# Statistical specific heat as energy fluctuation in spin glasses

Consider the specific heat (in statistical sense, as energy fluctuation in the canonical ensemble) of a complex model, something similar to a spin glass. Is the specific heat defined on fluctuations about a specific (local) equilibrium state, or on all possible fluctuations given values for the parameters of the Hamiltonian (including temperature)?

The question is not trivial because there are many possible local equilibria in a spin glass for a given temperature. Should I let the energy fluctuate between multiple ones?

Let me clarify using the computational point of view. Suppose I sample states from my spin glass, compute their energies and the variance of the latter. The histogram of energies looks like this:

Should I consider the peak on the right while computing the specific heat? I suspect more peaks may appear the more I sample, giving me unphysical jumps in the specific heat dependant on the number of sampling steps.

Please do not answer this question using specific spin glass models, as mine may be different, unless you want to use them as examples.

EDIT As it was pointed out, this depends on how we define the specific heat, and why we want to use it for, so I specify that I'm studying the behaviour of Cv to look for criticality-related peaks.

• just the common sense : more peaks with more samples ? this means first that you need to sample more, until you get a complete and stable picture. The interpretation of the physicality of the jumps would come after. – user46925 Jan 14 '16 at 18:38
• to get better distributions in the samples : Metropolis–Hastings algorithm – user46925 Jan 24 '16 at 2:24
• @igael I use Gibbs sampling, which is similar. – Bzazz Jan 25 '16 at 12:26

Regardless of the system, Cv will be proportional to the variance of energy. If you have peaks at higher energies, that will increase its value. But at high enough energies the occupation of those states will be so low they won't significantly affect the variance.

In this case the variance of the distribution isn't just the width² of one of the peaks, you have to calculate it from its definition. So take into account all of the peaks, but the most energetic ones will make a smaller contribution.

• So you're saying I should consider all the peaks... – Bzazz Jan 18 '16 at 13:10
• in the context of a saturation search ? there are 2 other recent questions around the same subject. Don't you need to add some context, ie on the sampling methods ? – user46925 Jan 24 '16 at 2:30

I think you have the wrong idea when you ask how specific heat is "defined". In computational physics, the starting point is an experimental measurement that one could measure, or at least, a physical quantity that one might care about ... and then the question is, "how do I compute it?" The wrong approach is to have in mind a certain formula. You should never start with a formula, you should start with a physical measurement / something in the real world.

So for example in a real glass (not a spin glass, I am just giving an everyday example), the "specific heat" that people normally talk about is: "If I take a certain glass and add heat, how much does the temperature go up?" If you're way colder than the glass transition, and you add heat at a reasonable rate (within an hour, as opposed to a trillion years), then you can be sure that the glass will not be sampling the whole phase space, but rather only the few states plausibly accessible from the starting configuration.

In other words, the only way to know which fluctuations "count" is to try to figure out what which fluctuations will actually occur in the physical context that you're asking about. In your case, you need to ask: "If I include the peak on the right, what is the physical meaning of the quantity that I'm calculating? And if I don't include it, what is the physical meaning?" And then decide which one you care about more (or maybe you care about both).

• Your point about the definition being relative is good. I use that formula because I'm checking whether Cv has a peak due to a critical point. So the question should rather be: do I have to take into account the other peaks to this end? – Bzazz Jan 19 '16 at 15:34