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Physics · lesson 3 of 3

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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Learn the ideas

Prerequisites: Probability, exponential functions and thermodynamics.

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

Example 1

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

Add weights 1+0.25.

1.25

Example 2

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

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

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

Show answer and explanation

1.25

Add weights 1+0.25.

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

Show answer and explanation

0.2

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

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

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g

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

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

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3.5

Z=2×1+3×0.5.

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

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Mean energy

Differentiating the canonical partition sum produces the weighted energy mean.

6 Why should probabilities be normalized after assigning Boltzmann factors?

Sample answer and marking guidance

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.

7 In a nondegenerate two-level model at the infinite-temperature limit, find excited-state probability.

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0.5

As β tends to zero, both weights tend to one.

8 Explain one distinction between microcanonical and canonical ensembles.

Sample answer and marking guidance

A microcanonical description fixes energy within its idealization for an isolated system; a canonical description fixes temperature through a reservoir while system energy can fluctuate.

Credit the different constraints and possible energy fluctuations.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.