Grand canonical ensemble convention

The convention that a system in contact with a heat and particle bath is described by the grand canonical ensemble.

Grand canonical ensemble convention

The grand canonical ensemble describes a system in thermal and chemical equilibrium with a reservoir at fixed temperature $T$ and chemical potential $\mu$ . The system can exchange both energy and particles with the reservoir.

Probability distribution

In the grand canonical ensemble, the probability of finding the system with $N$ particles in microstate $(q, p)$ is

\rho_N(q, p) = \frac{1}{\Xi} e^{-\beta (H_N(q, p) - \mu N)},

where:

$\beta = 1/(k_B T)$ is the inverse temperature
$H_N$ is the Hamiltonian for $N$ particles
$\Xi$ is the grand partition function

The natural thermodynamic potential for this ensemble is the grand potential . The bath is modeled as a particle reservoir . For the full formal definition, see grand canonical ensemble .