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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2018
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1806.02798 |
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| _version_ | 1866914068517879808 |
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| author | Ferrari, Pablo A. Nguyen, Chi Rolla, Leonardo T. Wang, Minmin |
| author_facet | Ferrari, Pablo A. Nguyen, Chi Rolla, Leonardo T. Wang, Minmin |
| contents | The Box-Ball System, shortly BBS, was introduced by Takahashi and Satsuma as a discrete counterpart of the KdV equation. Both systems exhibit solitons whose shape and speed are conserved after collision with other solitons. We introduce a slot decomposition of ball configurations, each component being an infinite vector describing the number of size $k$ solitons in each $k$-slot. The dynamics of the components is linear: the $k$-th component moves rigidly at speed $k$. Let $ζ$ be a translation invariant family of independent random vectors under a summability condition and $η$ the ball configuration with components $ζ$. We show that the law of $η$ is translation invariant and invariant for the BBS. This recipe allows us to construct a big family of invariant measures, including product measures and stationary Markov chains with ball density less than $\frac12$. We also show that starting BBS with an ergodic measure, the position of a tagged $k$-soliton at time $t$, divided by $t$ converges as $t\to\infty$ to an effective speed $v_k$. The vector of speeds satisfies a system of linear equations related with the Generalized Gibbs Ensemble of conservative laws. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1806_02798 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Soliton decomposition of the Box-Ball System Ferrari, Pablo A. Nguyen, Chi Rolla, Leonardo T. Wang, Minmin Mathematical Physics Dynamical Systems Probability The Box-Ball System, shortly BBS, was introduced by Takahashi and Satsuma as a discrete counterpart of the KdV equation. Both systems exhibit solitons whose shape and speed are conserved after collision with other solitons. We introduce a slot decomposition of ball configurations, each component being an infinite vector describing the number of size $k$ solitons in each $k$-slot. The dynamics of the components is linear: the $k$-th component moves rigidly at speed $k$. Let $ζ$ be a translation invariant family of independent random vectors under a summability condition and $η$ the ball configuration with components $ζ$. We show that the law of $η$ is translation invariant and invariant for the BBS. This recipe allows us to construct a big family of invariant measures, including product measures and stationary Markov chains with ball density less than $\frac12$. We also show that starting BBS with an ergodic measure, the position of a tagged $k$-soliton at time $t$, divided by $t$ converges as $t\to\infty$ to an effective speed $v_k$. The vector of speeds satisfies a system of linear equations related with the Generalized Gibbs Ensemble of conservative laws. |
| title | Soliton decomposition of the Box-Ball System |
| topic | Mathematical Physics Dynamical Systems Probability |
| url | https://arxiv.org/abs/1806.02798 |