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Chapter 7 Phase Transitions

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Chapter 7 Phase Transitions Chapter 7 Phase transitions In the last few chapters we have been preoccupied with various methods of calculating par- tition functions, both for ideal and for interacting systems. In principle, once we know a partition function we can calculate any thermodynamic properties that we might be inter- ested in. As we discussed a particular perturbative approach to calculating the partition function for a dilute gas { and then had to patch it up via Maxwell's construction in order to describe both a gas and a liquid state { you may be wondering: if it is even clear that a single partition function can actually describe multiple phases? In fact, perhaps you have an even deeper concern. Phase transitions are characterized by discontinuities in the free energy or its derivatives, and the free energy can be written as Z = e−βF = Tr e−βH ; where I'm using the trace notation to indicated a schematic sum over all of the degrees of freedom in the Hamiltonian, H. But most H that we study are non-singular functions of their degrees of freedom, so the partition function is just a sum of a large number of terms, each of which is exp (some analytic function). How could such a sum be non-analytic, and can a single partition function even describe a phase transition (let alone multiple phases)? A deep understanding of the answer to these questions, together with an understanding of how near critical points whole classes of disparate systems behave in the same way, has only been understood for, say, the last 50 years1. In this chapter we will discuss mean field models of phase transitions, critical points, Landau theory for phase transitions, the relationship between fluctuations and correlations, and the scaling hypothesis for behavior near critical points. This will lead us right up to the precipice of the renormalization group2; time permitting I will try to throw together a few bonus lectures on this. 7.1 Mean field condensation As a way of sliding from the last chapter to this one we'll return again to the van der Waals equation, but now with a somewhat different approach. In the section on interacting systems 1Perhaps making the subject the most \modern" of the graduate physics curriculum? Hmm... 2Or, since an anomalously large number of the professors I took graduate classes used British English, the \renormalisation group." 1 we were confronted with the problem of computing, say, the canonical partition function ! ! 1 Z Y X p2 X Z(V; T; N) = d3pd3q exp −β i exp −β φ(r ) ; (7.1) N!h3N 2m ij i i i<j and with a great deal of effort we constructed a perturbative expansion to do just that in the high-temperature / low-density regime. We then muddled along and argued we could work with the resulting equations even when the system was cooled below the Tc associated with the gas transitioning to a liquid. It felt dubious. Our goal is still to approximate the partition function, but let's consider instead a mean field calculation, in which we assume that the potential energy contribution of a particle is computed based on its interactions with an average { and uniform { background of particles. First, rather than specifying that we are working with a Lennard-Jones pairwise potential, let's just think of any potential which (1) has a \hard-core" component for r < σ, (2) is sufficiently short-ranged, and (3) has some attractive part of the potential, where the potential well has integrated area −u. In our approximation for the Lennard-Jones potential, for instance, we would have −u = −"Ω. Next, we first consider the actual (fluctuating) distribution of particle density, X n(r) = δ (r − ri) ; i in terms of which we could write the potential energy of our system as 1 Z U (fr g) = d3rd3r0 n(r)n(r0)φ(r − r0): (7.2) i 2 We now assume that the system has uniform density, and approximate n(r) ≈ n = N=V . Then the typical energy, U~, is n2 Z U~ = d3rd3r0φ(r − r0) 2 n2V Z = d3rφ(r) 2 −un2V = : (7.3) 2 We substitute this average energy into Eq. 7.1 for the canonical partition function, and we get 1 βuN 2 Z Y Z = exp d3r : (7.4) N!λ3N 2V i i We make one last approximation before we do the remaining integral over spatial coordinates, and that is a Ω=V expansion in how we treat the excluded volume effects. To first order in 2 (Ω=V ) we write Z Y 3 d ri = V (V − Ω) (V − 2Ω) ··· (V − (N − 1)Ω) i 1 = V N 1 − (Ω + 2Ω + ··· + (N − 1)Ω) + ··· V 1 N(N − 1) = V N 1 − Ω + ··· V 2 NΩN ≈ V − : (7.5) 2 Together with our mean-field estimate of the effect of the attractions, then, our partition function is N V − NΩ βuN 2 Z(V; T; N) = 2 exp ; (7.6) N!