Generalized hydrodynamic limit for the box-ball system

Abstract

We deduce a generalized hydrodynamic limit for the boxball system, which explains how the densities of solitons of different sizes evolve asymptotically under Euler space-time scaling. To describe the limiting soliton flow, we introduce a continuous state-space analogue of the soliton decomposition of Ferrari, Nguyen, Rolla and Wang (cf. the original work of Takahashi and Satsuma), namely we relate the densities of solitons of given sizes in space to corresponding densities on a scale of 'effective distances', where the dynamics are linear. 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.

Generalized hydrodynamics in box-ball system

“…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).…”

“…The equation above was obtained from (5.25) by replacing σ by ρ, which is legitimate thanks to the footnote 8 . In the simple case with l = 2 these equations can be solved directly, bypassing the use of y(ζ).…”

Hydrodynamic Equations for the Toda Lattice

“…The dynamics is given through a deterministic map iterated many times. The hydrodynamics of the box-ball system has been studied in considerable detail [34,65,96,98].…”