Thursday, July 30, 2009

More musing about phase transitions

Everyone has seen phase transitions - water freezing and water boiling, for example. These are both examples of "first-order" phase transitions, meaning that there is some kind of "latent heat" associated with the transition. That is, it takes a certain amount of energy to convert 1 g of solid ice into 1 g of liquid water while the temperature remains constant. The heat energy is "latent" because as it goes into the material, it's not raising the temperature - instead it's changing the entropy, by making many more microscopic states available to the atoms than were available before. In our ice-water example, at 0 C there are a certain number of microscopic states available to the water molecules in solid ice, including states where the molecules are slightly displaced from their equilibrium positions in the ice crystal and rattling around. In liquid water at the same temperature, there are many more possible microscopic states available, since the water molecules can, e.g., rotate all over the place, which they could not do in the solid state. (This kind of transition is "first order" because the entropy, which can be thought of as the first derivative of some thermodynamic potential, is discontinuous at the transition.) Because this kind of phase transition requires an input or output of energy to convert material between phases, there really aren't big fluctuations near the transition - you don't see pieces of ice bopping in and out of existence spontaneously inside a glass of icewater.

There are other kinds of phase transitions. A major class of much interest to physicists is that of "second-order" transitions. If one goes to high enough pressure and temperature, the liquid-gas transition becomes second order, right at the critical point where the distinction between liquid and gas vanishes. A second order transition is continuous - that is, while there is a change in the collective properties of the system (e.g., in the ferro- to paramagnetic transition, you can think of the electron spins as many little compass needles; in the ferromagnetic phase the needles all point the same direction, while in the paramagnetic phase they don't), the number of microscopic states available doesn't change across the transition. However, the rate at which microstates become available with changes in energy is different on the two sides of the transition. In second order transitions, you can get big fluctuations in the order of the system near the transition. Understanding these fluctuations ("critical phenomena") was a major achievement of late 20th century theoretical physics.

Here's an analogy to help with the distinction: as you ride a bicycle along a road, the horizontal distance you travel is analogous to increasing the energy available to one of our systems, and the height of the road corresponds to the number of microscopic states available to the system. If you pedal along and come to a vertical cliff, and the road continues on above your head somewhere, that's a bit like the 1st order transition. With a little bit of energy available, you can't easily go back and forth up and down the cliff face. On the other hand, if you are pedaling along and come to a change in the slope of the road, that's a bit like the 2nd order case. Now with a little bit of energy available, you can imagine rolling back and forth over that kink in the road. This analogy is far from perfect, but maybe it'll provide a little help in thinking about these distinctions. One challenge in trying to discuss this stuff with the lay public is that most people only have everyday experience with first-order transitions, and it's hard to explain the subtle distinction between 1st and 2nd order.

11 comments:

David Sanders said...

Hi,

Firstly, thanks for a very interesting blog and great explanations in general.

There's something that always seems strange to me with regards to 1st-order phase transitions. People often seem to define them by the existence of latent heat.

However, one of the most easily-understood (in my opinion) 1st-order transitions is that in the 2D Ising model with external field, when the field is switched from positive to negative.

In this case, as far as I can see, there is perfect symmetry between the two phases at the transition, and in particular they have equal (free) energies. Thus there is no latent heat, although the transition is 1st-order (below the critical temperature).

Of course, this is a bit of a special case due to the symmetry. Nonetheless, I think it's important to point it out.

Best wishes,
David.

Doug Natelson said...

Hi David - Thanks for the comment. Can you provide a reference that discusses how the 2d Ising in external field is first order? I found this, which argues that it is actually second order. It also seems important that there's a difference between a field-driven transition (as in your case, where after switching the field direction the system is no longer in equilibrium) and a temperature-driven transition. Is there something funny in the field-driven Ising case, just as it's a bit of a stretch (because it's a nonequilibrium situation) to talk about negative absolute temperatures in the post-field-reversal paramagnet case?

Anonymous said...

Interesting. Thanks for this discussion, I never got around to a thermo/stat mech class with discussion of this. Too much time spent on Maxwell relations and other mathematical tools, not enough physics...

Don Monroe said...

Thanks, Doug.

Another qualitative difference, perhaps more tangible than the latent heat, is that continuous transitions can't be supercooled, because the two phases are the same at the transition. Of course this is why fluctuations are important for continuous transitions. In contrast, a discontinuous transition like water condensation requires nucleation, which is another well known feature of familiar phase transitions.

David Sanders said...

