Lecture10Ch161Feb25

Lecture10Ch161Feb25 - Summary of last lecture...

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Summary of last lecture Joule-Thompson process (Throttling) reports on intermolecular interactions Canonical ensemble Boltzmann distribution Partition function and its relation to thermodynamics.
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Key Concepts and Lessons: a) Grand Canonical partition Function b) Relation of GCPF to thermodyanmics c) Paramagnetism d) Classical canonical PF e) Gibbs paradox Reading: Reif, Ch6
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Relation between Canonical Partition Function and Thermodynamics (cont) We get therefore: d (ln Z + β d E ) = βδ Q rev But we know from good old thermodynamics that q rev = δ q rev kT = dS And therefore we obtain the key relation of Statistical Mechanics: This is a truly remarkable result that shows how microproperties (Q) are related to TD quantities (A). Furthermore the last relation gives us the famous and very important Gibbs relation for entropy which is a foundation of modern information theory: S = k P v v ln P v S = E T + k ln Z + const or setting const = 0 : E TS = A = kT ln Z
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Relation to Microcanonical Ensemble In microcanonical ensemble - which we considered early on - the energy is fixed, I.e. it applies to thermally isolated systems In a more realistic canonical ensemble we allow energy to fluctuate because the system is in contact with heat bath. How great are fluctuations of energy then? In the last lecture we derived the relation: Δ E 2 ( ) = E ∂β = 2 ln Z ∂β 2 Recall that C v = E T N , V , i.e Δ E 2 ( ) = kT 2 C V Now note that Cv is extensive, I. .e Cv~N Δ E 2 ( ) E = O 1 N 10 11 i.e. it is indeed very very small
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Density of States Now we can ask the question: According to the Gibbs microcanonical distribution the lowest E microstate is most probable. But we claim that microcanonical and canonical ensembles are equivalent because E fluctuations are small. How comes? The answer is that we can consider summation over binned values of energy: For very large systems energy should be considered a continuous variable and we get: Z = Ω ( E ) e β E dE Now we know that is a very rapidly growing function of E (like E f ) and therefore the integrand has a well defined maximum at some value of E which determines the value of integral according to saddle-point method: Ω Z = Ω ( E ) e E dE = e E + ln Ω ( E ) dE = e E + S ( E ) dE The saddle point - equilibrum energy - is determined then by relation: S E = = 1 kT Which looks very familiar! Z = e E v v ( microstates ) = Ω ( E l ) l ( levels ) e E l
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Grand Canonical Ensemble and Grand Canonical Partition Function Now let us relax another condition: Constant number of particles. In other words consider a system which is in heat and particle exchange with reservoir. What is the probability to find our system in a microstate Ev with number of particles N’? We follow the same logic as before by focusing on the number of available
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Lecture10Ch161Feb25 - Summary of last lecture...

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