Scale invariance in sandpile model and forest fire model 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:


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*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.
 A: 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):


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*First order phase transition:


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*finite correlation length

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


*Second order phase transition:


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*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
