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Second class particles as microscopic characteristics in totally asymmetric nearest-neighbor $K$-exclusion processes


Author: Timo Seppäläinen
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
Published electronically: June 27, 2001
MathSciNet review: 1852083
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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.


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Additional Information

Timo Seppäläinen
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706-1388
Email: seppalai@math.wisc.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02872-0
Keywords: Exclusion process, second class particle, hydrodynamic limit, variational coupling method, characteristic, nonlinear conservation law
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.
Article copyright: © Copyright 2001 American Mathematical Society

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