January 6, 2015 I. INTRODUCTION FIRST ORDER PHASE TRANSITIONS. A. Basic phenomena

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1 1 January 6, 2015 I. INTRODUCTION If I need to summarize the focus of this coming quarter, I d say it is the topic of phase transitions. Phase transitions, however, are a manifestation of interactions between the particles in the systems we would like to analyze. The previous quarter concentrated on the statistical mechanics of non-interacting quantum particles. It is true, that more often than not, we can get a very good idea of phenomena without considering interactions. The ideal gas law is a great example. But as physicists we are interested in the special rather than the mundane. And in this quarter we will consider the more hard-to-understand phenomena that emerges when we consider many interacting particles. The most dramatic effects, and not uncommon in our nature are due to first order phase transitions. We will start with a discussion of those. We will continue to discuss the more rare, and more beutiful phase transition of higher orders. These are the symmetry breaking transitions which describe magnets, superconductors, and crystaline order. They also describe the universe - with its electro-weak symmetry breaking and the Anderson-Higgs phenomenon. Higher order phase transitions are also named critical phenomena. They are essentially always caused by symmetry breaking. The system chooses a direction although its hamiltonian is insensitive to direction. How could that be? In fact, one can mathematically prove it can not be. But mathematical proofs are not necessarily what describes nature. And nature becomes very beutiful at the points of syummetry breaking. It becomes scale invariant, and therefore can be analyzed with the renormalization group. Also, critical phenomena is universal. It is enough to understand it in one system, and obtain a description that transcends the specifics of the system, and applies very broadly to a large class of physical setups. II. FIRST ORDER PHASE TRANSITIONS The really dramatic phenomena associated with heat and thermodynamics are phase trasitions. Essentially all of our weather related experiences are associated with the fact that the surface temperature of the earth is comparable to the freezing and boiling temperatures of water. Think about it - in places other than california one can experience: fog, drizzle, rain, frozen rain sleet), hail, snow, and the most annoying form of water - slush. In boston, you can have them all on the same day while the sun is shining. On a more small scale - if you forget your coffee pot it may actually explode if the water evaporates too fast. Sweat - that s another thing that we can t really do without, and this is not a statement for you to not use deodorant - but by allowing water from our body to evaporate, we cool ourselves. A. Basic phenomena Let s focus on a very familiar phase transition - liquid gas. During this discussion it is useful to have water in mind. For a given pressure, for temperatures below the critical temperature a substance may be a dense fluid - liquid, while above the critical temperature the substance becomes a rather thin dilute fluid - gas. At the critical temperature, holding the pressure constant, as you try to heat the liquid, you expand it continuously, you hit the transition, and then heating doesn t seem to change the temperature, it just changes the volume: part of the liquid becomes a gas with a completely different density, and eventually the entire substance is gas, and the temperature continues to rise. The heat that went into the expansion is latent heat. If you go up to the mountains, wher the pressure is lower, and you try to cook, cooking pasta takes much longer. When we cook, we essentially boil water, and then wait for the heat from the water to cook our food. But cooking at high altitudes is longer since the boiling temperature of water is lower - DRAW ANOTHER POINT) - in fact, there is a critical line of coexistence the stretches for a while in the p-t graph. To consolidate your impressions with water - water can also freeze - ANOTHER LINE ) - and it also has a triple point, where liquid gas and solid can coexist. below the triple point in pressure or temperature, the substance can only be a gas or a solid, and the transition between the two, along this line, is called - sublimation. For water this is at p = 0.006A, T = 0.01C.

