus

View full text

Phase-ordering and persistence: relative effects of space-discretization, chaos, and anisotropy

Abstract

Introduction

Section snippets

Domain growth and interface dynamics in chaotic CMLs

Persistence at late times

Conclusion

Physica D

Adv. Phys.

Rev. Mod. Phys.

Prog. Theor. Phys.

Europhys. Lett.

Phys. Rev. Lett.

J. Stat. Phys.

Phys. Rev. Lett.

Current Science (India)

J. Phys. A

Eur. Phys. J. B