Generalized hydrodynamics in box-ball system

Kuniba, Atsuo; Misguich, Grégoire; Pasquier, Vincent

doi:10.1088/1751-8121/abadb9

Cited by 22 publications

(47 citation statements)

References 38 publications

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“…The results generalize the earlier work [11] for n = 2, where the cBBS itself reduces to the original one [1].…”

Section: Introductionsupporting

confidence: 85%

“…At present, time averaged current of the n-color cBBS on a circle is yet to be obtained. However we expect the result should coincide with (4.38) supported by the n = 2 case established in [11].…”

Section: Stationary Currentssupporting

confidence: 78%

“…Here we propose that it is done by restricting the state sum to the highest ones that are defined in Appendix C.1. Similar treatment has been done in [11,17,27], where more detailed justification was made especially in the first reference. Thus the GGE partition function of our concern is…”

Section: Generalized Gibbs Ensemblementioning

confidence: 93%

“…The first approximation we employ is the ballistic scaling which implies that the normal mode y i , against ζ 10 The associated time variable could also be denoted by t (r) l , which is however avoided for notational simplicity. 11 One can show that y 12 A short comment is here in order about the word state. So far we have been using state to represent a single microscopic ball configuration.…”

Section: Ballistic Picturementioning

confidence: 99%

“…Thanks to (5.12) a plateau can be labeled by a vector d ∈ ( ≥0 ) n−1 , which corresponds to the soliton content Example 5.1. Consider the cBBS with n = 2, which is nothing but the ordinary single color BBS studied in [11]. We have the time evolutions T (r=1) l with l = 1, 2, .…”

Section: Selectmentioning

confidence: 99%

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Generalized hydrodynamics in complete box-ball system for $U_q(\widehat{sl}_n)$

2021

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We introduce the complete box-ball system (cBBS), which is an integrable cellular automaton on 1D lattice associated with the quantum group U_q(\widehat{sl}_n)Uq(sl̂n). Compared with the conventional (n-1)(n−1)-color BBS, it enjoys a remarkable simplification that scattering of solitons is totally diagonal. We also submit the cBBS to randomized initial conditions and study its non-equilibrium behavior by thermodynamic Bethe ansatz and generalized hydrodynamics. Excellent agreement is demonstrated between theoretical predictions and numerical simulation on the density plateaux generated from domain wall initial conditions including their diffusive broadening.

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“…The results generalize the earlier work [11] for n = 2, where the cBBS itself reduces to the original one [1].…”

Section: Introductionsupporting

confidence: 85%

Section: Stationary Currentssupporting

confidence: 78%

Section: Generalized Gibbs Ensemblementioning

confidence: 93%

Section: Ballistic Picturementioning

confidence: 99%

Section: Selectmentioning

confidence: 99%

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Generalized hydrodynamics in complete box-ball system for $U_q(\widehat{sl}_n)$

2021

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Thermalization and Hydrodynamics in an Interacting Integrable System: The Case of Hard Rods

Singh,

Dhar,

Spohn

et al. 2024

J Stat Phys

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We consider the relaxation of an initial non-equilibrium state in a one-dimensional fluid of hard rods. Since it is an interacting integrable system, we expect it to reach the Generalized Gibbs Ensemble (GGE) at long times for generic initial conditions. Here we show that there exist initial conditions for which the system does not reach GGE even at very long times and in the thermodynamic limit. In particular, we consider an initial condition of uniformly distributed hard-rods in a box with the left half having particles with a singular velocity distribution (all moving with unit velocity) and the right half particles in thermal equilibrium. We find that the density profile for the singular component does not spread to the full extent of the box and keeps moving with a fixed effective speed at long times. We show that such density profiles can be well described by the solution of the Euler equations almost everywhere except at the location of the shocks, where we observe slight discrepancies due to dissipation arising from the initial fluctuations of the thermal background. To demonstrate this effect of dissipation analytically, we consider a second initial condition with a single particle at the origin with unit velocity in a thermal background. We find that the probability distribution of the position of the unit velocity quasi-particle has diffusive spreading which can be understood from the solution of the Navier–Stokes (NS) equation of the hard rods. Finally, we consider an initial condition with a spread in velocity distribution for which we show convergence to GGE. Our conclusions are based on molecular dynamics simulations supported by analytical arguments.

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Soliton Decomposition of the Box-Ball System

Ferrari

Nguyen

Rolla

et al. 2021

Forum of Mathematics, Sigma

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The box-ball system (BBS) was introduced by Takahashi and Satsuma as a discrete counterpart of the Korteweg-de Vries 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 kth component moves rigidly at speed k. Let $\zeta $ be a translation-invariant family of independent random vectors under a summability condition and $\eta $ be the ball configuration with components $\zeta $ . We show that the law of $\eta $ is translation invariant and invariant for the BBS. This recipe allows us to construct a large family of invariant measures, including product measures and stationary Markov chains with ball density less than $\frac {1}{2}$ . 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 generalised Gibbs ensemble of conservative laws.

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Generalized hydrodynamics in box-ball system

Cited by 22 publications

References 38 publications

Generalized hydrodynamics in complete box-ball system for $U_q(\widehat{sl}_n)$

Generalized hydrodynamics in complete box-ball system for $U_q(\widehat{sl}_n)$

Thermalization and Hydrodynamics in an Interacting Integrable System: The Case of Hard Rods

Soliton Decomposition of the Box-Ball System

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