## Monday, January 12, 2015

### 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.

Unknown said...

The main problem with this example is that even though in our real life experience of water gas and liquid seems very different, as you know, they are not. One can cross-over from one "phase" to the other with no clear demarcation of what is really liquid or gaseous...

What is for sure well defined is a phase transition, but that still does not help to define a phase (!?) for the same reason than above.

I have to say that I'm still looking for a definition that would avoid these caveats.

Douglas Natelson said...

Unknown, I think you have to be very careful when discussing the difference between a liquid and a gas. While neither phase has broken translational or rotational symmetry, they are certainly different. They have very different densities and compressibilities, and they are separated by a first-order phase transition line. It is true that you can go continuously from one phase to the other if you vary conditions (e.g., pressure, temperature) to go around the critical point. However, the same could be said for the ferromagnet/paramagnet transition. Indeed, through the lattice gas model, you can show in 2d that (a version of) the liquid-gas transition maps onto the Ising model for magnetism - they both have the same universality class. (This implies for that particular case that while the underlying Hamiltonian has Z2 symmetry when described in terms of the right variables, the equilibrium phase is a solution that breaks that Z2 symmetry.)

I was trying to improve the simple definition of a phase (a homogeneous region of matter with distinct, well-defined physical properties) by pointing out that the properties in question that distinguish phases arise from the collective response of the constituents.

Douglas Natelson said...

Sorry - meant to say that a good discussion of this point can be found in "Statistical Mechanics in a Nutshell" by Luca Peliti (see here).

Now, if you really want to get tricky, I can try to say something about systems that look in some ways like they're in different phases but formally aren't (like a liquid and a glass)....

Unknown said...

I completely agree with you in the case of magnetism, which is the paradigmatic model to discuss phase transition.

What I meant was the following. In the case of a pure cross-over, smooth changes in all quantities when tuning a parameter, we tend to call this "phase" a single phase, even though the properties of the system might change quite drastically as we change this parameter. A good example would be the BCS-BEC cross-over. At T=0, there is only one superfluid phase, but the properties of the system looks quite different in one extreme limit, say BCS, or the other (BEC).
Correct me if I'm wrong, but we only call this a single phase, because there is no "phase transition", which definition can be made independent of the one of "phase" (!?) through the non-analyticities of the free-energy.

This leads me to conclude that the definition of phase is by the separation by a phase transition, and thus liquid and gas are the same phase... The example of a magnet is even more striking, since both "phases" (upward or downward magnetization) separated by a 1st order line are definitely the same thing - phase -, up to the reversal of all spins.

Maybe the only consistent way to get two "phases" to be always separated by a phase transition is to have a change of symmetry, and thus liquid and solid are really two different phases...

Anonymous said...

Solid, liquid and gas are phases of materials whereas a plasma is a state of matter. The transition from one to another phase happens at constant temperature for the former whereas for plasma it does no follow this.

If there is some continuity then there is a phase change but a state change will demand discontinuity.

Douglas Natelson said...

Anon@11:16, I disagree. Changes from one phase to another do not have to happen at constant temperature - that's a feature only of first-order phase transitions.

Narasimha said...

Isn't NVE microcanonical ensemble?

Douglas Natelson said...

Narasimha, yes - thanks for catching that.