Hi Doug,

The reference you give is for the Ising model with fixed magnetisation. Thus there is always the same number of up and down spins. Thinking of the up spins as particles, and the down spins as holes, this can be mapped onto a lattice-gas in the canonical ensemble.

Usually when thinking about the Ising model, though, we think of spins which can flip, so that the magnetisation is not fixed: this is the "non-conserved order parameter" case, whereas fixed magnetisation is "conserved order parameter" -- so-called since the magnetisation is the order parameter for the Ising model. Confusingly, this is called the canonical ensemble for the Ising model (working with the system in contact with a heat bath at fixed temperature), but thinking again of the equivalent lattice-gas model, it is now the grand-canonical ensemble, since particles can appear and disappear!

In the case where magnetisation is not conserved, the spins want to align with the external field. So if the external field is positive, then the thermodynamically-stable phase has positive magnetisation; if the field is negative, the stable phase has negative magnetisation. For a system of finite size, plotting the magnetisation per spin against field gives a smooth curve which tends to +1 when h tends to +infinity, and to -1 when h tends to -infinity. However, when the system size grows, this curve becomes gradually more abrupt near h=0, and when the system size reaches infinity -- the thermodynamic limit -- the curve becomes discontinuous, which is the sign of a first-order transition. This is true when the temperature is less than the critical temperature T_c of the model.

Crossing the h=0 line, there is thus a discontinuity (in the thermodynamic limit) in the magnetisation in the stable phase. Since the magnetisation is the first derivative of the free energy F with respect to the field, we can call this a first-order transition, since some first derivative of F has a discontinuity.

There are then two coexisting phases exactly when the field h=0. This coexistence of different phases is one of the hallmarks of a first-order transition. It is this coexistence which leads to metastability and hysteresis when thinking about the dynamics of the model with a finite-size system. In general, first-order transitions involve all of coexistence of phases, metastability and hysteresis, and a discontinuity in some thermodynamic variable.

This is my understanding of the situation, as explained for example in "Lectures on Phase Transitions and the Renormalization Group", by N. Goldenfeld. There is also a discussion in "Statistical Mechanics of Phase Transitions", by J. Yeomans, which is less technical. I learnt this stuff mainly by doing Monte Carlo simulations and trying to understand what they were showing, for which I recommend "Monte Carlo Methods in Statistical Physics", by M. Newman and G. Barkema. Another very good recent book is "Statistical Mechanics: Entropy, Order Parameters, and Complexity", by J. Sethna, available online at
http://www.physics.cornell.edu/sethna/StatMech/EntropyOrderParametersComplexity.pdf
In general, however, this seems to be a subject that is often not well explained.

Anonymous said...

Actually I have found that it is easier to explain second order phase transitions to an audience. The idea that there is an order parameter ("little magnets that make up the big magnet") and it starts coming up ("aligning") at some temperature is something most people seem to get. I've found it much tougher to explain latent heat ("you heat it but it does not get hotter?").

Dale ritter said...

A few points about phase transitions from the RQT (Relative Quantum Topological) viewpoint may interest you. When a liquid is confined the typical heat required to elevate it's temperature by one kelvin is less than for an open vessel. The common isobaric monatomic heat capacity of 5/2 kT J becomes only 3/2 kT J when the gas is confined.
A phase transition is due to work applied, but more modern analysis can look at that event in quantum terms to give picoyoctometric definition to the atomic, or molecular model. Work of infrared or microwaves may be absorbed into an atom by conversion of electrical and magnetic force into energy waveparticles of the psi's internal heat capacity energy fields. The more detailed model accounts for energy, momentum, and force to explain the RQT physics which governs the phase changes' nano and picotechnical topological events. That concept leads to more advanced applications with greater safety and efficiency of work.
RQT analysis develops the pressure events by 3D imaging of spacons, which drive pressure as a force, and assigns picoyocto topology to all of the force and energy fields of a gas or liquid. This is achieved by solving the Schrodinger equation for one atom as a series expansion differential of nuclear transform of mass to forcons with valid joule values, by [ e = m(c^2) ] physics. That GT integral psi function succeeds by combination of the relativistic Lorenz-Einstein transform functions with the quantized wave functions for frequency and wavelength. When the series equation includes quantum symmetry numbers along the succession of nucleoplastic transfrom rates the result is a 3D animated video model image of the atom.
Views of the h-bar magnetic energy waveparticle of ~175 picoyoctometers are on display at http://www.symmecon.com.
(C) 2009, Dale B. Ritter, B.A.

Macksb said...

Let me suggest a new theory to explain phase transitions and phases of matter.

