(sorry, this post is going to get little bit technical… we’ll find a way of signalling when this is going to happen)
Yvi was reading Wilson’s article in SciAm on the RG… and the first page, although apparently naive, contained something shocking. It had three pictures which showed the idea of an ordered phase, critical point and disordered phase in an Ising ferromagnet. You have three (other) pics right here…
The left image shows the aspect of a ferromagnet below the critical temp Tc, the second at Tc and the third above Tc. And the funny thing is that we can recognize, more or less, the temperature from a single snapshot of the system. BUT we’re always told that temp is not a property of a single configuration, it is always a property of the ensemble!! How come?
We have two events which we’ll consider to be random: T is to have a certain temperature, and C is to get a certain configuration. Then, the conditional probability P(C|T) is just the Boltzmann factor, 1/Z exp(-beta E(C)). What about P(T|C)? I mean: what is the probability of the temperature being a given one, when we observe a single configuration? Somehow, this question makes sense. Otherwise, Wilson would not have put the pictures, would he?
We can use Bayes theorem to “swap” the conditional probability:
So, P(T|C) = P(C|T)· P(T)/P(C), the problem, as always in bayesian statistics, is the a priori probability for the temperature… Of course, you can start with any distribution P_0(T), and then iterate that prescription as you’re given more and more configurations: P(T|C_1,C_2…C_n). My hypothesis is that this process will have a single fixed point, independently of the initial distribution, with some mild restrictions on its support. What do you think?
Thanks to noema, flor and migeru. The latter pointed to maximal entropy methods, still I would like to find the relation… Ah, and see this nice post I found.