2 2 FIG. 1: phase diagram for water B. an-der-waals equation of state concentrating, as promised, on the liquid-gas phase transition. Let s build a model. So far we worked some with ideal gasses. They didn t have anything like it. But we did have one example of a phase transition - yeah - the Bose gas at high densities. We saw that the bosonic statistics leads to a weak attraction at low densities. This - as the density increases - beocmes more severe, and has to do with the condensation. But let s take the first piece - attraction. What would it do to us? How do we work it in? Well, pretty easy. Remember how we wroked in interactions to a noninteracting magnetic material? We added interactions to the free energy. Let s do that here. The free energy of a gas is: F T,, N) = T N log N If we try to get the pressure, we get: ) ) 2πmT ) d/2 + 1 = T N h d ) F = NT log ) N + log T d/2 + log 2πm)d /2 h d + 1 o.k.but what about interactions? What would interactions be like? If we take the dilute approximation, then the simplest interaction is: ) N 2 U int = a 3) the negative is for attraction - you don t expect anything special to appear with repulsion, do you? The free energy now becomes: F = T N log ) ) 2 N + log T d/2 + log 2πm)d /2 N h d + 1 a 4) and pressure gets an extra contribution: ) F = NT 1) 2) an2 2 5) The attraction reduces the pressure relative to an interacting gas. There is another thing we want to take into account - the interaction may be attractive, but only to a certain extent. Once atoms in a gas come very close together, they start repeling - otherwise all of matter would collapse into a single point. We can capture this by assuming that atoms see a hard wall once they heat other atoms. This is as though an atom has instead of volume to roam in, N 1)b space. This reduces the entropy of the gas. Where should we put it? thats right, in the log: log ) N log N b 6)

3 you can see that this is where we want it and not the interactions since the spatial entropy is given by the log term see by differentiating with respect to T), and not by the interaction, which doesn t contribute to entropy at all. Also, the attractive interactions should only depend on the distance between particles. Putting all this together, we calculate the pressure: ) F = T v b a 1 v 2. 7) This is the an der Waals equation, also written as: p + a v 2 ) v b) = T 8) As it turns out, this is an extremely useful equation for describing realistic gasses. It is essentially the first order virial correction to ideal gasses, plus finite volume. It is reproducing something which I m sure you heard about - the Lenard Jones potential of interactions between atoms: [ σ ) 12 σ ) ] 6 ur) = 4ɛ. 9) r r At low r, short distances, the 1/r 12 gives essentially a hard wall repulsion. But at longer distances, the 1/r 6 dominates, and we get a weak attraction. The parameters σ and ɛ determine the parameters a and b of dw equation. This also reproduces the liquid gas phase transition. Let s plot p vs. v. It is easy to see from the first version. At high T the T/v b) dominates, and we get an isothermic p-v curve that looks like this SEE FIGURE) the 1/v 2 gives a small correction, but nothing much. As temperature decreases, in regions where a v T 10) the 1/v 2 may dominate the behavior, and create this kink in the p-v diagram. As v decreases towards b, the pressure would increase again due to the short range repulsion. This kink, though, gets bigger and bigger as T decreases. SEE FIGURE) The outer limit on the kink is set by T, but the lower limit is set mostly buy b. This should look strange to you - pressure dropping as we make something more dense? Rings any bells? yeah! This is violating the stability rule that we derived from the Gibbs potential - 2 G 2 = p This must lead us to one conclusion - the gas can not be in this range of parameters. > 0. 11) 3 C. Liquid-gas phase boundary This forbidden range is exactly where the phase transition takes place. Let s think for a bit. In the forbidden range the gas is unstable. this means it can go to safety either to the right - low density - gas phase - or to the left - high density - liquid phase. Let s go back to the piston narative. Only make it in pressure LOOK AT THE P-T DIAGRAM). We move at constant T, but changing pressure. we start with a luquid under high pressure, we lower the pressure steadily at a constant T, the liquid expands, until at some point, when we hit the coexistence line, as we let the the pressure drop, the pressure freezes, and the gas evaporates, while taking heat out of the environment. v changes from the value here DENOTE ON THE P DIAGRAM) to here. let s call these points v 1 and v 2. now, the gas emerges on the other side, and the pressure can drop again, this time, with the isotherm of the gas, rather than the liquid. Looks okay, so where does it happen? For that we need to really understand the thermodynamics of this process. Thermodynamics - we have pressure, we have temperature, we have number. What is the right thermodynamic potential? Yeah - the Gibbs! Also - remember how we introduce the Gibbs? We always start with the free energy, and we do a legendre transformation: GN, p, T ; ) = F + p 12)

