The Annals of Probability

How to initialize a second class particle?

Márton Balázs and Attila László Nagy

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Abstract

We identify the ballistically and diffusively rescaled limit distribution of the second class particle position in a wide range of asymmetric and symmetric interacting particle systems with established hydrodynamic behavior, respectively (including zero-range, misanthrope and many other models). The initial condition is a step profile, which in some classical cases of asymmetric models, gives rise to a rarefaction fan scenario. We also point out a model with nonconcave, nonconvex hydrodynamics, where the rescaled second class particle distribution has both continuous and discrete counterparts. The results follow from a substantial generalization of Ferrari and Kipnis’ arguments (Ann. Inst. H. Poincaré 31 (1995) 143–154) for the totally asymmetric simple exclusion process. The main novelty is the introduction of a signed coupling measure as initial data, which nevertheless results in a proper probability initial distribution for the site of the second class particle and makes the extension possible. We also reveal in full generality a very interesting invariance property of the one-site marginal distribution of the process underneath the second class particle which in particular proves the intrinsicality of our choice for the initial distribution. Finally, we give a lower estimate on the probability of survival of a second class particle–antiparticle pair.

Article information

Source
Ann. Probab., Volume 45, Number 6A (2017), 3535-3570.

Dates
Received: January 2016
Revised: August 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1511773658

Digital Object Identifier
doi:10.1214/16-AOP1143

Mathematical Reviews number (MathSciNet)
MR3729609

Zentralblatt MATH identifier
06838101

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C22: Interacting particle systems [See also 60K35]

Keywords
Second class particle limit distribution rarefaction fan shock hydrodynamic limit collision probability

Citation

Balázs, Márton; Nagy, Attila László. How to initialize a second class particle?. Ann. Probab. 45 (2017), no. 6A, 3535--3570. doi:10.1214/16-AOP1143. https://projecteuclid.org/euclid.aop/1511773658


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