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On Linear Hypocoercive BGK Models Read chapter Read first chapter

2016 | OriginalPaper | Chapter

Duality Relations for the Periodic ASEP Conditioned on a Low Current

Author : G. M. Schütz

Published in: From Particle Systems to Partial Differential Equations III

Publisher: Springer International Publishing

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Abstract

We consider the asymmetric simple exclusion process (ASEP) on a finite lattice with periodic boundary conditions, conditioned to carry an atypically low current. For an infinite discrete set of currents, parametrized by the driving strength \(s_K\), \(K \ge 1\), we prove duality relations which arise from the quantum algebra \(U_q[\mathfrak {gl}(2)]\) symmetry of the generator of the process with reflecting boundary conditions. Using these duality relations we prove on microscopic level a travelling-wave property of the conditioned process for a family of shock-antishock measures for \(N>K\) particles: If the initial measure is a member of this family with K microscopic shocks at positions \((x_1,\dots ,x_K)\), then the measure at any time \(t>0\) of the process with driving strength \(s_K\) is a convex combination of such measures with shocks at positions \((y_1,\dots ,y_K)\), which can be expressed in terms of K-particle transition probabilities of the conditioned ASEP with driving strength \(s_N\).

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previous chapter Derivation of the Boltzmann Equation: Hard Spheres, Short-Range Potentials and Beyond
Appendix
Available only for authorised users
Footnotes
1
We mention that the deep link between duality of Markov processes and symmetries of its generator, first noted in [32], that we exploit here was given a systematic abstract treatment in [21]. More recently many concrete symmetry-based dualities for interacting particle systems were derived using this approach [7, 10, 12‐15, 17, 25, 29].
 
2
When the summation is over \(\varOmega =\mathbb {S}^L\) we shall usually omit the set \(\mathbb {S}^L\) under the summation symbol and simply write \(\sum _{\eta }\).
 
3
This is equivalent to Eq. (2.14) in [33], which, however, has a sign error and should read \(H^T = V^{-2} H V^2\).
 
4
Notice a sign error in front of the term \(2k_i\) in Eq. (3.12) of [33] and pay attention to the different convention \(q \leftrightarrow q^{-1}\).
 
5
Eqs. (2.62a) and (2.62b) of [30] have some sign errors which are corrected in Proposition (1).
 
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Metadata
Title
Duality Relations for the Periodic ASEP Conditioned on a Low Current
Author
G. M. Schütz
Copyright Year
2016
Book
From Particle Systems to Partial Differential Equations III

Print ISBN: 978-3-319-32142-4

Electronic ISBN: 978-3-319-32144-8

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-32144-8_16

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