λ3N 2V from which we can compute the pressure in the canonical ensemble as @ log Z N u N 2 βP = = − β canonical @V NΩ 2 V 2 V − 2 Nk T u N 2 ) P = B − : (7.7) canonical NΩ 2 V V − 2 This is precisely the van der Waals equation again! But here, rather than derive it in the context of a perturbative expansion in the density and the temperature, we have derived it in the context of an approximation of uniform density. And certainly a uniform density approximation is the sort of thing we expect to be equally valid in both the gas and liquid states, hence why we expect that we ought to be able to describe multiple phases with the same Hamiltonian and the same partition function. 7.1.1 Maxwell construction, once again Let's pause to better understand how to connect this picture with the liquid-gas phase transition. Recall that one of the problems in the canonical picture is understanding what happens in the coexistence region. We've already seen several examples of how moving from the canonical to the grand canonical ensemble can make our life easier, so for practice (and variety), let's see an example of another Legendre transform and work in the Gibb's canonical ensemble, i.e., the isobaric ensemble, in which we imagine our system coupled to a piston which keeps the pressure controlled. Our partition function in this ensemble becomes Z 1 Z 1 Z(P; T; N) = dV e−βP V Z(V; T; N) = dV e (V ); (7.8) 0 0 where (V ) = −βP V + log Z(V; T; N). We now approximate this expression by the saddle point method we saw back in Chapter 2, and say there is some particular value of V which 3 Figure 7.1: Saddle-point picture for the condensation isotherm Left shows a T < Tc isotherm, right schematically sketches the function (V ) for different points along the isotherm. The critical pressure is when the heights of the two purple peaks are precisely equal. maximizes (V ), which we will call Vmax. Thus Z(P; T; N) ≈ e (Vmax): We can find this Vmax by extremizing : @ @Z = 0 ) 0 = −βP + = −β(P − P (V )): (7.9) @V @V canonical If T > Tc this is fine, but we know that for T < Tc sometimes there are three possible V to check for the canonical pressure. In this regime one V will correspond to a local minimum of , so there are two candidate Vmax, as sketched in Fig. 7.1. We would say Z ≈ e (Vgas) + e (Vliquid); and we again say that Z is dominated by exponential with the largest argument. Thus, as the temperature changes and the relative ordering of (Vgas) and (Vliquid) switch we can get a discontinuity in our system3. Notice, by the way, that by extensivity we can write exp [−βP V − βE + βT S] = exp (−βµN) ; so as we anticipated last chapter, choosing V to minimize at fixed N is like choosing the lowest value of µ. Additionally, what is the critical value of pressure at which the discontinuity we sketched above can be found? It must be when (Vgas) = (Vliquid) ) (Vliquid) − (Vgas) = 0: Writing the difference as an integral of a derivative: Z Vliquid d 0 = (Vliquid) − (Vgas) = dV Vgas dV Z Vliquid = β dV (Pcanonical − Pc); (7.10) Vgas In that last integral we recognize precisely the Maxwell equal-area construction for finding the isotherm. 3Note, again, that it is only a discontinuity in the assumption that N ! 1; otherwise everything is smooth 4 7.2 The law of corresponding states Having seen two different derivations of the van der Waals equation, let's write it in a simplified form, k T a P = B − ; (7.11) v − b v2 where v = V=N is just the volume per particle, and a and b are parameters that depend on the microscopic details of the systems under study (generically related to the characteristic size of the interacting particles, and how strong the attractive interactions are). We have seen that this equation goes from being monotonic to non-monotonic at some critical value of temperature, and that at Tc there is a point where the critical isotherm is flat (with both the first and second derivative vanishing. Where does is this critical point? We find it by4 writing @P k T 2a = 0 = − B + (7.12) @v (v − b)2 v3 @2P 2k T 6a = 0 = B − : (7.13) @v2 (v − b)3 v4 vc−b vc Dividing the first equation by the second equation gives us 2 = 3 ) vc = 3b, and we can then plug this critical volume per particle into the above expressions to find the critical point on the critical isotherm: 8a a v = 3b; k T = ;P = : (7.14) c B c 27b c 27b2 Suppose we wanted to know if the van der Waals equation was actually any good, and we convinced someone to do an experiment.
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