Phase Diagram of the ABC Model on an Interval

Ayyer, Arvind; Cárlen, Eric A.; Lebowitz, Joel L.; Mohanty, P. K.; Mukamel, David; Speer, Eugène R.

doi:10.1007/s10955-009-9834-x

Cited by 29 publications

(142 citation statements)

References 25 publications

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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%

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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Section: Introductionmentioning

confidence: 99%

Aging processes in systems with anomalous slow dynamics

Afzal

Pleimling²

2013

Phys. Rev. E

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“…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 numerically harder to measure, except that they vary as power laws of the system size, with exponents which seem to depend on the geometry [31].Here we consider the ABC model [32,33], a diffusive system which is known to exhibit a phase transition [34][35][36][37][38]: we study the fluctuations of the current near this transition. Generically, outside the transition the cumulants have a diffusive scaling (3).…”

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confidence: 99%

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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“…The conclusions drawn from the above generic analysis are demonstrated explicitly for the ABC model [36][37][38]. This is a three-species exclusion model defined on a discrete one-dimensional lattice.…”

Section: Introductionmentioning

confidence: 85%

“…Here we consider the model on an interval, where its dynamics satisfies detailed balance [38]. Each of the L lattice sites can be occupied by one particle (either A,B or C) or remain empty.…”

Section: Abc Model On An Interval: Canonical Ensemblementioning

confidence: 99%

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Ensemble inequivalence: Landau theory and the ABC model

Cohen¹,

Mukamel²

2012

J. Stat. Mech.

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Abstract. It is well known that systems with long-range interactions may exhibit different phase diagrams when studied within two different ensembles. In many of the previously studied examples of ensemble inequivalence, the phase diagrams differ only when the transition in one of the ensembles is first order. By contrast, in a recent study of a generalized ABC model, the canonical and grand-canonical ensembles of the model were shown to differ even when they both exhibit a continuous transition. Here we show that the order of the transition where ensemble inequivalence may occur is related to the symmetry properties of the order parameter associated with the transition. This is done by analyzing the Landau expansion of a generic model with long-range interactions. The conclusions drawn from the generic analysis are demonstrated for the ABC model by explicit calculation of its Landau expansion.

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Phase Diagram of the ABC Model on an Interval

Cited by 29 publications

References 25 publications

Aging processes in systems with anomalous slow dynamics

Aging processes in systems with anomalous slow dynamics

Current fluctuations at a phase transition

Ensemble inequivalence: Landau theory and the ABC model

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