Grand canonical ensemble

In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium (thermal and chemical) with a reservoir. The system is said to be open in the sense that the system can exchange energy and particles with a reservoir, so that various possible states of the system can differ in both their total energy and total number of particles. The system's volume, shape, and other external coordinates are kept the same in all possible states of the system.

The thermodynamic variables of the grand canonical ensemble are chemical potential (symbol: µ) and absolute temperature (symbol: T). The ensemble is also dependent on mechanical variables such as volume (symbol: V), which influence the nature of the system's internal states. This ensemble is therefore sometimes called the µVT ensemble, as each of these three quantities are constants of the ensemble.

In simple terms, the grand canonical ensemble assigns a probability P to each distinct microstate given by the following exponential: $${\displaystyle P=e^{{(\Omega +\mu N-E)}/{(kT)}},}$$ where N is the number of particles in the microstate and E is the total energy of the microstate. k is the Boltzmann constant.

The number Ω is known as the grand potential and is constant for the ensemble. However, the probabilities and Ω will vary if different µ, V, T are selected. The grand potential Ω serves two roles: to provide a normalization factor for the probability distribution (the probabilities, over the complete set of microstates, must add up to one); and, many important ensemble averages can be directly calculated from the function Ω(µ, V, T).

In the case where more than one kind of particle is allowed to vary in number, the probability expression generalizes to $${\displaystyle P=e^{{(\Omega +\mu _{1}N_{1}+\mu _{2}N_{2}+\dots +\mu _{s}N_{s}-E)}/{(kT)}},}$$ where µ₁ is the chemical potential for the first kind of particles, N₁ is the number of that kind of particle in the microstate, µ₂ is the chemical potential for the second kind of particles and so on (s is the number of distinct kinds of particles). However, these particle numbers should be defined carefully (see the note on particle number conservation below).

The distribution of the grand canonical ensemble is called generalized Boltzmann distribution by some authors.

Grand ensembles are apt for use when describing systems such as the electrons in a conductor, or the photons in a cavity, where the shape is fixed but the energy and number of particles can easily fluctuate due to contact with a reservoir (e.g., an electrical ground or a dark surface, in these cases). The grand canonical ensemble provides a natural setting for an exact derivation of the Fermi–Dirac statistics or Bose–Einstein statistics for a system of non-interacting quantum particles (see examples below).

The grand canonical ensemble is the ensemble that describes the possible states of an isolated system that is in thermal and chemical equilibrium with a reservoir (the derivation proceeds along lines analogous to the heat bath derivation of the normal canonical ensemble, and can be found in Reif). The grand canonical ensemble applies to systems of any size, small or large; it is only necessary to assume that the reservoir with which it is in contact is much larger (i.e., to take the macroscopic limit).