Grand canonical ensemble

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Statistical mechanics
Microcanonical ensemble
Canonical ensemble
Grand canonical ensemble
Isothermal–isobaric ensemble
Isoenthalpic–isobaric ensemble
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In statistical mechanics, the grand canonical ensemble is a statistical ensemble. The grand canonical ensemble loosens the assumptions of the canonical ensemble. A grand canonical ensemble can be viewed as a system in contact with a reservoir with which it can exchange energy and particles. Or, equivalently, a grand canonical ensemble is a collection consisting of copies of a given system. The number of particles and total energy of the collection remain constant while energy and particles are allowed to flow between members of the collection.

The grand canonical ensemble frequently provides the most convenient avenue for calculations.

The partition function of the grand canonical ensemble, called the grand partition function, is given by

${\mathcal Z} (V,T,\mu) = \sum_i \sum_j \exp {-\beta(E_i - \mu N_j)} ,$

where

β is the thermodynamic beta,

μ is the chemical potential of the system,

E_i denotes the energy value indexed by i,

N_j denotes the energy value indexed by i, and

The indices i, j in the summation runs over all available (E_i, N_j) states of the system.

An ensemble of quantum mechanical systems is described by a density matrix. In a suitable representation, a density matrix ρ takes the form

$\rho = \sum_k p_k |\psi_k \rangle \langle \psi_k|$

where p_k is the probability of a system chosen at random from the ensemble will be in the microstate

$|\psi_k \rangle.$

So the trace of ρ, denoted by Tr(ρ), is 1. This is the quantum mechanical analoge of the fact that the accessible region of the classical phase space has total probability 1.

It is also assumed that the ensemble in question is stationary, i.e. it does not change in time. Therefore, by Liouville's theorem, [ρ, H] = 0, i.e. ρH = Hρ where H is the Hamiltonian of the system. Thus the density matrix describing ρ is diagonal in the energy representation.

Suppose

$H = \sum_n E_i |\psi_i \rangle \langle \psi_i|$

where E_i is the energy of the i-th energy eigenstate. If a system i-th energy eigenstate has n_i number of particles, the corresponding observable, the number operator, is given by

$N = \sum_n n_i |\psi_i \rangle \langle \psi_i|.$