Statistical thermodynamics MOC

Canonical ensemble

The canonical ensemble represents an otherwise closed system in thermal equilibrium with a heat bath at constant temperature. As such, its natural variables are

• — Number of particles in the system
• — System volume
• — Temperature of the heat bath (and thus the system)

and microstates of all energies are accessible to the system, with probabilities at equilibrium given by the Boltzmann distribution

where is the canonical partition function, related to the Helmholtz free energy . The determination of depends on the statistics used.

• Maxwell-Boltzmann statistics
• Modified Boltzmann approximation
• Fermi-Dirac statistics
• Bose-Einstein statistics

Derivation

derivation

The ratio of probabilities of two states and is given by the Boltzmann factor

To calculate the probability of the system having a given energy , it is necessary to include the energy degeneracy , hence

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