2020/03/29 by Kazuki Kondo, Kondo, Kazuki · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Probability (math.PR) #Quantum chaos and dynamical systems #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR
paper · pdf · doi:10.48550/arxiv.2003.12974
arxiv created 2020/03/29 · openalex publication_date 2020/03/29 · arxiv updated 2020/03/31 · openalex created_date 2020/04/03 · openalex updated_date 2026/07/28
The Box-Ball System (BBS) is a cellular automaton introduced by Takahashi and Satsuma in the 1990s. The system is a discrete counterpart of the KdV equation and exhibits solitonic behavior. Recently, the BBS started from a random two-sided infinite particle configuration has been studied, by encoding the particle configuration to a certain discrete path on Z and defining the dynamic of the BBS in terms of the path. In this paper, we extend some results of the previous study in this direction to a generalization of the BBS with \kappa-color balls (particles), called the multicolor BBS. We first introduce an encoding of the \kappa-color particles configuration to a discrete path in \kappa-dimensional Euclidean space. Then we show that the dynamics of the multicolor BBS is expressed by the composition of the Pitman's transformation and a permutation operator. Applying this expression, we characterize the set of configurations for which the dynamics are well-defined and reversible for all times. Then, we give a simple class of random initial conditions which are invariant in distribution under the dynamics of the multicolor BBS. Finally, we introduce a continuous version of the multicolor BBS, which is defined for continuous paths on R in \kappa-dimensional Euclidean space, and show that \kappa-dimensional Brownian motion with a proper drift is invariant under the dynamics of the continuous version of the multicolor BBS.