Integrable hydrodynamics of Toda chain: case of small systems

TL;DR

This paper demonstrates that generalized hydrodynamics accurately describes the fluctuation spreading in small integrable Toda chains, aligning with large-system results and emphasizing the relevance of integrable hydrodynamics in experimental microscopic systems.

Contribution

The study numerically verifies that integrable hydrodynamics applies to small systems, bridging microscopic simulations and continuum descriptions in the Toda chain.

Findings

Small systems' fluctuation spreading matches thermodynamic predictions.

Generalized hydrodynamics effectively describes microscopic system dynamics.

Results support the validity of integrable hydrodynamics in laboratory-scale experiments.

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 spatiotemporal 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.