I have a system with a number of measurables (in time). Some measurables are discrete some are continuous (within the measurement accuracy). How can I determine whether my system experiences criticality or not?

I am looking for many different ways to (dis)proof criticality.

See here for critical phenomena http://en.wikipedia.org/wiki/Critical_phenomena.

  • 4
    $\begingroup$ Can you tell us a little more about your research? Linking to single wikipedia pages is a bit on the light side. Also, do you have some specific qualm about criticality that leads you to believe it is fundamentally flawed, in spite of decades of reasonable application? This would make for a better question and discussion than "I [want to] (dis)prove criticality". $\endgroup$
    – Jen
    Aug 2, 2011 at 15:48
  • 1
    $\begingroup$ There are various ways of seeing that system might be critical (absence of scale, singularities with certain behavior in correlators or macroscopic quantities, etc.). In general it's very hard to determine whether system is critical rigorously. So there's no way we can help you unless you tell us what system you study. That is, if you want something more concrete than the very general information in spirit of my parenthetical. $\endgroup$
    – Marek
    Aug 2, 2011 at 16:42
  • $\begingroup$ ah, this actually was my question or part of it at least. If given a random phenomena, can you say that it is critical or not by just looking some measurables. Do you have to know something about the measurables (that they are e.g. energy release rate vs. time)? hmm, I'll have to think my question again... but lets treat this system as an earthquake or sand pile like phenomena (events, avalanches, etc.). The activity rate increases as a power law (scale invariance) before major (catastrophic, final) event. Is this enough for criticality? Is there other means to verify criticality? $\endgroup$
    – Juha
    Aug 3, 2011 at 20:23
  • $\begingroup$ I think true criticality is tricky; most critical field theories are unstable in some directions (in the renormalisation sense), so phenomena are only approximated well by a critical field theory over a finite range of scales. So technically, one might never have a "proper" critical phenomenon (whatever that might be); nevertheless, it is clear that many things of interest are in a critical regime, and that is good enough. $\endgroup$
    – genneth
    Aug 26, 2012 at 18:10

2 Answers 2


You can think about some critical phenomena is in terms of analytic continuation and Fisher zeros. As you probably know, the Taylor series expansion an analytic function can only converge within a disk that does not contain singularities. However, you can find Taylor expansions by 'working around' the singularity by means of analytic continuation.

An example of analytic continuation, from wikipedia

Fisher (and others) realized that the boundary between two phases is separated by a line of zeros. Even if you know a thermodynamic function exactly in one part of the phase diagram, you cannot analytic continuate into another. See Fig. 1.

I mention this because it sounds similar to a paper and talk I recently heard by Anatoli Polkovnikov, who was asking similar questions in regard to a dynamic phase transition.

If that doesn't help, other signs to look for are:

  1. Critical slowing: it takes longer and longer for dynamics to converge. This applies to simulations as well, which is a good hint!
  2. A point where correlations are algebraic, not exponential.
  3. Scaling of dynamical phenomena, such as quenches or ramps. This is another one where simulations might help out.

You can say that a critical phenomenon is present when your system show no characteristics scale. Namely mean or variance, of some physical quantity, are not finite. For example in the Ising model magnetization is $M=<\sigma_i>$ and suceptibility is $\chi \sim <\sigma^2>-<\sigma>^2$, where $\sigma_i=\pm 1$ is the spin value at the lattice point $i$. Near the critical temperature $T_c$, $\chi\rightarrow +\infty$. See "Conformal Field Theory" P. de Francesco

  • $\begingroup$ Is non-finiteness always indicator of criticality? $\endgroup$
    – Juha
    May 29, 2012 at 10:43
  • $\begingroup$ i think so. the problem is that to show that the mean or variance in the data are not finite is not trivial. $\endgroup$ May 29, 2012 at 15:50
  • $\begingroup$ Are first order transitions not critical for purposes of this question? $\endgroup$
    – Ron Maimon
    Jul 28, 2012 at 7:34
  • $\begingroup$ @EmanueleLuzio: Why? Newton made stupid blunders all the time--- there are no superhumans in this world. The question is whether you include first order transitions as "critical". Is it standard to call only scale-invariant limits critical? $\endgroup$
    – Ron Maimon
    Jul 28, 2012 at 8:00
  • 2
    $\begingroup$ @EmanueleLuzio: That's not a good criterion--- you could always change coordinates to make the mean or variance of anything infinite. The right criterion is that the physical fluctuations are scale invariant in units/scale where the leading derivative term in the energy function is normal form and has coefficient set to 1. $\endgroup$
    – Ron Maimon
    Jul 28, 2012 at 20:24

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.