[Paper Review] Hydrodynamic Equations for the Toda Lattice

Herbert Spohn|arXiv (Cornell University)|Jan 16, 2021

Random Matrices and Applications142 references19 citations

TL;DR

This paper derives hydrodynamic equations for the classical Toda lattice using generalized hydrodynamics (GHD), establishing a framework for non-equilibrium dynamics in integrable systems. It introduces a Navier-Stokes-type correction to the Euler-level hydrodynamics, with a closed-form expression for entropy production that ensures positivity and satisfies the second law of thermodynamics.

ABSTRACT

1. Introduction, 2. Dynamics of the classical Toda lattice, 3. Static properties, 4. Mean-field Dyson Brownian motion, 5. Hydrodynamics for hard rods, 6. Generalized hydrodynamic equations, 7. Linearized hydrodynamics and GGE dynamical correlations, 8. Domain wall initial states, 9. Toda fluid, 10. Hydrodynamics for the Lieb-Liniger delta-Bose gas, 11. Quantum Toda lattice, 12. Beyond the Euler time scale.

Motivation & Objective

• To develop a hydrodynamic description for the classical Toda lattice, an integrable many-body system with infinitely many conserved quantities.
• To extend generalized hydrodynamics beyond the Euler scale by incorporating viscous, non-dissipative corrections via a Navier-Stokes-type framework.
• To derive a closed-form expression for entropy production that guarantees thermodynamic consistency and positivity.
• To establish a connection between the Toda lattice and quantum integrable systems through hydrodynamic analogies.
• To validate the hydrodynamic framework using exact sum rules and form factor expansions, particularly for the entropy current and production terms.

Proposed method

• Uses generalized Gibbs ensemble (GGE) formalism to describe local equilibrium states with infinitely many conserved charges.
• Applies action-angle variables and scattering theory to characterize the integrable structure of the Toda lattice.
• Derives average currents and hydrodynamic equations by taking local averages of conserved field currents in the GGE framework.
• Introduces a Navier-Stokes correction to the Euler-level hydrodynamics using a dissipative term derived from the Gaudin matrix and covariance structure.
• Employs a variational principle and thermodynamic Bethe ansatz (TBA) to compute the generalized free energy and density of states.
• Derives the entropy current and entropy production via a fluctuation-dissipation-type relation, with the production term expressed as a quadratic form involving the inverse covariance matrix and the diffusion operator.

Experimental results

Research questions

• RQ1 How can hydrodynamic equations be consistently derived for the classical Toda lattice, an integrable system with infinitely many conserved fields?
• RQ2 What is the form of the Navier-Stokes correction to the Euler-level hydrodynamics in the Toda lattice, and how does it ensure thermodynamic consistency?
• RQ3 How is the entropy production rate expressed in terms of the system's dynamical correlations and covariance structure?
• RQ4 What is the role of the Gaudin matrix and the Lax operator in constructing the hydrodynamic transport coefficients?
• RQ5 How does the entropy production formula ensure positivity and satisfy the second law of thermodynamics?

Key findings

• The entropy production rate is derived as $\sigma = \langle (\partial_x \rho_p), C^{-1} L C^{-1} (\partial_x \rho_p) \rangle$, which is manifestly non-negative and ensures thermodynamic consistency.
• The entropy current is expressed as $\mathfrak{j}_\mathrm{s} = -\langle (\log \rho_n) v^{\mathrm{eff}} \rho_p \rangle + \langle (\log \rho_n + (1 - T\rho_n)) D \partial_x \rho_p \rangle$, combining flow and diffusive contributions.
• The Navier-Stokes correction is derived from a form factor expansion and matches known results for hard rods, validating the framework.
• The dissipative term arises from a non-local correction to the current, with the diffusion coefficient $D = L C^{-1}$, where $L$ and $C$ are the Lax and covariance matrices.
• The framework extends to quantum integrable systems, such as the Lieb-Liniger model, via the Bethe ansatz and GGE formalism.
• The entropy balance equation $\partial_t s + \partial_x \mathfrak{j}_\mathrm{s} = \sigma$ is rigorously established, with $\sigma \geq 0$ ensuring the second law of thermodynamics.