What is a phase (of matter)?

Defining "a phase of matter" for a popular audience is a tricky business, with choices ranging from the overly simplistic and therefore vacuous (a collection of matter that has homogeneous, uniform, well-defined physical properties that are distinct from other such phases), to the very technical, to sophistry (like the famous definition of obscenity).

A critical ingredient missing from the simple definition above is the deep, profound point that phases of matter only make sense as emergent from the collective behavior of many constituents (the dynamics of which are often governed by simple rules).  A single water molecule is not a solid, a liquid, or a gas - it is just a single molecule, with a structure and some mechanical, electronic, and optical properties that can be calculated with pretty good accuracy through "ab initio" techniques like density functional theory and its relatives.  (Note:  Even doing that is bloody hard, given that ten electrons is actually a lot from the standpoint of quantum chemistry.)

However, if you take a collection of \(N\) water molecules and stick them in a box of a fixed volume \(V\), with a certain amount of kinetic energy \(E\), and let them bounce around and do their thing, interacting with each other via van der Waals and longer-ranged (dipolar, since water is a polar molecule) forces, something interesting will happen.  To avoid difficult conceptual issues about reversibility, let's imagine you have a whole bunch of boxes like this, all prepared with the same \(N, V\) and \(E\) but with the microscopic initial conditions like molecular positions and velocities scrambled.  (This is the "microcanonical ensemble", for experts.)  Wait an unspecified long while. What you will find is that as \(N\) increases from 1 to a large number, at some point you will start being able to classify the emergent, "coarse-grained" properties of these boxes.  For a sufficiently low \(E\), you will find that the vast majority of the boxes contain a blob of water molecules that have arranged themselves in a spatially ordered way, with spatially periodic positions and orientations.  There will be a few leftover molecules bouncing around, and the blob will have a certain amount of jiggling going on.  If you shook the box, you would see that the blob moves rigidly, exhibiting some resistance to deformation, though the molecules at the edges would move more easily, and would be constantly exchanging with the few leftover molecules bouncing around the rest of the box.  Somehow, the molecules in those boxes have spontaneously broken a bunch of symmetries (picking out spatial locations that exhibit some periodicity and rotational symmetry), and what we think of as "bulk" properties have emerged, like density, some kind of elastic modulus, a speed of sound, etc.  There is now some interface as well, between the solid and the mostly unoccupied void. 

For higher \(E\), you will probably find that the vast majority of boxes contain a blob of water molecules that are very close together, bumping into each other all the time, but tumbling around with no particular relative orientation.  This blob of water has an interface with the remaining "gas", and does not respond rigidly if it bumps into a wall of the box.  If you could look at all the molecules, you could add up how much energy it takes to expand the surface of that blob - this is proportional to the surface tension.

At still higher \(E\), you will find that the water molecules are roughly homogeneously distributed throughout each box, bumping into each other and the walls.  You could still think about an average density for this gas, and if you banged on the wall of the box to impart momentum to the molecules that happen to be hitting that wall, you could watch the propagation of a density wave (sound!) through the molecules.   In the really high \(E\) limit, the molecules decompose and the constituent atoms ionize - this is a plasma.

Each of these arrangements that you would find in a very large percentage of such imaginary boxes, with its emergence of well-defined "bulk" physical properties (including more subtle ones I haven't mentioned, like magnetic order or electrical conductivity) as \(N\) grows to a statistically large value, is a thermodynamic phase of matter.    Why are these the particular ones that occur?  Why do water molecules tend to form particular solid structures?  Why don't we see the spontaneous appearance of phases that look very different, like long 1-d chains of water molecules, for instance?  It's not at all obvious!  That's the fun of condensed matter physics:  The answer somehow lies in the microscopic properties of the molecules and their interactions - it's latent in there as soon as you have one molecule, but somehow cannot emerge and be realized except through the collective response of a large ensemble.  More soon.