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I asked a similar question but the wrong way here. Because my intention was to ask about non thermodynamic system, I will be more specific:

  • What is the relation between critical behaviour and the scale invariance in these two model (sandpile, forest fire)?

What I can't figure out is the meaning of "infinity correlation length" in these two model.

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  • $\begingroup$ Dear emanuele, check e.g. the page and picture at en.wikipedia.org/wiki/… - The correlation length is what it always is - the thickness of the boundary between well-defined phases. On the picture, there are no such well-defined phases occupying a big volume so everything is the boundary and the corr. length is infinite. The sandpile model is interesting for creating "self-organized criticality": one gets scale invariance i.e. infinite corr. length even without fine-tuning any parameters, unlike in phase transitions in solids/liquids etc. $\endgroup$
    – Luboš Motl
    Commented May 5, 2012 at 15:28
  • $\begingroup$ Dear Lubos, do you mean that the phases, in the sand pile, are represented by the slopes at the different points of the sand pile? $\endgroup$
    – Emanuele Luzio
    Commented May 5, 2012 at 15:50
  • $\begingroup$ I think it's right. A feature of the model is, of course, that there are never qualitatively separated phases, but that's one of the "virtues" of the model. One doesn't have a sharp definition of a phase here because it doesn't exist, regardless of the parameters. The correlation length should have a well-defined definition but I wouldn't be able to define it. At the end, it's infinite, anyway, so there's no finite number to calculate. ;-) So the only thing that can really be meaningfully discussed is why the behavior is analogous to other critical points and CFTs. $\endgroup$
    – Luboš Motl
    Commented May 5, 2012 at 18:22
  • $\begingroup$ Thanx for the reply. Do you have references of these analogies? $\endgroup$
    – Emanuele Luzio
    Commented May 5, 2012 at 20:10
  • $\begingroup$ Regarding the fine tuning of parameters, I recommend this paper doi.org/10.1590/S0103-97332000000100004 $\endgroup$
    – Juan I. Perotti
    Commented Aug 9, 2021 at 16:44

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This is a tricky question and in my opinion some elder physicists are deliberately try to confuse students by using euphemisms when characterizing phase transitions.

Roughly speaking (there might be counter examples, please comment. I am interested of finding all of them):

  1. First order phase transition:

    • finite correlation length
    • scales as e.g. $k^\alpha, \alpha = 2$ (short or finite range interaction) in fourier space (1D)

  2. Second order phase transition:

    • infinite correlation length
    • scales as e.g. $k^\alpha, \alpha = 1$ (long or infite range interaction) in fourier space (1D)
    • scale invariance
    • (I think SOC goes here...)

Note that there is a continuous transition with exponent $\alpha$ that escapes my mind. Also $\alpha$ is dependent of dimension and probably something else (see e.g. anne tanguy et al. From individual to collective pinning: effect of long range interactions, PRE 1998). Also Daniel Fisher has a nice paper, Collective transport in random media: from superconductors to earthquakes.

Also I just stumbled upon: http://www.tcm.phy.cam.ac.uk/~bds10/phase/introduction.pdf which has a nice overview.

In general the purpose of these correlation lengths, roughness exponents and orders of phase transitions is just to find universality classes. The goal is to group the phenomena together and say "look, all these systems have properties of X,Y and Z. This simple model has the same properties. So by explaining the simple mode, I explain all these systems."

Simple characterization of phase transitions can be found at: http://link.aps.org/doi/10.1103/RevModPhys.76.663

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  • $\begingroup$ What is that k-space alpha exponent nonsense in your answer? $\endgroup$
    – Ron Maimon
    Commented Aug 31, 2012 at 17:21
  • $\begingroup$ Could you be more spesific? Alpha = 2 means nearest neighbour interaction (= laplace, short range) and alpha = 1 means long range, infinity range (if I recall right, it means 1/r^2 interaction => infinite). Just fourier transform it and you get the exact formulation. The point is that k-space formulation is essentially saying short or long range interaction (and possibly something extra). This is what I meant by "elder physicists are deliberately try to confuse students by using euphemisms when characterizing phase transitions". $\endgroup$
    – Juha
    Commented Sep 1, 2012 at 20:06
  • $\begingroup$ Ok, -1. This is categorically false. Second order phase transition can scale with any $\alpha$ less than two, and even bigger than 2 (as in a Lifschitz point). Please erase this answer, as it is wrong. $\endgroup$
    – Ron Maimon
    Commented Sep 2, 2012 at 13:06
  • $\begingroup$ 1. do you have a reference? 2. Read the second paragraph. $\endgroup$
    – Juha
    Commented Sep 2, 2012 at 18:07
  • $\begingroup$ I don't use references to back up what I am saying, I can give you a model. Please delete this answer, there are not just counterexamples, it's like saying "every function has a derivative with is cos(x). Are there counterexamples?" There are hardly any examples! You are possibly confusing a dynamic exponent with a critical exponent. The correlation functions in k space fall off as a nontrivial power, there is no "1" "2" dichotomy in phase transitions. You made this answer up, and you made it up wrong. $\endgroup$
    – Ron Maimon
    Commented Sep 2, 2012 at 20:43

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