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arXiv:2101.06528

Hydrodynamic Equations for the Toda Lattice

Herbert E. Spohn

Published Jan 16, 2021• 19 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2101.06528

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We obtain the exact generalized 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 factorized scattering of Toda particles, we obtain the generalized 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 factorized scattering of Toda particles.
Abstract Passing from a microscopic discrete lattice system with many degrees of freedom to a mesoscopic continuum system described by a few coarse-grained equations is challenging. The common folklore is to take the thermodynamic limit so that the physics of the discrete lattice describes the continuum results. The analytical procedure to do so relies on defining a small length scale (typically the lattice spacing) to coarse grain the microscopic evolution equations. Moving from the microscopic scale to the mesoscopic scale then requires careful approximations. In this work, we numerically test the coarsening in a Toda chain, which is an interacting integrable system, i.e., a system with a macroscopic number of conserved charges. Specifically, we study the spreading of fluctuations by computing the spatio-temporal thermal correlations with three different methods: (a) using microscopic molecular dynamics simulation with a large number of particles; (b) solving the generalized hydrodynamics equation; (c) solving the linear Euler scale equations for each conserved quantities. Surprisingly, the results for the small systems (c) match the thermodynamic results in (a) and (b) for macroscopic systems. This reiterates the importance and validity of integrable hydrodynamics in describing experiments in the laboratory, where we typically have microscopic systems.
We show that the equations of generalized hydrodynamics (GHD), a hydrodynamic theory for integrable quantum systems at the Euler scale, emerge in full generality in a family of classical gases, which generalize the gas of hard rods. In this family, the particles, upon colliding, jump forward or backward by a distance that depends on their velocities, reminiscent of classical soliton scattering. This provides a "molecular dynamics" for GHD: a numerical solver which is efficient, flexible, and which applies to the presence of external force fields. GHD also describes the hydrodynamics of classical soliton gases. We identify the GHD of any quantum model with that of the gas of its solitonlike wave packets, thus providing a remarkable quantum-classical equivalence. The theory is directly applicable, for instance, to integrable quantum chains and to the Lieb-Liniger model realized in cold-atom experiments.
Abstract We report on molecular dynamics simulations of spacetime correlations of the Toda lattice in thermal equilibrium. The correlations of stretch, momentum, and energy are computed numerically over a wide range of pressure and temperature. Our numerical results are compared with the predictions from linearized generalized hydrodynamics on the Euler scale. The system size is N=3000,4000N=3000,4000N=3000,4000 N = 3000 , 4000 and time t=600t =600t=600 t = 600 , at which ballistic scaling is well confirmed. With no adjustable parameters, the numerically obtained scaling functions agree with the theory within a precision of less than 3.5%.