Grand canonical ensemble - Definition

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 is

<math>Z=\mathop{Tr} \left[\exp\left(-\beta(\hat{H}-\mu\hat{N}\right)\right].<math>

Here <math>\mu<math> is the chemical potential, <math>\beta<math> the inverse temperature, sometimes also adorned with the inverse of the Boltzmann constant. <math>\hat{H}<math> is the Hamiltonian of the system class considered, <math>\hat{N}<math> the operator that counts the total number of particles in one system.