# 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.

Course: [Analytical and statistical mechanics](https://studybrix.com/en/courses/analytical-and-statistical-mechanics/index.md)
Level: advanced
Prerequisites: Probability, exponential functions and thermodynamics.

## 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.

## Practice questions

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

Answer: 1.25

Explanation: 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.

Answer: 0.2

Explanation: 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?

A. g² always
B. g
C. No factor in any case
D. 1/g always

Answer: g

Explanation: 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.

Answer: 3.5

Explanation: Z=2×1+3×0.5.

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

A. Mean position always
B. Volume always
C. Mean energy
D. Zero in all ensembles

Answer: Mean energy

Explanation: Differentiating the canonical partition sum produces the weighted energy mean.

### 6. Why should probabilities be normalized after assigning Boltzmann factors?

Sample answer / marking guidance: Individual weights need not sum to one. Dividing by Z ensures a valid probability distribution across all included microstates.

Explanation: Credit Z as the normalization sum.

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

Answer: 0.5

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

### 8. Explain one distinction between microcanonical and canonical ensembles.

Sample answer / 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.

Explanation: Credit the different constraints and possible energy fluctuations.

## Further reading

- https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/
- https://openstax.org/details/books/university-physics-volume-3

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.