4 now we minimize with respect to. Let s look at the dw free energy with this regards, and let s look at the Gibbs per particles. BTW - remember that the giibs per particle is just the chemical potential. g = pv T lnv b) a 1 + const 13) v Lets plot here on the side the function f = F/N DRAW F) The log diverges at b, but saturates pretty quickly. Then the 1/v kicks in, and if T is low enough, it gives us another dive. Add to that the pv of the Gibbs, and you get this: a double well configuration. DRAW THE G GRAPH). Thermodynamic equilibria we used to find by differentiating the G with respect to, hoping to find a minimum. But here - hell, this way we get three solutions for the volume, one of which is a maximum - definitely unstable. That s exactly what we see with the dw equation, if we just look for extremum of g, we find three. These correspond to the three possible volumes that the DW equation has in teh p-v diagram. But now we can look: Generically, between the two wells, the extremum is a maximum of G, which tehrefore should not be counted towards thermal equilibrium. This is where the compressibility κ = 1 ) 14) p is negative by the naive calculation. Now we can see exactly what unstable means - the system, given the option, won t sit here in the top of the potential, it ll fall to one of the two minimas. DRAW THESE ON THE DW P DIAGRAM) here or here. But another thing - to be completely stable, the system must be in an absolute minimum. When the pressure is high, the density is high, the specific volume is low, we are in the gas phase. And indeed, the F will get translated to a G that looks like this:draw G) The low v minimum is definitely lower.for low p, F is the same, but G - completely different. The minimum that is lower now, is in the far side - hgih v. Indeed- a gas. There is a special point in between, in which the two minima are at the same gibbs level. That is the critical point! In fact - this is straight forward - we have a pahse transition at the point that the Gibbs per particles for the gas and liquid are equals - ie - where the chemical potential of the gas and liquid are equal. T 4 In equations we also have: D. Maxwell construction fv 1, T ) fv 2, T ) = pv 2 v 1 ) 15) This is the condition for the critical point in volumes. This has a very simple geometric interpretation, called the Maxwell construction. We can write the above is: F 1, T, N) F 2, T, N) = 2 1 ) v F 2 = p dw )d = p 2 1 ) 16) v 1 This requires an explanation. We follow an isotherm, and we need to know when the gibbs is equal here, and on the other side of the volume divide. It happens when the area under the dw curve, is the same as the area under the constant-p curver. Or, alternatively, when the between the dw and the constant-p cancel exactly. This is the Maxwell construction. E. Clausius-Clayperon Equation This is great - we can now understand how a gas - liquid phase transition comes about. An interaction gives an instability, the Gibbs develops two minimas, rather than one, and at any temperature there is a pressure such that the two are with the same Gibbs, the same chemical potential. At that point there is a phase transition between the two phases. But at which pressure does it happen for each temperature? For this we have the maxwell construction, or, more directly, the phase equilibrium condition: Gp, N, T ; 1 ) p, N, T ; 2 ) = 0 17)

5 1, 2 here are dummy variables, only denoting which one of the two minimas we want to consider. Otherwise, it is a function of p and T which is double valued at the phase transition. Now, we want to knwo how p c and T c at the coexistence line depend on eachother. So we take a differential: ) Gp, N, T ; 1 ) p NT ) Gp, N, T ; 2 ) p NT ) + dt Gp, ) N, T ; 1 ) T Np ) Gp, N, T ; 2 ) T but phase-transitions or no phase-transitions, we know what these are. the natural derivatives of G, therefore: Np ) = ) 1 2 ) = dt S 1 S 2 ) 19) where S 1 and S 2 are the entropies in the two phases. Since they are at completely different volumes, no wonder that they are different - just look at F, the volume is different between the two phases, even though the temperature and the number are the same. The difference in the entropy is exactly the latent heat over T : and putting things together, we get the C-C equation: S 1 S 2 ) = L 12 T dt = L 12 T 1 2 ) 20) 21) 1=G, 2=L. This applies not only for gas and liquid, but for any First order phase transition. And here is a definition for first order phase transitions: A TD process in which a first derivative of a TD potential such as gibbs or helmholtz free-energy) changes discontinuously - has a jump. By extension - second order - a second derivative of a potential has a jump. etc. An example for the use of the C-C equation is the water-ice phase transition at atmospheric pressure. We know that we need to give ice heat to make it melt. But ice also is less dense than water. So: and so: L water ice > 0 22) water ice < 0 23) dt < 0 24) As is indeed the case. Higher pressure - lower freezing temperature. This implies that by increasing pressure on ice, you can melt it.