Abstract
The three species asymmetric ABC model was initially defined on a ring by Evans, Kafri, Koduvely, and Mukamel, and the weakly asymmetric version was later studied by Clincy, Derrida, and Evans. Here the latter model is studied on a one-dimensional lattice of N sites with closed (zero flux) boundaries. In this geometry the local particle conserving dynamics satisfies detailed balance with respect to a canonical Gibbs measure with long range asymmetric pair interactions. This generalizes results for the ring case, where detailed balance holds, and in fact the steady state measure is known, only for the case of equal densities of the different species: in the latter case the stationary states of the system on a ring and on an interval are the same. We prove that in the limit N→∞ the scaled density profiles are given by (pieces of) the periodic trajectory of a particle moving in a quartic confining potential. We further prove uniqueness of the profiles, i.e., the existence of a single phase, in all regions of the parameter space (of average densities and temperature) except at low temperature with all densities equal; in this case a continuum of phases, differing by translation, coexist. The results for the equal density case apply also to the system on the ring, and there extend results of Clincy et al.
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Lieb, E.H., Mattis, D.C. (eds.): Mathematical Physics in One Dimension. Academic Press, New York (1966)
Percus, J.K.: Exactly solvable models of classical many-body systems. In: Lebowitz, J.L. (ed.) Simple Models of Equilibrium and Nonequilibrium Phenomena. Amsterdam, North-Holland (1987)
Privman, V. (ed.): Nonequilibrium Statistical Mechanics in One Dimension. Cambridge University Press, Cambridge (1997)
Schütz, G.M.: Exactly solvable models for many-body systems far from equilibrium. In: Domb, C., Lebowitz, J.L. (eds.) Phase Transitions and Critical Phenomena, vol. 19. Academic Press, London (2000)
Evans, M.R., Kafri, Y., Koduvely, H.M., Mukamel, D.: Phase separation in one-dimensional driven diffusive systems. Phys. Rev. Lett. 80, 425–429 (1998)
Evans, M.R., Kafri, Y., Koduvely, H.M., Mukamel, D.: Phase separation and coarsening in one-dimensional driven diffusive systems: Local dynamics leading to long-range Hamiltonians. Phys. Rev. E 58, 2764–2778 (1998)
Lahiri, R., Barma, M., Ramaswamy, S.: Strong phase separation in a model of sedimenting lattices. Phys. Rev. E 61, 1648–1658 (2000)
Clincy, M., Derrida, B., Evans, M.R.: Phase transition in the ABC model. Phys. Rev. E 67, 066115 (2003)
Bodineau, T., Derrida, B., Lecomte, V., van Wijland, F.: Long range correlations and phase transition in non-equilibrium diffusive systems. J. Stat. Phys. 133, 1013–1031 (2008)
Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Towards a nonequilibrium thermodynamics: a self-contained macroscopic description of driven diffusive systems. J. Stat. Phys. 135, 857–872 (2009)
Fayolle, G., Furtlehner, C.: Stochastic deformations of sample paths of random walks and exclusion models. In: Drmota, M., Flajolet, P., Gardy, D., Gittenberger, B. (eds.) Mathematics and Computer Science III: Algorithms, Trees, Combinatorics and Probabilities. Trends in Mathematics. Birkhäuser, Basel (2004)
Fayolle, G., Furtlehner, C.: Stochastic dynamics of discrete curves and multi-type exclusion processes. J. Stat. Phys. 127, 1049–1094 (2007)
Sandow, S., Schütz, G.: On U q [SU(2)]-symmetric driven diffusion. Europhys. Lett. 26, 7–12 (1994)
Blythe, R.A., Evans, M.R.: Nonequilibrium steady states of matrix-product form: a solver’s guide. J. Phys. A 40, R333 (2007)
Aizenmann, M., Lieb, E.: The IIIrd law of thermodynamics and the degeneracy of ground states in lattice systems. J. Stat. Phys. 24, 279–297 (1981)
Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin (1999)
Hewitt, E., Savage, L.J.: Symmetric measures on Cartesian products. Trans. Am. Math. Soc. 80, 470–501 (1955)
Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften, vol. 320. Springer, Berlin (1999)
Lebowitz, J.L.: Coexistence of phases in Ising ferromagnets. J. Stat. Phys. 16, 463–476 (1977)
Ellis, R.S.: Entropy, Large Deviations, and Statistical Mechanics. Grundlehren der Mathematischen Wissenschaften, vol. 271. Springer, Berlin (1985)
Dunford, N., Pettis, B.J.: Linear operators on summable functions. Trans. Am. Math. Soc. 47, 323–392 (1940)
Mazur, S.: Über konvexe Menge in linearen normierten Raumen. Stud. Math. 4, 70–84 (1933)
Kafri, Y., Biron, D., Evans, M.R., Mukamel, D.: Slow coarsening in a class of driven systems. Eur. Phys. J. B 16, 669–676 (2000)
Armitage, J.V., Eberlein, W.F.: Elliptic Functions. London Mathematical Society Student Texts, vol. 67. Cambridge University Press, Cambridge (2006)
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An erratum to this article is available at http://dx.doi.org/10.1007/s10955-011-0287-7.
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Ayyer, A., Carlen, E.A., Lebowitz, J.L. et al. Phase Diagram of the ABC Model on an Interval. J Stat Phys 137, 1166–1204 (2009). https://doi.org/10.1007/s10955-009-9834-x
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DOI: https://doi.org/10.1007/s10955-009-9834-x