# What is the hydrodynamic limit exactly and why is it called that?

Hydrodynamics is one of those words which are used everywhere in the literature, but I can not seem to find a clear definition! My idea (which could be wrong) is it is a continuum limit of a theory where space and time become continuous. Is this right?

Like I can formulate the thermodynamic limit as taking the particle number and volume of a system to infinity while holding the density fixed. Is there some sort of statement like this?

Also, if you know, does the name come from Navier-Stokes equation? I recall reading something on Boltzmann to Navier-Stokes to hydrodynamics. Some history would just be nice!

Hydrodynamics is an effective (emergent) theory that describes the long-time, long-distance (small frequency and wave-number) dynamics of most interacting many-body systems. The hydrodynamic limit is the limit $$\omega\to 0$$, $$\vec{k}\to 0$$. The hydrodynamic description is systematically improvable order by order in $$\omega$$ and $$k$$.
If the system is confined to a finite volume, then as $$t\to\infty$$ complete equilibration takes place, and hydrodynamics reduces to thermodynamics. Hydrodynamics is then a dynamic, time-dependent, generalization of thermodynamics. In the hydrodynamic limit systems are in approximate local equilibrium, but not necessarily in global equilibrium.
There are many kinds of many-body systems, and many hydrodynamic theories. These fall into classes, dependning on the symmetries, the dimensionality, and the number and type of conserved or quantities. Historically, the first system to be studied is the theory of non-relativistic many body systems (like water or air, which is why the theory is called hydrodynamics) that are governed by conserved particle number, energy, and momentum. The corresponding theory is called the Euler equation, and including the first correction in $$k$$ leads to the Navier-Stokes equation.