I have a toy model of $\rho N$ particles in $N$ boxes and Hamiltonian $H = \sum_{i=1}^N\log{(1+n_i)}$
$$P(\underline{n})=\frac{1}{Z(T,\rho,N)}e^{-H(\underline{n})/T}$$
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 $\mu \to \infty$