Thermodynamics neglects fluctuations and deals with average macroscopic quantities (which is also important for the extensity of extensive thermodynamics coordinates e.g., the internal energy ${\rm U}$, entropy ${\rm S}$ etc). On the other hand, the fluctuations cannot be neglected at the critical point. How far can we trust thermodynamic results at the critical point?

How do we understand whether thermodynamics fails or remains valid at the critical point?

  • $\begingroup$ In general, aren't all (most?) derivatives of thermodynamic quantities singular at critical points? Doesn't that suggest they aren't usable anymore? I know that's not a statistical mechanics argument, and might not even be correct in general. $\endgroup$ – tpg2114 May 13 '17 at 18:06
  • $\begingroup$ @tpg2114 For 2nd order phase transition, 2nd derivative of free energy is discontinuous. Higher derivatives diverge. But don't they also diverge when you calculate from statistical mechanics? Which thermodynamic prediction(s) do not match with experiments at the critical point? $\endgroup$ – SRS May 13 '17 at 18:17
  • $\begingroup$ My experience is with gases, and I know at critical points between vapor and supercritical liquid, things like heat capacities are discontinuous. I'm not sure about other systems or states though. I've also never looked at it from a statistical mechanics perspective, so I don't know. It's an interesting question. $\endgroup$ – tpg2114 May 13 '17 at 18:38
  • $\begingroup$ @tpg2114 Recently I came to know that at the critical point specific heats actually diverge making Ehrenfest classification less useful. $\endgroup$ – SRS Mar 25 '18 at 15:01

How far can we trust thermodynamic results at the critical point?

If the system is in equilibrium one can fully trust thermodynamics, even at the critical point. The derivation of the main thermodynamic formulas (e.g., in Chapter 9 of my online book) does not refer to the critical point, hence are universally valid. The singularities at the critical point appear only in the response functions (i.e., the behavior under systematic small changes), which poses no conceptual problem. The theoretically predicted asymptotic power laws at the critical point have been quantitatively verified by experiment.

But as one approaches the critical point, measurements get more and more difficult and hence more inaccurate, due to the increasingly long range fluctuations. On the other hand, the thermodynamic models used in practice for real substances usually have an inadequate (often mean field) analytic form to capture the correct singularity structure and hence are inaccurate, too.

  • $\begingroup$ Though I had accepted the question, I have one more query. Does the extensivity of the extensive variables spoiled at the critical point? @ArnoldNeummaier $\endgroup$ – SRS Mar 25 '18 at 14:59
  • $\begingroup$ @srs: no. All resulrs of equilibrium thermodynamics are valid at the critical point. But the extensive variables become infinitely sensitive to changes in the intensive variables since the local dependence is a power law instead of being linear. $\endgroup$ – Arnold Neumaier Mar 26 '18 at 6:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.