The Canonical Ensemble: a derivation using a 4 spin system
The Canonical Ensemble is an important concept in statistical mechanics. By knowing the partition function for a canonical ensemble of a particular system, one can calculate the probability of any energetic state for that system, as well as many thermodynamic properties, such as the Helmholtz free energy, entropy, internal energy, and specific heat.
0:00 - introduction
0:54 - prerequisite video: negative absolute temperature
1:14 - starting from coin flips
2:05 - finding the number of ways to get different configurations
3:25 - calculating probability from number of ways
3:54 - magnetic spins in a magnetic field
5:17 - microcanonical vs. canonical ensemble
6:00 - system vs. bath for the canonical ensemble
7:32 - joint probabilities of all states possible for 4 spins in a bath
9:57 - combining joint probabilities with number of ways
10:55 - the partition function using probabilities only
13:02 - the partition function using energy
15:48 - replacing joint probabilities with an energy function (Boltzmann weight)
18:34 - demonstrating the energy-based partition function on our 4 spin system
23:49 - writing the partition function for the canonical ensemble (quantum and classical)
25:42 - finding the energy-based partition function for 75% aligned spins
28:52 - relation to magnetic field strength in NMR