Statistical Thermodynamics Lecture 2 Canonical Ensemble and the

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

Probability Distribution Observable Probability Average of all Single observation 1

Ensemble and Canonical 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. systems energy E 1 E 2 E 1 (NVT ) E 2 E 1 In principal, these are energy eigenvalues of quantum states 2

Most probable distribution Constraints on the 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

Most probable distribution 5

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

Supplementary material Pressure on system Work done on the system Change of ensemble internal energy Work 9

Supplementary material 10