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Physics · Advanced university

Canonical ensembles and partition functions

For a system at inverse temperature β=1/(kBT), the canonical probability of a microstate of energy E is exp(−βE)/Z.

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Prerequisites

Probability, exponential functions and thermodynamics.

Study notes

Canonical weights

For a system at inverse temperature β=1/(kBT), the canonical probability of a microstate of energy E is exp(−βE)/Z.

Partition function

Z sums Boltzmann weights over microstates. Degeneracy must be included when summing over distinct energy levels.

Averages

Mean energy is −∂ln Z/∂β when the energy spectrum is independent of β. Thermodynamic identities inherit the ensemble assumptions.

Physical scope

Canonical equilibrium assumes thermal contact with an appropriate reservoir. It is not a universal model for every evolving system.

Worked examples

Two nondegenerate levels have energies 0 and ε with exp(−βε)=0.25. Find Z.

Add weights 1+0.25.

1.25

For a nondegenerate two-level canonical model with energies 0 and ε and exp(−βε)=0.25, find the excited-state probability.

Divide the excited weight by total weight: 0.25/1.25.

0.2

Check your understanding

  1. What extra factor enters a level's weight when it has degeneracy g?

    1. g² always
    2. g
    3. No factor in any case
    4. 1/g always

  2. A ground level has degeneracy 2 and excited level degeneracy 3. With excited weight per microstate 0.5, find Z.

  3. What quantity is −∂ln Z/∂β for a temperature-independent spectrum?

    1. Mean position always
    2. Volume always
    3. Mean energy
    4. Zero in all ensembles

  4. Why should probabilities be normalized after assigning Boltzmann factors?

Answers and marking guidance

Answers and marking notes
  1. g

    There are g microstates at that energy, each contributing a Boltzmann weight.

  2. 3.5

    Z=2×1+3×0.5.

  3. Mean energy

    Differentiating the canonical partition sum produces the weighted energy mean.

  4. Individual weights need not sum to one. Dividing by Z ensures a valid probability distribution across all included microstates.

    Credit Z as the normalization sum.

Further reading and review references

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