Lecture 2 ： Canonical Ensemble and the Partition Function Dr. Ronald M. Levy Statistical Thermodynamics

Probability Observable Probability Distribution Single observation Average of all 1

Ensemble and Canonical Ensemble: Ensemble: An idealization consisting of a large number of similar systems Canonical Ensemble: An ensemble with the same Number of molecules, Volume and Temperature, but different Energy per system. (N, V, T) Step 1: Ensemble placed in heat bath to equilibrate at T. Step 2: Ensemble surrounded by thermal insulation: Total energy fixed. Ensemble and Canonical Ensemble E1E1 E2E2 E1E1 (NVT) E2E2 E1E1 systems energy In principal, these are energy eigenvalues of quantum states 2

Constraints on the distribution Most probable distribution Total number of members of the ensemble fixed Total energy of the ensemble fixed Any distribution which satisfies constraints above is a possible distribution for the ensemble If each energy level is distinct Eliminating over counting 3

Most probable distribution 4

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Canonical partition function Average statistical mechanical energy of the system Stat Mech: Average statistical mechanical energy Thermodynamics Internal Energy 6

Canonical partition function 7

Supplementary material 8

Pressure on system Work done on the system Change of ensemble internal energyWork 9

Supplementary material 10