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Second-order or continuous transitions are usually identified with non-analyticies within the free energy (which is proportional to the logarithm of the sum of exponentials). Such singularities are only possible within infinite systems, id est., taking TD Limit (thermodynamic limit).

Real systems are finite. How do we usually explain such an apparent paradox? I'm aware of theories studying the finite size effects, especially the scaling laws for correlation functions. But still, I can't grasp a convenient solution of the problem. How is this problem usually approached?

Some useful bibliography:

http://books.google.es/books/about/The_Theory_of_Critical_Phenomena.html?id=lvZcGJI3X9cC&redir_esc=y http://philsci-archive.pitt.edu/8340/1/Phase_transitions_in_finite_systems.pdf

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  • $\begingroup$ Think of a function $f(L,g)$ which (as a function of $g$) is singular at some point $g=g_0$ in the limit $L\rightarrow \infty$. Then if you take $L$ to be some very large number (say $L=10^{23}$) you will see that the given function is already "almost singular" even though $L$ is not really infinite. $\endgroup$ – user10001 Jul 5 '13 at 16:51
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    $\begingroup$ I'm aware of that fact, but as I have read somewhere, it does not constitute a legitimate phase-transition. It seems that the very concept of continuous transitions is rather impossible: $\endgroup$ – cacosomoza Jul 5 '13 at 17:02
  • $\begingroup$ 2nd paper on bibliography: "For the definition of a phase transition is an all-or- nothing singularity in the free energy, which in no clear sense can be “approached” as N becomes very large. And it is important to realise that the theories really do require a genuine singularity; vague appeals to “steepness” or an “extreme gradient” will not do. For we can find finite systems with extreme gradients in the relevant thermodynamic variables which do not become a singularity as the TD limit is taken: these do not represent phase transitions." $\endgroup$ – cacosomoza Jul 5 '13 at 17:05
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I do not think Mainwood makes any argument against what he calls the "theoreticians case", much less a compelling one. The "theoretician's case" is that phase transitions do not exist in finite size systems but only as features which become infinitely sharp in the infinite size limit (also user10001's comment). In fact Mainwood briefly dismisses the case and then basically turns around and makes it again himself. As far as I can tell the appropriate combination the arguments in (Sec 3.3) and (Sec 3.4) is correct:

  1. What we experience as phase transition do not require singularities in the partition function
  2. Thermodynamic singularities cannot occur in finite $N$ systems

Point one is obvious - if we do not measure finely we do not distinguish between sharp features and singularities, so the apparent experience of singular features does not require singularities in the statistical description. In fact if we do measure finely we find that the singularities smooth out so singularities are not only unecessary but experimentally excluded. Point two is a well known mathematical fact.

The only remaining question is what is the connection between the mathematical singularities in the infinite limit and the sharp behavior in the finite $N$ limit. This is simply that if we want to calculate some thermodynamic observable $O$, which is a function of $T$ in some system with which happens to have $N$ particles. Statistical mechanics gives us a prescription to do this - it produces an umbiguously defined function $\bar{O}_N(T)$, which agrees with experiment. As a mathematical fact we can perform the following manipulation:

$$\bar{O}_N(T) = \bar{O}_\infty(T) - E$$

where $O_\infty(T)$ is the same calculation in infinite size limit and $E$ is the error. We have mathematical control over the error term $E$ and can bound it to be extremely small when $N$ is large. Note there is no limit being taken or anything. Its just a controlled approximation like saying $3.1 < \pi <3.2$.

Its totally possible for a thing to have a sharp feature without a decomposition like the above working. Whether or not you wanted to call that a phase transition would be up to you but I don't see any fundamental philosophical issue

Mainwood is upset about this, because he wants to be able to point at something and say "aha! the phase transition". He wants the phase transition to have nearly fundamental ontological status. But it doesn't. So what?

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Whether finite systems may feature a mathematical singularity or not is an old controversial issue. In fact, at the Van der Waals memorial meeting in 1937, the audience could not agree on the question, whether partition function for a finite system could or could not explain a sharp phase transition. So the chairman of the session, Kramers, put it to a vote!

At quantum level, for sure, a statistical partition function for a finite system is an analytic function of (typically) temperature and volume. This implies, e.g. the specific heat at constant volume, Cv, is finite, never diverges. However, if we consider alternatively the specific heat at constant `pressure', this quantity, Cp, may become singular, as shown first here and then here.

According to them, the ideal Bose gas confined in a cubic box, (the standard textbook quantum system), may undergo a liquid-gas-type phase ``transition" under constant pressure condition, even though it consists of a finite number of particles.

The punchline is to choose an alternative section condition on the domain of the analytic function (temperature-volume plane), such as the constant pressure condition, and to realize a singularity.

I would invite you to watch a short YouTube Video and to think of the gedenken experiment therein.

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