Understanding Critical Exponents and Their Application in Systems: A Guide

[Talker1500]

Hi,

I've been reading about critical exponents and how they're related in any system. I've seen how, for example, several exponents can be extrapolated from charts using a log-log fitting.

I would like to know how this procedure works exactly, I know it's a silly question but I've been trying to use it to estimate some exponents in the Ising model and I get impossible results, so any help would be appreciated.

Thanks

 

[Timo]

You question is a bit vague, so let me give two comments that may or may not help:

1) The reason critical exponents are inferred from a log-log plot of A and B is purely mathematical: The assumption is that A = a * B^E, where E is the critical exponent and "a" some proportionality constant. If you take the log of sides of the equation you straightforwardly arrive at log(A) = E * log(B) + log(a). That is, the logarithms of A and B are linearly related with a slope equal to the critical exponent E. So you plot the data, fit the best line through them, and read off E from that.

2) The critical relations hold true only very close to the critical point, a unique state in the phase diagram. If you relate A and B away from the critical point, you do not expect A = a*B^E to hold in the first place. So it is not expected that plotting log(A) vs. log(B) would result in a linear relation.

Oh well, a third comment:
3) As always in physics there is the possibility that your data are not sufficiently accurate or that you are in a regime where the expected relation does not hold true very well (that 2nd point is related to my comment 2, but a bit more general). For example, if you got your data from computer simulations, apart from your simulation being buggy you might have not reached the required accurancy (e.g. not simulated long enough) or have simulated an Ising system that is too small to show the behavior of the "real" Ising system, which is infinitely large (interesting keyword, but somewhat advanced: Finite-size scaling).

If these comments do not help you and you want to devote serious time to the extraction of critical exponents in the Ising model, I recommend you add an explanation of exactly what you were trying to do, what results you got, and why you think they are wrong. In principle, the task should be sufficiently basic for someone on PF to be able to discuss the issue with you.