# Statistical Mechanical Ensembles

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The central application of statistical mechanics rests on the assumption that the average of a property over a large number of systems will give the same value as the thermodynamic quantity of interest. We can distinguish between mechanical properties such as pressure, energy, volume etc. and non-mechanical properties such as entropy. Although there are a large number of particles and an extremely large number of quantum states accessible to even a small system, the state of the system can be characterized by just a few thermodynamic variables.

• Introduction to Ensembles
The central application of statistical mechanics rests on the assumption that the average of a property over a large number of systems will give the same value as the thermodynamic quantity of interest. We can distinguish between mechanical properties such as pressure, energy, volume etc. and non-mechanical properties such as entropy.
• The Canonical Ensemble
The most practical ensemble is the canonical ensemble with N, V, and T fixed. We can imagine a collection of boxes with equal volumes, and equal number of particles. The entire collection is kept in thermal equilibrium.
• The Grand Canonical Ensemble
To consider theories for fluctuations in the number of particles we require an ensemble that keeps V, T, and the chemical potential, m constant, a grand canonical ensemble.

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