Anomalous long-range correlations at a non-equilibrium phase transition

Gerschenfeld, A.; Derrida, Bernard

doi:10.1088/1751-8113/45/5/055002

Cited by 5 publications

(8 citation statements)

References 42 publications

(155 reference statements)

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“…This leads (as will be shown in the longer [41]) to a second cumulant which diverges at β = β * . This divergence can be computed exactly thanks to the knownledge of the steady-state profiles (12) for β ↓ β * , leading to…”

Section: Fluctuation Theorysupporting

confidence: 54%

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Current fluctuations at a phase transition

Gerschenfeld

Derrida

2011

EPL

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The ABC model is a simple diffusive one-dimensional non-equilibrium system which exhibits a phase transition. Here we show that the cumulants of the currents of particles through the system become singular near the phase transition. At the transition, they exhibit an anomalous dependence on the system size (an anomalous Fourier's law). An effective theory for the dynamics of the single mode which becomes unstable at the transition allows one to predict this anomalous scaling. 05.70 Ln, A lot of work has been devoted recently to the study of the fluctuations of the current of heat or of particles through nonequilibrium one dimensional systems [1][2][3][4][5][6][7][8][9][10][11][12]. In such studies the basic quantity one considers is the total flux Q(t) of energy or of particles through a section of the system during time t. In the steady state this flux Q(t) fluctuates due to the randomness of the initial condition for purely deterministic models and due to the noisy dynamics in stochastic models (here we only discuss classical systems: see [13][14][15][16] for the quantum case). If one assumes that the total energy or the total number of particles in the system remains bounded, the average current lim t→∞ Q(t) t as well as the higher cumulants lim t→∞ Q(t) n c t of the flux Q(t) do not depend on the section of the system where this flux is measured.For a one dimensional system of length L, a central question is the size dependence of these cumulants [17]. In particular one would like to know whether a given system satisfies Fourier's law, meaning that, for large L, the average current scales like 1/L:where the prefactor A 1 depends on the temperatures T 1 and T 2 of the two heat baths or on the chemical potentials µ 1 and µ 2 of the two reservoirs of particles at the ends of the system. At equilibrium (T 1 = T 2 or µ 1 = µ 2 ) the prefactor A 1 in (1) vanishes but the question of the validity of Fourier's law remains. One then wants to know whether the second cumulant of Q(t) scales like 1/L.One can show that (2) holds for diffusive systems such as the SSEP (symmetric simple exclusion process) [1,4,18,19] or the KMP (Kipnis-Marchioro-Presutti) model [20]. The macroscopic fluctuation theory developed by Bertini et al. [2,21,22] allows one also to determine[1] all the cumulants of the flux Q(t), with the result that they all scale with system size as 1/L.Even corrections of order 1/L 2 have been computed in some cases [6,7].For mechanical systems with deterministic dynamics, in particular systems which conserve momentum, the average current scales as a non-integer power of the system size:The exponent α takes the value 1/2 for some exactly soluble special models [3]. Values ranging from 0.25 to 0.4 have also been reported in simulations depending on the model considered [23][24][25][26][27]. Theoretical predictions based on a mode coupling approach [28,29] or on renormalization group calculations [30] confirm this anomalous Fourier's law. Less is known on the size dependence of the higher cumulants, which are nu...

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Section: Fluctuation Theorysupporting

confidence: 54%

“…The anomalous current fluctuations of the ABC model at the phase transition are accompanied by anomalous long range density fluctuations which can also be understood in terms of the slow noisy dynamics of the first Fourier mode (the density fluctuations will be discussed in the forthcoming longer version of the present letter [41]). …”

Section: Discussionmentioning

confidence: 92%

Current fluctuations at a phase transition

Gerschenfeld

Derrida

2011

EPL

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“…The ABC model has recently yielded a flurry of interesting studies [20][21][22][23][24][25][26][27][28][29][30][31][32][33] that helped in establishing it as a paradigm for systems far from equilibrium. Not only is the ABC model characterized by its anomalous slow dynamics, making it a representative for a larger class of systems with a similar coarsening process [34][35][36][37][38], it also exhibits a variety of interesting nonequilibrium phase transitions whose properties change dramatically when breaking certain conservation laws.…”

Section: Introductionmentioning

confidence: 99%

“…The models discussed in the following are the so-called ABC model [18], a driven diffusive system composed of three different types of particles that swap places asymmetrically, and a related domain model [19] whose simplified dynamics is supposed to capture the dynamics of the ABC model at the later stages of the coarsening process. The ABC model has recently yielded a flurry of interesting studies [20][21][22][23][24][25][26][27][28][29][30][31][32][33] that helped establishing it as a paradigm for systems far from equilibrium. Not only is the ABC model characterized by its anomalous slow dynamics, making it a representative for a larger class of systems with a similar coarsening process [34][35][36][37][38], it also exhibits a variety of interesting non-equilibrium phase transitions whose properties change dramatically when breaking certain conservation laws.…”

Section: Introductionmentioning

confidence: 99%

Aging processes in systems with anomalous slow dynamics

Afzal

Pleimling²

2013

Phys. Rev. E

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Recently, different numerical studies of coarsening in disordered systems have shown the existence of a crossover from an initial, transient, power-law domain growth to a slower, presumably logarithmic, growth. However, due to the very slow dynamics and the long-lasting transient regime, one is usually not able to fully enter the asymptotic regime when investigating the relaxation of these systems toward equilibrium. We here study two simple driven systems-the one-dimensional ABC model and a related domain model with simplified dynamicsthat are known to exhibit anomalous slow relaxation where the asymptotic logarithmic growth regime is readily accessible. Studying two-times correlation and response functions, we focus on aging processes and dynamical scaling during logarithmic growth. Using the time-dependent growth length as the scaling variable, a simple aging picture emerges that is expected to also prevail in the asymptotic regime of disordered ferromagnets and spin glasses.

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“…We finally analyze the behavior of the drift near the second order phase transition, showing that, in contrast to the variance, it does not diverge. In this respect, we also quote [18,19], where current fluctuations and long-range correlations at the phase transition are analyzed.…”

Section: Introductionmentioning

confidence: 99%

Drift of Phase Fluctuations in the ABC Model

Bertini

Buttà

2013

J Stat Phys

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In a recent work, Bodineau and Derrida analyzed the phase fluctuations in the ABC model. In particular, they computed the asymptotic variance and, on the basis of numerical simulations, they conjectured the presence of a drift, which they guessed to be an antisymmetric function of the three densities. By assuming the validity of the fluctuating hydrodynamic approximation, we prove the presence of such a drift, providing an analytical expression for it. This expression is then shown to be an antisymmetric function of the three densities. The antisymmetry of the drift can also be inferred from a symmetry property of the underlying microscopic dynamics

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Anomalous long-range correlations at a non-equilibrium phase transition

Cited by 5 publications

References 42 publications

Current fluctuations at a phase transition

Current fluctuations at a phase transition

Aging processes in systems with anomalous slow dynamics

Drift of Phase Fluctuations in the ABC Model

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