Lecture 2 ： Canonical Ensemble and the Partition Function

Dr. Ronald M. Levy

ronlevy@temple.edu

Statistical Thermodynamics

ProbabilityObservable

Probability Distribution

Single observation

Average of all

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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

E1

E2 E1

(NVT)

E2

E1

In principal, these are energy

eigenvalues of quantum states

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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 distinctEliminating over counting

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Most probable distribution

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Most probable distribution

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Canonical partition function

Average statistical mechanical energy

of the system

Stat Mech:Average statistical mechanical energy

ThermodynamicsInternal Energy

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Canonical partition function

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Supplementary material

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Supplementary material

Pressure on system

Work done on the system

Change of ensemble internal energy Work

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Supplementary material

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