Abstract
We present new results for the current as a function of transmission rate in the one-dimensional totally asymmetric simple exclusion process (TASEP) with a blockage that lowers the jump rate at one site from one tor<1. Exact finitevolume results serve to bound the allowed values for the current in the infinite system. This proves the existence of a nonequilibrium “phase transition,” corresponding to an “immiscibility” gap in the allowed values of the asymptotic densities which the infinite system can have in a stationary state. A series expansion inr, derived from the finite systems, is proven to be asymptotic for all sufficiently large systems. Padé approximants based on this series, which make specific assumptions about the nature of the singularity atr=1, match numerical data for the “infinite” system to 1 part in 104.
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References
H. Sphon,Large-Scale Dynamics of Interacting Particles (Springer-Verlag, 1991); P. A. Ferrari; Shocks in the Burgers equation and the asymmetric simple exclusion process, inAutomata Networks, Dynamical Systems and Statistical Physics, E. Goles and S. Martinez, eds (Kluwer, 1992) and references therein.
K. Kawasaki, Diffusion constants near the critical point for time-dependent Ising models. I.Phys. Rev. 145:224–230 (1966).
S. Katz, J. Lebowitz, and H. Spohn, Nonequilibrium steady states of stochastic lattice gas models of fast ionic conductors,J. Stat. Phys. 34:497–537 (1984).
A. de Masi and E. Presutti,Mathematical Methods for Hydordynamic Limits, (Springer-Verlag, 1991), and references therein.
S. A. Janowsky and J. L. Lebowitz, Finite size effects and shock fluctuations in the asymmetric simple exclusion process,Phys. Rev. A 45:618–625 (1992).
W. Dietrich, P. Fulde, and I. Peschel, Theoretical models for superionic conductors,Adv. Phys. 29:527–605 (1980), and references therein.
K. Nagel and M. Schreckenberg, A cellular autamaton model for freeway traffic,J. Phys. (Paris)I 2:2221 (1992).
D. E. Wolf and L.-H. Tang, Inhomogeneous growth processes,Phys. Rev. Lett. 65:1591–1594 (1990).
D. Kandel and D. Mukamel, Defects, interface profile and phase transitions in growth models,Europhys. Lett. 20:325–329 (1992).
F. J. Alexander, Z. Cheng, S. A. Janowsky, and J. L. Lebowitz, Shock fluctuations in the two-dimensional asymmetric simple exclusion process,J. Stat. Phys. 68:761–785 (1992).
M. Bramson, Personal communication
B. Schmittman, Critical behavior of the driven diffusive lattice gas,Int. J. Mod. Phys. B 4:2269–2306 (1990).
P. Garrido, J. L. Lebowitz, C. Maes, and H. Spohn, Long-range correlations for conservative dynamics,Phys. Rev. A 42:1954–1968 (1990).
R. Bhagavatula, G. Grinstein, Y. He, and C. Jayaprakash Algebraic correlations in conserving chaotic systems,Phys. Rev. Lett. 69, 3483–3486 (1992).
B. Derrida, E. Domany, and D. Mukamel, An exact solution of a one dimensional asymmetric exclusion model with open boundaries,J. Stat. Phys. 69:667–687 (1992).
J. Krug, Steady state selection in driven diffusive systems, inSpontaneous Formation of pace-Time Structures and Criticality, T. Riste and D. Sherrington, eds. (Plenum Press, 1991).
B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation.J. Phys. A 26:1493–1517 (1993).
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Janowsky, S.A., Lebowitz, J.L. Exact results for the asymmetric simple exclusion process with a blockage. J Stat Phys 77, 35–51 (1994). https://doi.org/10.1007/BF02186831
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DOI: https://doi.org/10.1007/BF02186831