# First order phase transition in a classical system

I've never liked discontinuous quantities in classical physics, so I find the discontinuity in heat capacity weird.

My question is, do first order phase transitions ever really exist? Or are our discontinuous experimental $C_v$ vs. $T$ graphs just really steep curves? Discontinuous for all practical and theoretical purposes.

I understand that the theoretical model of a pure component system has a first order phase transition - and therefore a discontinuity in $C_p$. But its theoretically impossible to have a pure component (the chemical potential for any impurity in $A$ is infinite for a pure $A$). So effectively we only ever have multicomponent systems.

In classical thermodynamics (the best verified physical theory we have), phase transitions are true discontinuities, obtained from a continuous thermodynamic potential by taking derivatives (or even second derivatives for heat capacities) at points where these do not exist.

Thus there is nothing weird about them. It is no more weird than that the derivative of the absolute value function has a jump at zero.

First order phase transitions also appear in impure materials, though at slightly different volume, pressure and temperature than in the pure case.

Note that the discontinuities are inherited from statistical mechanics (Lee-Yang theorem).

 Note that thermodynamics applies to matter regarded as a continuum. Once one looks at atoms or molecules inside a system one has entered in the realm of statistical mechanics, a game with different rules. In general, as one looks at any physical system in more detail, the previously useful description starts to become approximate if not invalid. In particular, questions about continuity or differentiability lose their meaning (or regain it in a very different way).

[Edit2] Traditional thermodynamics in its usual axiomatic form (e.g., Callen) is valid for finitely extended matter. From a microscopic point of view, thermodynamics is usually justified in terms of statistical mechanics, based on the thermodynamic limit of infinite volume. But this is necessary only if the derivation is based on the microcanonical or canonical ensemble. In the grand canonical ensemble, thermodynamics follows without a thermodynamic limit (see Chapter 9 of my book http://lanl.arxiv.org/abs/0810.1019). In the original derivation of the Lee-Yang theorem (for ferromagentism), there is no phase transition without a thermodynamic limit, as the number of particles in an Ising system is bounded. However, real fluids treated from the most basic level, relativistic quantum field theory, have no upper bound on the number of particles, so that the original Lee-Yang statement no longer applies.

• Very interesting Arnold. My question, I guess, is more about the seam between theory and reality. I can imagine a theoretical system having a discontinuity at it's nth derivative, just like I can imagine a set of (classical) atoms standing perfectly still and achieving 0K. What happens when I add features to the model? Let's say I have a finite set of (classical) atoms at 0K in a rigid box larger than their crystal volume. I start to heat the atoms. Will the jump in Cv be perfectly discontinuous? Won't vacancies (holes) enter the solid and act like a "second component"? – Lenzuola Nov 19 '12 at 16:59
• This is wrong. First-order phase transitions, in the sense of being discontinuous curves, only exist if you take the limit of a system of infinite size. Real systems that we measure are not of infinite size, they are merely very large indeed, and consequently any $C_V$ vs $T$ curve for a real system is not discontinuous but just very steep, as the OP suggests. – Nathaniel Nov 26 '12 at 11:40
• Yes, but traditional axiomatic thermodynamics is an idealisation made nowadays only for practical and pedagogical purposes. The OP was asking whether the curves are "really" discontinuous, or only for all practical purposes. The answer is the latter, because axiomatic thermodynamics only holds for practical purposes. All real systems obey statistical mechanics instead. – Nathaniel Nov 26 '12 at 14:42
• @Nathaniel: Even continuity is an idealization. Without idealization, there is ''really'' only a physics we do not understand including quantum gravity), and all our current physics is only for all practical purposes. The standard model holds only for practical purposes, and so does statistical mechanics. Questions about what is ''really'' the case become therefore unanswerable, unless one settles on a particular basis. The notion of a phase trasition is well-defined only in thermodynamics, hence this is the correct basis. I had already mentioned in my answer what happens on lower levels. – Arnold Neumaier Nov 26 '12 at 16:31
• Maybe you're right - I shouldn't try to guess the OP's intentions. It still seems odd to me to say that the discontinuous curves are "real" when we know where that idealisation breaks down, though. (By the by, it is possible to calculate heat capacity for a statistical ensemble, so mentioning it doesn't automatically imply that we're talking about the infinite-$k_B$ limit.) – Nathaniel Nov 27 '12 at 2:16

In a sense you are right. The heat capacity only becomes discontinuous for a system of infinite extent. For all others it is continuous.

But that's a theoretical concern only. At the theoretical point of discontinuity the slope of the heat capacity is infinite. Pick any finite value for the slope, no matter how large and I can find a finite system where the slope is larger than that.

As for "impurities", there is no problem handling multicomponent systems. Chemists do it all the time. The results are only slightly more complex than for single component systems and my remarks about the phase transition above are still true.