What is a liquid?

I wrote recently about phases of matter (and longer ago here).  The phase that tends to get short shrift in the physics curriculum is the liquid, and this is actually a symptom indicating that liquids are not simple things. 

We talk a lot about gases, and they tend to be simple in large part because they are low density systems - the constituents spend the overwhelming majority of their time far apart (compared to the size of the constituents), and therefore tend to interact with each other only very weakly.  We can even look in the ideal limit of infinitesimal particle size and zero interactions, so that the only energy in the problem is the kinetic energy of the particles, and derive the Ideal Gas Law.  

There is no such thing as an Ideal Liquid Law.  That tells you something about the complexity of these systems right there.

A classical liquid is a phase of matter in which the constituent particles have a typical interparticle distance comparable to the particle size, and therefore interact strongly, with both a "hard core repulsion" so that the particles are basically impenetrable, and usually through some kind of short-ranged attraction, either from van der Waals forces and/or longer-ranged/stronger interactions.  The kinetic energy of the particles is sufficiently large that they don't bond rigidly to each other and therefore move past and around each other continuously.  However, the density is so high that you can't even do very well by only worrying about pairs of interacting particles - you have to keep track of three-body, four-body, etc. interactions somehow.    

The very complexity of these strongly interacting collections of particles leads to the emergence of some simplicity at larger scales.  Because the particles are cheek-by-jowl and impenetrable, liquids are about as incompressible as solids.  The lack of tight bonding and enough kinetic energy to keep everyone moving means that, on average and on scales large compared to the particle size, liquids are homogeneous (uniform properties in space) and isotropic (uniform properties in all directions).  When pushed up against solid walls by gravity or other forces, liquids take on the shapes of their containers.  (If the typical kinetic energy per particle can't overcome the steric interactions with the local environment, then particles can get jammed.  Jammed systems act like "rigid" solids.)

Because of the constant interparticle collisions, energy and momentum get passed along readily within liquids, leading to good thermal conduction (the transport of kinetic energy of the particles via microscopic, untraceable amounts we call heat) and viscosity (the transfer of transverse momentum between adjacent rough layers of particles just due to collisions - the fluid analog of friction).  The lack of rigid bonding interactions means that liquids can't resist shear; layers of particles slide past each other.  This means that liquids, like gases, don't have transverse sound waves.   The flow of particles is best described by hydrodynamics, a continuum approach that makes sense on scales much bigger than the particles.   

Quantum liquids are those for which the quantum statistics of the constituents are important to the macroscopic properties.  Liquid helium is one such example.  Physicists have also adopted the term "liquid" to mean any strongly interacting, comparatively incompressible, flow-able system, such as the electrons in a metal ("Fermi liquid").  

Liquids are another example emergence that is deep, profound, and so ubiquitous that people tend to look right past it.  "Liquidity" is a set of properties so well-defined that a small child can tell you whether something is a liquid by looking at a video of it; those properties emerge largely independent of the microscopic details of the constituents and their interactions (water molecules with hydrogen bonds; octane molecules with van der Waals attraction; very hot silica molecules in flowing lava); and none of those properties are obvious if one starts with, say, the Standard Model of particle physics.