Second class particles as microscopic characteristics in totally asymmetric nearest-neighbor $K$-exclusion processes
HTML articles powered by AMS MathViewer
- by Timo Seppäläinen PDF
- Trans. Amer. Math. Soc. 353 (2001), 4801-4829 Request permission
Abstract:
We prove laws of large numbers for a second class particle in one-dimensional totally asymmetric $K$-exclusion processes, under hydrodynamic Euler scaling. The assumption required is that initially the ambient particle configuration converges to a limiting profile. The macroscopic trajectories of second class particles are characteristics and shocks of the conservation law of the particle density. The proof uses a variational representation of a second class particle, to overcome the problem of lack of information about invariant distributions. But we cannot rule out the possibility that the flux function of the conservation law may be neither differentiable nor strictly concave. To give a complete picture we discuss the construction, uniqueness, and other properties of the weak solution that the particle density obeys.References
- Donald P. Ballou, Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions, Trans. Amer. Math. Soc. 152 (1970), 441–460 (1971). MR 435615, DOI 10.1090/S0002-9947-1970-0435615-3
- C. M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26 (1977), no. 6, 1097–1119. MR 457947, DOI 10.1512/iumj.1977.26.26088
- B. Derrida, S. A. Janowsky, J. L. Lebowitz, and E. R. Speer, Exact solution of the totally asymmetric simple exclusion process: shock profiles, J. Statist. Phys. 73 (1993), no. 5-6, 813–842. MR 1251221, DOI 10.1007/BF01052811
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
- Pablo A. Ferrari, Shock fluctuations in asymmetric simple exclusion, Probab. Theory Related Fields 91 (1992), no. 1, 81–101. MR 1142763, DOI 10.1007/BF01194491
- P. A. Ferrari and L. R. G. Fontes, Shock fluctuations in the asymmetric simple exclusion process, Probab. Theory Related Fields 99 (1994), no. 2, 305–319. MR 1278887, DOI 10.1007/BF01199027
- P. A. Ferrari and L. R. G. Fontes, Current fluctuations for the asymmetric simple exclusion process, Ann. Probab. 22 (1994), no. 2, 820–832. MR 1288133
- P. A. Ferrari, L. R. G. Fontes, and Y. Kohayakawa, Invariant measures for a two-species asymmetric process, J. Statist. Phys. 76 (1994), no. 5-6, 1153–1177. MR 1298099, DOI 10.1007/BF02187059
- P. A. Ferrari and C. Kipnis, Second class particles in the rarefaction fan, Ann. Inst. H. Poincaré Probab. Statist. 31 (1995), no. 1, 143–154 (English, with English and French summaries). MR 1340034
- P. A. Ferrari, C. Kipnis, and E. Saada, Microscopic structure of travelling waves in the asymmetric simple exclusion process, Ann. Probab. 19 (1991), no. 1, 226–244. MR 1085334
- Hitoshi Ishii, Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations, Indiana Univ. Math. J. 33 (1984), no. 5, 721–748. MR 756156, DOI 10.1512/iumj.1984.33.33038
- Claude Kipnis and Claudio Landim, Scaling limits of interacting particle systems, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 320, Springer-Verlag, Berlin, 1999. MR 1707314, DOI 10.1007/978-3-662-03752-2
- N. Kruzkov, First order quasilinear equations in several independent variables. Math. USSR Sb. 10 (1970), 217–243.
- Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR 0350216
- Thomas M. Liggett, Stochastic interacting systems: contact, voter and exclusion processes, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 324, Springer-Verlag, Berlin, 1999. MR 1717346, DOI 10.1007/978-3-662-03990-8
- O. A. Oleĭnik, Discontinuous solutions of non-linear differential equations, Amer. Math. Soc. Transl. (2) 26 (1963), 95–172. MR 0151737, DOI 10.1090/trans2/026/05
- Fraydoun Rezakhanlou, Hydrodynamic limit for attractive particle systems on $\textbf {Z}^d$, Comm. Math. Phys. 140 (1991), no. 3, 417–448. MR 1130693
- Fraydoun Rezakhanlou, Microscopic structure of shocks in one conservation laws, Ann. Inst. H. Poincaré C Anal. Non Linéaire 12 (1995), no. 2, 119–153. MR 1326665, DOI 10.1016/S0294-1449(16)30161-5
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
- H. Rost, Nonequilibrium behaviour of a many particle process: density profile and local equilibria, Z. Wahrsch. Verw. Gebiete 58 (1981), no. 1, 41–53. MR 635270, DOI 10.1007/BF00536194
- Timo Seppäläinen, A microscopic model for the Burgers equation and longest increasing subsequences, Electron. J. Probab. 1 (1996), no. 5, approx. 51 pp.}, issn=1083-6489, review= MR 1386297, doi=10.1214/EJP.v1-5,
- T. Seppäläinen, Hydrodynamic scaling, convex duality and asymptotic shapes of growth models, Markov Process. Related Fields 4 (1998), no. 1, 1–26. MR 1625007
- T. Seppäläinen, Coupling the totally asymmetric simple exclusion process with a moving interface, Markov Process. Related Fields 4 (1998), no. 4, 593–628. I Brazilian School in Probability (Rio de Janeiro, 1997). MR 1677061
- Timo Seppäläinen, Existence of hydrodynamics for the totally asymmetric simple $K$-exclusion process, Ann. Probab. 27 (1999), no. 1, 361–415. MR 1681094, DOI 10.1214/aop/1022677266
- T. Seppäläinen, A variational coupling for a totally asymmetric exclusion process with long jumps but no passing. Hydrodynamic Limits and Related Topics, Eds. S. Feng, A. Lawnicsak and S. Varadhan, Fields Institute Communications, Volume 27, 2000, 117–130. American Mathematical Society.
- Richard L. Wheeden and Antoni Zygmund, Measure and integral, Pure and Applied Mathematics, Vol. 43, Marcel Dekker, Inc., New York-Basel, 1977. An introduction to real analysis. MR 0492146
Additional Information
- Timo Seppäläinen
- Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706-1388
- Email: seppalai@math.wisc.edu
- Received by editor(s): October 27, 2000
- Received by editor(s) in revised form: March 28, 2001
- Published electronically: June 27, 2001
- Additional Notes: Research partially supported by NSF grant DMS-9801085.
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 4801-4829
- MSC (2000): Primary 60K35; Secondary 82C22
- DOI: https://doi.org/10.1090/S0002-9947-01-02872-0
- MathSciNet review: 1852083