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In Statistical Mechanics , the grand canonical ensemble is a Statistical Ensemble , that means a set of identically prepared systems,
each of which is in equilibrium with an external bath with respect to Particle and Energy exchange.

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


PARTITION FUNCTION


The Partition Function of the grand canonical ensemble with a matrix/operator formalism is:

:\Xi=\mathop{Tr} \left[\exp\left(-\beta(\hat{H}-\mu\hat{N} ight) ight].

Here \mu is the Chemical Potential , \beta the inverse temperature, sometimes also
adorned with the inverse of the Boltzmann Constant . \hat{H} is the Hamiltonian of the
system class considered, \hat{N} the operator that counts the total number of particles
in one system.


DISCRETE SUMMATION FORMALISM

The partition function of the grand canonical ensemble is given by

:\Xi(V,T,\mu) = \sum_i \sum_j \exp {-\beta(E_i - \mu N_j)} \;\,

The sum of the index ''i'' is over all the energy states of the system. The sum over the index ''j'' is over all the number of partitions, where N_j gives the number of particles in partition j.


CHARACTERISTIC STATE FUNCTION

The characteristic state function for the grand canonical ensemble is the quantity PV. This is because the ensemble satisfies the property

:\Xi(T,V,\mu) = e^{\beta P V} \,\;