Phase transition in the ABC model.

Recent studies have shown that one-dimensional driven systems can exhibit phase separation even if the dynamics is governed by local rules. The ABC model, which comprises three particle species that diffuse asymmetrically around a ring, shows anomalous coarsening into a phase separated steady state. In the limiting case in which the dynamics is symmetric and the parameter q describing the asymmetry tends to one, no phase separation occurs and the steady state of the system is disordered. In the present work, we consider the weak asymmetry regime q=exp(-beta/N), where N is the system size, and study how the disordered state is approached. In the case of equal densities, we find that the system exhibits a second-order phase transition at some nonzero beta(c). The value of beta(c)=2pi square root 3 and the optimal profiles can be obtained by writing the exact large deviation functional. For nonequal densities, we write down mean-field equations and analyze some of their predictions.

[1]  Michael E. Fisher,et al.  Critical Exponents for Long-Range Interactions , 1972 .

[2]  Eytan Domany,et al.  An exact solution of a one-dimensional asymmetric exclusion model with open boundaries , 1992 .

[3]  K. Mallick,et al.  Shocks in the asymmetry exclusion model with an impurity , 1996 .

[4]  Anomalous coarsening and glassy dynamics , 2001, cond-mat/0111171.

[5]  Evans,et al.  Spontaneous symmetry breaking in a one dimensional driven diffusive system. , 1995, Physical review letters.

[6]  K. Binder,et al.  Monte Carlo Simulation in Statistical Physics , 1992, Graduate Texts in Physics.

[7]  H. Hinrichsen.  Non-equilibrium critical phenomena and phase transitions into absorbing states , 2000, cond-mat/0001070.

[8]  Krug,et al.  Boundary-induced phase transitions in driven diffusive systems. , 1991, Physical review letters.

[9]  E. R. Speer,et al.  Spatial particle condensation for an exclusion process on a ring , 2000 .

[10]  B. Derrida,et al.  Free energy functional for nonequilibrium systems: an exactly solvable case. , 2001, Physical review letters.

[11]  G. Parisi,et al.  Replica field theory for deterministic models: I. Binary sequences with low autocorrelation , 1994, hep-th/9405148.

[12]  M. R. Evans,et al.  Bose-Einstein condensation in disordered exclusion models and relation to traffic flow , 1996 .

[13]  J. Bernasconi.  Low autocorrelation binary sequences : statistical mechanics and configuration space analysis , 1987 .

[14]  Cumulants of the three-state Potts model and of nonequilibrium models with C3v symmetry , 2002, cond-mat/0205592.

[15]  Are Steadily Moving Crystals Unstable , 1996, cond-mat/9610022.

[16]  Strong phase separation in a model of sedimenting lattices , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  B Derrida,et al.  Exact free energy functional for a driven diffusive open stationary nonequilibrium system. , 2002, Physical review letters.

[18]  B. Derrida,et al.  Exact solution of the totally asymmetric simple exclusion process: Shock profiles , 1993 .

[19]  D. Mukamel,et al.  PHASE SEPARATION IN ONE-DIMENSIONAL DRIVEN DIFFUSIVE SYSTEMS , 1998 .

[20]  Herbert Spohn,et al.  Nonequilibrium steady states of stochastic lattice gas models of fast ionic conductors , 1984 .

[21]  Kurt Binder,et al.  Monte Carlo Simulation in Statistical Physics , 1992, Graduate Texts in Physics.

[22]  Zia,et al.  Spontaneous structure formation in driven systems with two species: Exact solutions in a mean-field theory. , 1994, Physical review letters.

[23]  D. Mukamel,et al.  Phase Separation and Coarsening in One-Dimensional Driven Diffusive Systems: Local Dynamics Leading to Long-Range Hamiltonians , 1998 .

[24]  B. Schmittmann,et al.  Long-range order in a quasi one-dimensional non-equilibrium three-state lattice gas , 1999 .

[25]  E. Domany,et al.  Phase transitions in an exactly soluble one-dimensional exclusion process , 1993, cond-mat/9303038.

[26]  B. Derrida,et al.  Exact solution of a 1d asymmetric exclusion model using a matrix formulation , 1993 .