  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.
