Abstract
We derive an ODE for the macroscopic evolution of a tagged particle in models such as asymmetric simple exclusions and zero range processes. The right-hand side of the ODE is discontinuous and its solutions are understood in the Filippov sense. We establish the uniqueness of the ODE, and explore its relationship with the hydrodynamic equation of the particle density.
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References
Cocozza, C.T.: Processus des misanthropes. Z. Wahrs. Verw. Gebiete70, 509–523 (1985)
Dafermos, C.M.: Generalized characteristics and the structure of solutions of hyperbolic conservation laws. Indiana U. Math. J.26, 1097–1119 (1977)
Filippov, A.F.: Differential equations with discontinuous right-hand side. Mat. Sbornik (N.S.)51 (93), 99–128 (1960)
Guo, M.Z., Papanicolaou, G.C., Varadhan, S.R.S.: Nonlinear diffusion limit for a system with nearest neighbor interactions. Commun. Math. Phys.118, 31–59 (1988)
Kipnis, C.: Central limit theorems for infinite series of queues and applications to simple exclusion. Ann. Probab., Vol.14, No. 2, 397–408, (1986)
Kipnis, C., Varadhan, S.R.S.: Central limit theorem for additive functionals for reversible Markov processes and applications to simple exclusions. Commun. Math. Phys.104, 1–19 (1986)
Kružkov S.N.: First order quasilinear equations in several independent variables. Math. USSR-Sb10, 217–243 (1970)
Liggett, T.M.: Interacting Particle Systems. Berlin, Heidelberg, New York: Springer, 1985
Quastel, J.: Diffusion of colour in the simple exclusion process. Comm. Pure Appl. Math.
Rezakhanlou, F.: Hydrodynamic limit for attractive particle systems on ℤd. Commun. Math. Phys.140, 417–448 (1991)
Rezakhanlou, F.: Propagation of chaos for symmetric simple exclusions. To appear in Comm. Pure Appl. Math.
Rezakhanlou, F.: Microscopic structure of shocks in one conservation laws, preprint
Saada, E.: A limit theorem for the position of a tagged particle in a simple exclusion process. Ann. Probab., Vol.15, 375–381 (1987)
Saada, E.: Processus de zero-range avec particule marquée. Ann. Inst. Henri Poincaré, Vol26, No. 1, 5–17 (1990)
Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Berlin, Heidelberg, New York: Springer, 1980
Spohn, H.: Large Scale Dynamics for Interacting Particles. Berlin, Heidelberg, New York: Springer, 1991.
Sznitman, A.: Topics in propagation of chaos. Lecture Notes in Mathematics1464, Berlin, Heidelberg, New York: Springer, (1989)
Zeimer, W.: Weakly Differentiable Functions. Berlin, Heidelberg, New York: Springer, 1989
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Communicated by J.L. Lebowitz
Research partially supported by National Science Foundation grant DMS-9208490.
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Rezakhanlou, F. Evolution of tagged particles in non-reversible particle systems. Commun.Math. Phys. 165, 1–32 (1994). https://doi.org/10.1007/BF02099734
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DOI: https://doi.org/10.1007/BF02099734