Grand canonical ensemble

In statistical mechanics, a grand canonical ensemble is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that is being maintained 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:

where $N$ is the number of particles in the microstate and $E$ is the total energy of the microstate. $k$ is Boltzmann's 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 $Ω(µ, V, T)$ .

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