Hi everyone,
I'm doing some research in applied math (modelling with large datasets), and I've been drawn into the realm of statistical thermodynamics. Unfortunately, I am not a physicist. (I somehow managed to get a bachelor's and masters in math applied math without taking a physics class.) This means that I'm unclear on some definitions that physicists probably consider very basic. I'm hoping that the wise people at xkcd can help me out, so here goes:
If I have a thermodynamic system in equilibrium with a temperature bath, I can define pretty much everything interesting through the canonical partition function Z(T). In particular, I can define the Helmholtz free energy as A(T) = kT ln(Z(T)). My questions are about the terminology associated with criticality and phase transitions.
First, do these two terms refer to the same thing, or am I missing some nuance? That is, is a critical temperature of a system the same as the temperature at which a phase transition occurs? I usually see the term phase transition used when dealing with grand canonical ensembles. I.e., where other quantities like volume, pressure or N are allowed to vary. Is it still kosher to use the term for a canonical ensemble?
Second, is the critical point (or phase transition, if they're different) defined as a temperature where the free energy is nonanalytic?
Thanks in advance for your help. If this proves useful, I'll probably come back to the thread with more questions.
Criticality of Thermodynamic Systems
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 eta oin shrdlu
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Re: Criticality of Thermodynamic Systems
Rather than type up a big wall of text which may all be obvious to you, let me start with what may be a very simple question.
The terminology of phase transitions and critical behavior is generalized from thermodynamics. Are you familiar with PT phase diagrams? There's one as the first diagram ("Figure 11.22") here.
(If you don't know what I'm talking about, it's just a diagram of the equilibrium phase versus pressure and temperature. Phase transitions occur at the white boundary lines (between solid and liquid, solid and gas, and liquid and gas); the critical point is the point labeled "B", at the endpoint of the liquidgas boundary line. Looking at that diagram should make it clear that phase transitions and the critical point are different.)
The terminology of phase transitions and critical behavior is generalized from thermodynamics. Are you familiar with PT phase diagrams? There's one as the first diagram ("Figure 11.22") here.
(If you don't know what I'm talking about, it's just a diagram of the equilibrium phase versus pressure and temperature. Phase transitions occur at the white boundary lines (between solid and liquid, solid and gas, and liquid and gas); the critical point is the point labeled "B", at the endpoint of the liquidgas boundary line. Looking at that diagram should make it clear that phase transitions and the critical point are different.)

 Posts: 154
 Joined: Sun Oct 22, 2006 4:29 am UTC
Re: Criticality of Thermodynamic Systems
First, thanks for your reply. I am fairly familiar with PT phase diagrams, but I am not entirely clear how to apply them to my problem. For reference, I am studying systems discussed in this paper: http://link.springer.com/article/10.100 ... 01102294
If that's behind a paywall for you, I'll summarize briefly. If you have a probability distribution over some discrete state space, you can always formally write it as a Boltzmann distribution. That is,
p(x;T) = 1/Z(T)*e^{E(x)/kT},
where
E(x) = ln(p(x)).
If kT=1, you get your original distribution. If T is allowed to vary, you get a family of distributions. Several papers, including the one I linked to, argue that when these models are used for biological systems, they appear "critical". Unfortunately, there doesn't seem to be a particularly unified definition of the term critical. I have seen these phase diagrams before, but my issue is that these models don't have an intrinsic volume parameter (and therefore no intrinsic pressure). As far as I can tell, the closest analogue to the PT phase plane in these systems would be a "T phase line", so to speak. (I think the analogue to volume would be an arbitrary extensive parameter in p, and pressure would arise from a Legendre transform, but none of the models I've looked at include anything like this.)
Given this context (and please let me know if I'm not explaining or understanding something well), maybe I should clarify my question. Is there a physical analogue to PT phase diagrams when my model includes temperature, but not pressure? If there is, then would such a phase transition correspond to a point where the free energy is not analytic? If there is not, does "a point where the free energy is not analytic" sound like a reasonable extension of the term "phase transition" or "critical behavior"?
Again, thank you for taking the time to help out.
If that's behind a paywall for you, I'll summarize briefly. If you have a probability distribution over some discrete state space, you can always formally write it as a Boltzmann distribution. That is,
p(x;T) = 1/Z(T)*e^{E(x)/kT},
where
E(x) = ln(p(x)).
If kT=1, you get your original distribution. If T is allowed to vary, you get a family of distributions. Several papers, including the one I linked to, argue that when these models are used for biological systems, they appear "critical". Unfortunately, there doesn't seem to be a particularly unified definition of the term critical. I have seen these phase diagrams before, but my issue is that these models don't have an intrinsic volume parameter (and therefore no intrinsic pressure). As far as I can tell, the closest analogue to the PT phase plane in these systems would be a "T phase line", so to speak. (I think the analogue to volume would be an arbitrary extensive parameter in p, and pressure would arise from a Legendre transform, but none of the models I've looked at include anything like this.)
Given this context (and please let me know if I'm not explaining or understanding something well), maybe I should clarify my question. Is there a physical analogue to PT phase diagrams when my model includes temperature, but not pressure? If there is, then would such a phase transition correspond to a point where the free energy is not analytic? If there is not, does "a point where the free energy is not analytic" sound like a reasonable extension of the term "phase transition" or "critical behavior"?
Again, thank you for taking the time to help out.
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