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Lecture 6 - Stat Mech Postulate I f you can calculatea m...

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β β - - = j j E j E e P e j Now that we know the probability of finding the quantum state with E j at a given N,V We can calculate other mechanical thermodynamic properties M j = j j j M P M Stat. Mech. Postulate: If you can calculate a mechanical property X i consistent with the macroscopic parameters, then, <X i > =macroscopic thermodynamic X β β - = PARTITION Q(N,V, ) FUNCTION j E e j ( 29 β - - = = 1 1 2 2 for states 1 and 2 separated by =E, E P E E e P 1 2 a a
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β β β β - - = = ( , ) ( , ) For the internal energy: ( , , ) ( , , ) j j E N V j j E N V E N V e E E N V e j β β β - = ( , ) ( , , ) ( , , ) j E N V j j E N V e Q N V β β β β β β β - ׳ = - ׳ ׳ ׳ = - ( , ) , , ( , , ) considering that ( , , ) ( , , ) ( , , ) j E N V j j N V N V Q N V E N V e Q N V E Q N V β β ׳ = - ׳ , ln ( , , ) N V Q N V E
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U - = ׳ = For the pressure: the work done ON the system to induce in a canonical ensemble - j j j N j dV dE p E V V p d β β - - - ׳ = ( , ) ( , ) j j E N V j j N E N V E e V p e j With these tools in hand , we can now combine mechanical properties with thermodynamics and extract information about nonmechanical functions (like S or T) Going back to the calculation of mechanical properties…
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Consider 2 ensembles A and B that become in thermal contact, with no change in volume o dV = 0 ; d A = 0 d B = 0 Ensemble A o β A Ensemble B o β
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