2019
DOI: 10.1063/1.5095622
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Invariant measures for the box-ball system based on stationary Markov chains and periodic Gibbs measures

Abstract: The box-ball system (BBS) is a simple model of soliton interaction introduced by Takahashi and Satsuma in the 1990s. Recent work of the authors, together with Tsuyoshi Kato and Satoshi Tsujimoto, derived various families of invariant measures for the BBS based on two-sided stationary Markov chains [4]. In this article, we survey the invariant measures that were presented in [4], and also introduce a family of new ones for periodic configurations that are expressed in terms of Gibbs measures. Moreover, we show … Show more

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Cited by 12 publications
(12 citation statements)
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“…For smooth initial conditions, we further show that the resulting evolution of the soliton densities in space can alternatively be characterised by a partial differential equation, which naturally links the time-derivatives of the soliton densities and the 'effective speeds' of solitons locally. We highlight that our results contribute to the growing literature concerning the randomization of the BBS [1,2,3,8,9,10,13,14,15,17,18], and that it also connects to current work on generalized hydrodynamics [4,16,19].…”
Section: Introductionsupporting
confidence: 74%
See 1 more Smart Citation
“…For smooth initial conditions, we further show that the resulting evolution of the soliton densities in space can alternatively be characterised by a partial differential equation, which naturally links the time-derivatives of the soliton densities and the 'effective speeds' of solitons locally. We highlight that our results contribute to the growing literature concerning the randomization of the BBS [1,2,3,8,9,10,13,14,15,17,18], and that it also connects to current work on generalized hydrodynamics [4,16,19].…”
Section: Introductionsupporting
confidence: 74%
“…We will demonstrate that det(M (I) ) > 0. Clearly det(M (2) ) ≥ d 2 > 0. Hence we will now assume I ≥ 3, and expand the determinant along the bottom row of the matrix to deduce that (5.1) det 1) ,…”
Section: Effective Speedsmentioning
confidence: 99%
“…Recently there has been a renewed interest on BBS from the perspectives of statistical physics and probability theory in and out of equilibrium [6,7,8,18,19,29,30,25]. Our aim in this paper is to explore such features further in the light of generalized hydrodynamics (GHD).…”
Section: Arxiv:200401569v2 [Math-ph] 16 Apr 2020mentioning
confidence: 99%
“…See [LPS21,Sak14a,Sak14b] for some other combinatorial developments. The paper [HMO01] shows that stationary Markov chains are invariant measures for the Pitman transformation [Pit75], a dynamics equivalent to BBS; see [CKST18,CS19,CS20] for extensions.…”
Section: Overviewmentioning
confidence: 99%