Abstract
We study driven 1d lattice gas models with two types of particles and nearest neighbor hopping. We find the most general case when there is a shock solution with a product measure which has a density-profile of a step function for both densities. The position of the shock performs a biased random walk. We calculate the microscopic hopping rates of the shock. We also construct the hydrodynamic limit of the model and solve the resulting hyperbolic system of conservation laws. In case of open boundaries the selected steady state is given in terms of the boundary densities.
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REFERENCES
P.A. Ferrari, Shocks in one-dimensional processes with a drift,in Probability and Phase Transition,G.Grimmett,ed.(Dordrecht:Kluwer),1994.
B. Derrida, S.A. Janowsky, J.L. Lebowitz,and E.R. Speer,J.Stat.Phys. 73:813 (1993).
B. Derrida,J.L. Lebowitz,and E.R. Speer,J.Stat.Phys. 89:135–167(1997).
C. Pigorsch and G.M.Schütz,J.Phys.A 33:7919 (2000).
M. Balázs,J.Stat.Phys. 105:511–524(2001).
V. Belitsky and G.M. Schütz,El.J.Prob. 7Paper No.111–21 (2002).
K. Krebs, F.H. Jafarpour,and G.M. Schütz,New J.Phys. 5:145.1–145.14 (2003).
M. Balázs,math.PR/0401053,to appear in JSP.
V. Popkov and G.M. Schütz,J.Stat.Phys 112:523–540(2003).
V. Popkov,J.Phys.A 37:1545–1557(2004).
B. Tóth and B. Valkó,J.Stat.Phys. 112:497–521(2003).
J. Fritz and B. Tóth,math.PR/0304481.
G.M. Schütz,J.Phys.A 36:R339–R379(2003).
S. Grosskinsky and H. Spohn,Bull.Braz.Math.Soc. 34(3):489–507(2003).
T. Hanney and M.R. Evans,J.Phys.A 36:L441–L447(2003).
G.M. Schütz,in Phase Transitions and Critical Phenomena Vol 19,C. Domb and J. Lebowitz,eds.(Academic,London),2001.
P.F. Arndt, T. Heinzel,and V. Rittenberg,J.Phys.A 31:833–843(1998).
M.R. Evans,Braz.J.Phys. 30:42 (2000).
D. Mukamel,Phase transitions in nonequilibrium systems in Soft and Fragile Mat-ter:Nonequilibrium Dynamics,Metastability and Flow M.E. Cates and M.R. Evans,eds. (Bristol: Institute of Physics Publishing),2000.
G.B. Whitham,Linear Nonlinear Waves (New York: John Wiley & Sons),1974.
A.B. Kolomeisky, G.M. Schütz, E.B. Kolomeisky,and J.P. Straley,J.Phys.A 31:6911–6919 (1998).
V. Popkov and G.M. Schütz,Europhys.Lett. 48:257–263(1999).76 R ´akos and Sch ¨utz
B. Derrida, M.R. Evans, V. Hakim,and V. Pasquier,J.Phys.A 26:1493 (1993).
G.M. Schütz and E. Domany,J.Stat.Phys. 72:277 (1993).
T.M. Liggett,Trans.Amer.Math.Soc. 179:433 (1975).
H. Spohn,J.Phys A 16:4275 (1983).
J. Krug,Phys.Rev.Lett. 67:1882 (1991).
M.R. Evans, D.P. Foster, C. Godrèche,and D. Mukamel,Phys.Rev.Lett. 74:208–211 (1995).
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Rákos, A., Schütz, G.M. Exact Shock Measures and Steady-State Selection in a Driven Diffusive System with Two Conserved Densities. Journal of Statistical Physics 117, 55–76 (2004). https://doi.org/10.1023/B:JOSS.0000044064.62295.29
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DOI: https://doi.org/10.1023/B:JOSS.0000044064.62295.29