Derivation of ensemble distribution

Question

I heard that you can derive the canonical ensemble by maximizing

$L = \sum_i p_ilog( p_i ) + \alpha (\sum_i p_iE_i-E)$

or for the grand-canonical ensemble

$L = \sum_i p_ilog( p_i ) + \alpha (\sum_i p_iE_i-E) +\beta (\sum_i p_iN_i-N) ,$

where $\alpha,\beta$ are Lagrange multipliers. I mean, I see that the first term is just the entropy, but I don't understand the conditions here. THe point in the canonical ensemble should be that the energy is not constant, so why do we want to have $(\sum_i p_iE_i-E)=0$ and what does this $E$ represent?-Similarly, for the particle number in the grand canonical ensemble, it is not clear to me, why we want to have $(\sum_i p_iN_i-N)=0$.

Answer 1

For a system in the grand-canonical ensemble, the symbols $E$ and $N$ are not fixed values of the energy and particle number respectively. They are specified values for the ensemble averages of these quantities. Explicitly: \begin{align} E = \langle \hat H\rangle, \qquad N = \langle \hat N\rangle, \end{align} where angled brackets denote ensemble averages, and $\hat H$ and $\hat N$ are the hamiltonian and number operator of the system respectively. Even more explicitly, in the density matrix formalism, the ensemble average of an observable $\hat O$ given a state $\hat \rho$ of the system is defined as \begin{align} \langle \hat O\rangle = \mathrm{tr}(\hat \rho\hat O). \end{align} In the thermodynamic limit, namely in the limit of a large number of constituent subsystems, the distribution of energy and particle number will generically be extremely sharply peaked around their ensemble averages, and in that case, there essentially is a fixed amount of energy or a fixed number of particles in the system, so this often leads to abuse of notation and language, but the abuse isn't so bad since the distributions are indeed extremely sharp.

Answer 2

E is the total energy. The energy $E_i$ is the energy of the $i$-th ensemble. The quantity $p_i E_i$ is like $\langle E_i\rangle$, which is the average energy of the $i$-th ensemble. The same sort of argument is true for the particle number. The total energy E across all the sub-systems has to be fixed and some finite-value. The same is true for the particle number and hence, the constraints for $E$ and $N$.