Drift of Phase Fluctuations in the ABC Model
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Drift of Phase Fluctuations in the ABC Model Lorenzo Bertini · Paolo Buttà
Received: 7 November 2012 / Accepted: 18 April 2013 / Published online: 25 April 2013 © Springer Science+Business Media New York 2013
Abstract 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. Keywords ABC model · Phase fluctuations · Fluctuating hydrodynamics
1 Introduction The ABC model, introduced by Evans et al. [12, 13], is a one-dimensional stochastic conservative dynamics with local jump rates, whose invariant measure undergoes a phase transition. It is a system consisting of three species of particles, traditionally labeled A, B, and C, on a discrete ring with L sites. The system evolves by nearest neighbor particles exchanges with the following rates: AB → BA, BC → CB, CA → AC with rate q and BA → AB, CB → BC, AC → CA with rate 1/q. In particular, the total number of particles Nα , of each species α ∈ {A, B, C}, are conserved and satisfy NA + NB + NC = L. When q = 1, Evans et al. [12, 13] argued that in the thermodynamic limit L → ∞ with Nα /L → rα the system segregates into pure A, B, and C regions, with translationally invariant distribution of the phase boundaries. In the equal densities case NA = NB = NC = L/3 the dynamics is reversible and its invariant measure can be explicitly computed. As shown in [14, 15], the L. Bertini · P. Buttà () Dipartimento di Matematica, Sapienza Università di Roma, P.le Aldo Moro 5, 00185 Rome, Italy e-mail: [email protected] L. Bertini e-mail: [email protected]
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L. Bertini, P. Buttà
ABC model can be reformulated in terms of a dynamic of random walks on the triangular lattice. As discussed by Clincy et al. [10], the natural scaling to investigate the asymptotic behavβ }, where the parameter ior of the ABC model is the weakly asymmetric regime q = exp{− 2L β plays the role of an inverse temperature. With this choice, the reversible measure of the equal densities case rA = rB = rC = 1/3 becomes a canonical Gibbs measure with √ a mean field Hamiltonian, which undergoes a second order phase transition at βc = 2π 3. More precisely, for β ≤ βc the typical densities profiles are homogeneous while for β > βc the three species segregate. For unequal densities the invariant measure of the ABC dynamics on a ring is not reversible and cannot be computed explicitly. The asymptotic of the twopoint correlation functions in the homogeneous phase is obtained in [6, 10], where the large deviation rat
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