The Microcanonical and Canonical Ensembles
An isolated system at fixed energy and a system held at fixed temperature are described by different ensembles, and each has its own way of assigning probabilities. For large systems the two agree on everything measurable.
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
1 · Read
The microcanonical rulebook covers an isolated system with fixed energy, volume, and particle count. Every allowed microstate gets equal odds, and entropy is the log of that count. Few real benches are isolated, but the idea anchors everything.
The canonical rulebook covers a system touching a heat bath, with temperature fixed and energy free to wobble. Odds follow the Boltzmann factor, normalized by the partition function. The bath is huge, so giving or taking a little energy barely shifts its temperature.
A sealed flask sitting on a bench trades heat with the room, so it belongs to the canonical book at the room temperature. A perfectly insulated twin that trades nothing belongs to the microcanonical book at its fixed energy. Ask what is fixed: insulation points one way, a thermostat points the other.
For a handful of particles the two books disagree, but for a vast crowd the energy wobbles shrink to nothing beside the total. Both books then predict the same measurable averages. Large numbers wash out the difference.
Fixed energy means equal odds, fixed temperature means Boltzmann odds, and huge crowds make both agree.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.