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

  • 𝑁 β€” Number of particles in the system
  • 𝑉 β€” System volume
  • 𝑇 β€” 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

𝑃𝑖=exp⁑(βˆ’πΈπ‘–π‘˜π΅π‘‡)𝑍𝑍=βˆ‘π‘–exp⁑(βˆ’πΈπ‘–π‘˜π΅π‘‡)=exp⁑(βˆ’πΉπ‘˜π΅π‘‡)

where 𝑍 is the canonical partition function, related to the Helmholtz free energy 𝐹 = βˆ’π‘˜π΅π‘‡ln⁑𝑍. The determination of 𝑍 depends on the statistics used.

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

β„™(𝑖)β„™(𝑗)=exp⁑(πΈπ‘—βˆ’πΈπ‘–π‘˜π΅π‘‡)

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

β„™(𝐸)=𝑔(𝐸)𝑍exp⁑(βˆ’πΈπ‘˜π΅π‘‡)

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