Kinetic theory of quantum and classical Toda lattices

Bulchandani, Vir B.; Cao, Xiangyu; Moore, Joel E.

doi:10.1088/1751-8121/ab2cf0

Cited by 44 publications

(52 citation statements)

References 36 publications

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“…An example is shown in Figure 1. As observed in [22], the numerical solution of (4.6) by iteration with fixed µ converges systematically to the low pressure branch, which should be considered an artifact of the iteration scheme. Instead, the two solutions can be reliably found by a Fokker-Planck or Dyson Brownian motion approach, which will be outlined below.…”

Section: Free Energy Random Matrix Modelmentioning

confidence: 68%

“…for the conserved fields and correspondingly for the currents, As will be explained, variational type formulas for F toda (P, V ) and ρ Q (x) [20] are available. There is also a prediction for ρ J (x) based on the collision rate assumption [21,22]. So our strategy will be to sample the quantities in (3.13) and to compare them with the theoretical predictions.…”

Section: Lax Matrix and Gge Averaged Fieldsmentioning

confidence: 99%

“…In our paper we investigate the collision rate assumption for the classical Toda lattice [20,21,22]. The Toda lattice has several simplifying features.…”

Section: Introduction Collision Rate Assumptionmentioning

confidence: 99%

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The GGE averaged currents of the classical Toda chain

Cao

Bulchandani

Spohn

2019

J. Phys. A: Math. Theor.

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The Toda chain with random initial data is studied. Of particular interest are generalized Gibbs ensembles, their averaged conserved fields, and the averages of the corresponding currents. While averaged fields are well-understood, the description of averaged currents has hitherto relied on the collision-rate assumption. For the Toda chain, the rate assumption can be investigated numerically. Here, we provide convincing evidence for the validity of the rate assumption. This lends further support to the idea that generalized Euler-type equations have a structure common to all integrable extensive systems.arXiv:1905.04548v1 [cond-mat.stat-mech]

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Section: Free Energy Random Matrix Modelmentioning

confidence: 68%

Section: Lax Matrix and Gge Averaged Fieldsmentioning

confidence: 99%

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The GGE averaged currents of the classical Toda chain

Cao

Bulchandani

Spohn

2019

J. Phys. A: Math. Theor.

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“…This therefore explains purely from classical scattering the semiclassical result, in particular identifying the quasiparticles of classical TBA as the velocity tracers of the Toda particles themselves. The analysis of the semi-classical limit provided in [41] (which appeared shortly after the first version of the present paper) discusses further its relation to the quasiparticles.…”

Section: Resultsmentioning

confidence: 95%

“…The gas GGE we have obtained agrees with this semi-classical result. A more in-depth semi-classical analysis has been provided in [41], which appeared shortly after the first version of the present paper.…”

Section: Introductionmentioning

confidence: 99%

Generalized hydrodynamics of the classical Toda system

Doyon

2019

Journal of Mathematical Physics

107

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We obtain the exact generalised hydrodynamics for the integrable Toda system. The Toda system can be seen in a dual way, both as a gas and as a chain. In the gas point of view, using the elastic and factorised scattering of Toda particles, we obtain the generalised free energy and exact average currents, and write down the Euler hydrodynamic equations. This is written both as a continuity equation for the density of asymptotic momenta, and in terms of normal modes. This is based on the classical thermodynamic Bethe ansatz (TBA), with a single quasiparticle type of Boltzmann statistics. By explicitly connecting chain and gas conserved densities and currents, we then derive the thermodynamics and hydrodynamics of the chain. As the gas and chain have different notions of length, they have different hydrodynamics, and in particular the velocities of normal modes differ. We also give a derivation of the classical TBA equations for the gas thermodynamics from the factorised scattering of Toda particles.

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Generalized Gibbs Ensembles of the Classical Toda Chain

Spohn

2019

J Stat Phys

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Version 3 of this preprint is published online by Journal of Statistical Physics. Unfortunately, on the right hand side of the two equations in (2.27) a factor 2 is missing. To properly correct, factors of 2 dis-and reappear in various formulas throughout Sections 2 and 3. In this version the text, including references, remains untouched, only the factors of 2 are properly taken care of.Abstract. The Toda chain is the prime example of a classical integrable system with strictly local conservation laws. Relying on the Dumitriu-Edelman matrix model, we obtain the generalized free energy of the Toda chain and thereby establish a mapping to the one-dimensional log-gas with an interaction strength of order 1/N. The (deterministic) local density of states of the Lax matrix is identified as the object, which should evolve according to generalized hydrodynamics.It is my great pleasure to dedicate this article to Joel Lebowitz as teacher, as guide to yet unexplored scientific territories, and with gratitude for a lasting friendship. 25.11.2019

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Kinetic theory of quantum and classical Toda lattices

Cited by 44 publications

References 36 publications

The GGE averaged currents of the classical Toda chain

The GGE averaged currents of the classical Toda chain

Generalized hydrodynamics of the classical Toda system

Generalized Gibbs Ensembles of the Classical Toda Chain

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