Art Winfree did some brilliant work in the 1960's on coupled oscillators. He showed that limit cycle oscillators have a tendency to couple, and when they do, they couple in certain exact and precise ways that can be predicted mathematically. The simplest example is two oscillator systems, which couple either synchronously (kangaroo legs) or anti-synchronously (legs of a human walking). Three and four oscillator systems offer more options, but always the combinations are exact and precise.
Winfree applied his theory mainly to the biological world; he never applied it to physics. He viewed himself as a bio-mathematician.

In my judgment, Winfree's theory of coupled oscillators can be applied with even greater success in physics. It can serve as the organizational law that explains phases of matter and their transitions.

As we all know, temperature and pressure are two highly relevant parameters in phases of matter and their transitions. Similarly, Winfree's theory involves the frequency of oscillations, and the proximity of oscillating units. The primary variables in Winfree's theory are thus directly analogous to the primary variables in phase transitions.

As far as I know, no one has ever proposed a general law or mechanism to explain why there are phases of matter and why there are phase transitions. I believe that Winfree's theory may with minor modifications provide an overarching explanation.

For a good summary of Winfree's theory, see the December 1993 Scientific American, where there is an article by Steven Strogatz. Prof. Strogatz, now of Cornell, worked for Winfree as a post doc.

venus said...

Interesting. Thanks for this discussion, I never got around to a thermo/stat mech class with discussion of this.
--
Venus
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Macksb said...

A recent announcement (April 22, 2010)by Riken, the Japanese research institute, provides experimental confirmation of my coupled oscillator theory, which I describe two posts above.

The Riken announcement concerns superconductivity in a simple pnictide. They display a picture showing that Cooper pairs adopted a characteristic S + - wave structure that is unique to a material with two sets of electrons.

My interpretation of the evidence is that there are two sets of waves that intersect perfectly. Wave set A emanates from point A in the lattice. Wave set B emanates from point B in the lattice. The A and B sets merge together in a perfectly synchronized seiche. (It's a nautical term, mainly.) The up dots in the picture are the wave crests, and the down dots in the picture are wave troughs. They are aligned exactly. The waves are electromagnetic--they are the oscillations that synchronize in this case.

My interpretation is consistent with other experiments showing that the angles in the pnictide structures must be exactly right to create superconductivity. Slight variations in structural angles destroy the superconductivity.

This experimental observation is consistent with my coupled oscillator theory--which is really Art Winfree's coupled oscillator theory (see also Huygens, Kuramoto and Steven Strogatz, author of Sync) redirected from biology to physics. All limit cycle oscillators have a tendency to couple, and when they do, they couple in certain exact, precise patterns.

As applied to superconductivity, the law of coupled oscillators appears to explain all three forms of superconductivity: BCS, pnictide, and the cuprates. More on that in my next post.

Macksb said...

Here's my theory as to the common denominator in the three forms of superconductivity: BCS, pnictide and cuprate.

In each case, oscillators couple and synchronize, providing absolutely perfect order within the material insofar as is required for the passage of an electric current. In each of the three cases, the oscillators that couple are different, but in all cases oscillations synchronize.

In BCS, the Cooper pairs are themselves coupled oscillators. Electrons are directly coupled to each other by orbit (antisynchronous) and spin (antisynchronous).

In pnictides, we see from the Riken evidence that the oscillations in question are electromagnetic waves, emerging from two different points. These oscillating waves synchronize at the point where the waves cross each other. It's a perfectly organized seiche of electromagnetic waves, in time and place. Unlike BCS, while this is a necessary condition for superconductivity, it is not necessarily sufficient. Again, the synchrony in question is the type that emerges in two oscillator systems--exactly antisynchronous. (This is like a human walking, as contrasted with a kangaroo. Our legs are exactly antisynchronous when we walk, while theirs are synchronous.)

Now for the cuprates. Assuming the d wave symmetry observations are correct--and I believe they are--the oscillations that synchronize do so in Pair A and in Pair B, and then a Pair A and a Pair B synchronize, joining in the middle so to speak. That produces the four lobes. I am not certain about the oscillations that couple, but I believe they involve certain synchronized lattice vibrations, much like the four legs of a horse moving at a trot, which organize pairs of excitons. Whether this is correct or not, I am confident that the D wave symmetry involves antisynchronous pairing in Pair A; antisynchronous pairing in Pair B; and antisynchronous pairing in the AB pair.

So: all forms of superconductivity can be explained on the basis of Art Winfree's law of coupled oscillators. Some experimental evidence provides direct confirmation, at least as to the pnictides and as to BCS. So Ockham's Razor suggests that this is highly likely to be the fundamental explanation.