I have a toy model of $\rho N$ particles in $N$ boxes and Hamiltonian $H = \sum_{i=1}^N\log{(1+n_i)}$


The canonical partition function is obtained by summing $e^{-H \left (\underline{n}\right )/T}$ over all the states with $\rho N$ particles.

And here is the first sentence I don't understand in my notes:

this corresponds to looking at large deviations where $\langle n_i\rangle = \rho$ ...

What corresponds to what? Large deviations from what?

Then it says a simpler way to study the system would be to introduce the grand canonical ensemble and the chemical potential.

$$\mathcal{Z}(T,\mu,N) = \sum_{M=0}^{+\infty} e^{-M\mu/T} Z(T,\rho = M/N,N)$$

removing the restriction on the density.

And there is this second sentence I can't understand:

The grand canonical trick is biasing a priori probabilities ($\mu = 0$) on the distribution of particles in each box to recover states with a given density as large deviations, i.e. as typical outcome under the biased distribution

What are the a priori probabilities?


Turns out this was wrong.

Canonical ensemble can be recovered as large deviation of the Canonical ensemble itself at $T \to \infty$ and Grand Canonical as large deviation of the Canonical for $\mu \to \infty$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.