What is the Difference Between Canonical and Grand Canonical Ensemble?

The canonical and grand canonical ensembles are both statistical ensembles in thermodynamics, but they differ in the conditions they describe.

A canonical ensemble represents a mechanical system's possible states in thermodynamic equilibrium with a heat reservoir. The system can exchange energy with the heat bath, making the states of the system differ in total energy. The principal thermodynamic variable of the canonical ensemble is the absolute temperature, T.

On the other hand, a grand canonical ensemble describes a system in contact with both a heat reservoir and a particle reservoir. This means that the system can exchange both energy and particles with its surroundings. The grand canonical ensemble is denoted as µVT ensemble, where µ is the chemical potential, V is the volume, and T is the temperature.

In summary, the key difference between canonical and grand canonical ensembles is that the canonical ensemble describes a system in contact with a heat reservoir, while the grand canonical ensemble describes a system in contact with both a heat reservoir and a particle reservoir.

Comparative Table: Canonical vs Grand Canonical Ensemble

The main difference between the canonical and grand canonical ensemble lies in the constraints they impose on the system. Here is a table summarizing the differences between the two ensembles:

The canonical ensemble represents the possible states of a mechanical system in thermodynamic equilibrium with a heat bath, with fixed particle number, volume, and temperature. The probability distribution of the system's states depends on the absolute temperature. On the other hand, the grand canonical ensemble describes the possible states of a system in contact with both a heat reservoir and a particle reservoir, allowing for changes in the number of particles. The grand canonical ensemble is typically used when modeling systems with variable particle numbers, such as those in thermal and chemical equilibrium with a reservoir.