One of the first things that we learn in thermodynamics is that temperature is the property of an ensemble, not of a single configuration. But is it true? Can we make sense of the idea of the “temperature of a single configuration“?
I became sure that a meaning could be given to that phrase when I read Kenneth Wilson’s article about the renormalization group in Scientific American long long back. There he gave three pics describing the state of a ferromagnet at low, critical and high temperature. He gave just a single pic for each state!! No probability distributions, no notions of ensemble. Just pictures, that looked like these ones:
Was Wilson wrong? No, he wasn’t! Black spins are down, and green ones are up. So, he wanted to show that, at low temperatures (left pic) you have large domains. At high temperatures (right pic), it is all random. And at the critical temperature, the situation becomes interesting: you have patches within patches, of all sizes… But that is another story, I may tell it some other day.
So, you see: Wilson’s pics make the point, so it is true that a single configuration can give the feeling for the temperature at which it was taken.
In statistical mechanics, each possible configuration C for a system has a certain probability, given by the Boltzmann factor:
where is the energy of the configuration, is the temperature and is Boltzmann’s constant. The proportionality is a technical thing: probabilities have to be normalized. In terms of conditional probability, we can say that, given a temperature, we have a probability:
which means: given that the temperature is , the probability is such and such. Our question is, therefore, what is the probability for each temperature, given the configuration?
Now, remember Bayes theorem? It says that you can reverse conditional probabilities:
So, we can say:
Great, but… what does that mean? We need the a priori probability distribution for the temperatures and for the configurations. That’s a nice technical problem, which I leave now. But see my main point: given the temperature, you have a probability distribution for the configurations and, given the configuration, you have a probability distribution for the temperatures.
Of course, that distribution might be quite broad… Imagine that you have a certain system at a certain unknown temperature . You get one configuration and, from there, try to estimate the probability. You will get a certain probability distribution , presumably broad. OK, now get more configurations and iterate: . As you get more and more, your distribution should narrow down and you should finally get a delta peak on the right temp! So, you get a sort of visual thermometer…
The idea is in a very alpha release… so comments are very welcome and, if you get something nice and publish, please don’t forget where you got the idea! :)
(Note: I already made an entry of this here, but this one is explained better)