The Microcanonical and Canonical Ensembles
Microcanonical systems have fixed energy, volume and particle number; canonical systems have fixed temperature, volume and particle number. For ordinary short-range systems in a regular thermodynamic limit, matched ensembles agree on local thermodynamic averages, but not on energy fluctuations.
What a learner can do afterwards
- States what is held fixed in each ensemble and how probabilities are assigned
- Explains why the canonical ensemble suits a system in a heat bath
- Argues why the two agree in the limit of a large number of particles
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The microcanonical ensemble fixes energy E, volume V and particle number N in an isolated system. Allowed microstates in the specified energy shell are equally probable. With W allowed states, each microstate has probability 1/W. The system's entropy is S=k_B ln W. States outside the shell are excluded.
The canonical ensemble fixes T,V,N in equilibrium with a large heat bath. Here volume is fixed, unlike the preceding fixed-pressure Gibbs criterion. Heat exchange lets energy fluctuate without appreciably changing the bath's temperature. Each probability is exp(-E/(k_B T))/Z. Since Z sums all weights, their normalized sum is Z/Z=1.
A sealed, rigid flask equilibrated with a large room at steady temperature is canonical. Sealing fixes N; rigidity fixes V. Thermal contact sets equilibrium T and permits energy fluctuations. A flask still warming or cooling is not yet in this equilibrium description.
A sealed, rigid, perfectly insulated box with no other energy transfer is microcanonical: E,V,N stay fixed. Insulation blocks heat, but moving walls or electrical connections can transfer work. Check heat, work and particle exchange.
For ordinary short-range systems away from singular regimes, mean energy and variance usually scale with N. Root-mean-square fluctuation scales with sqrt(N), giving relative size 1/sqrt(N). At matched mean energy, ensembles approach the same local thermodynamic averages, but their energy variances remain different.
At positive finite T, canonical states of different energies generally have different probabilities. For states at 0 and D with exp(-D/(k_B T))=1/2, their weights are 1 and 1/2, giving probabilities 2/3 and 1/3. Equal probabilities apply within a microcanonical energy shell, not across all canonical energies.
These heights are weights. Dividing by their sum gives probabilities 2/3 and 1/3.
A small particle count does not justify a large-system equivalence approximation. Some observables may agree even in a small system, but agreement is not guaranteed. Large size alone is also insufficient for every system: long-range interactions or singular limits can require further analysis. State the ensemble's constraints and which observable is being compared.
Microcanonical: fixed E,V,N and equal probabilities within an energy shell. Canonical: fixed T,V,N and Boltzmann probabilities. Large-system equivalence has limits and excludes energy variance.
Original lesson, questions and flashcards written for this learning dataset.
Original readings and flashcards written for this learning dataset.
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