3. The Canonical Ensemble

1. 3. The Canonical Ensemble Equilibrium between a System & a Heat Reservoir A System in the Canonical Ensemble Physical Significance of Various Statistical Quantities in the Canonical Ensemble Alternative Expressions for the Partition Function The Classical Systems Energy Fluctuations in the Canonical Ensemble: Correspondence with the Microcanonical Ensemble Two Theorems: the “Equipartition” & the “Virial” A System of Harmonic Oscillators The Statistics of Paramagnetism Thermodynamics of Magnetic Systems: Negative Temperatures

2. Reasons for dropping the microcanonical ensemble: • Mathematical: Counting states of given E is difficult. • Physical: Experiments are seldom done at fixed E. Canonical ensemble : System at constant T through contact with a heat reservoir. • Let r be the label of the microstates of the system. • Probablity Pr( Er) can be calculated in 2 ways: • Pr # of compatible states in reservoir. • Pr~ distribution of states in energy sharing ensemble.

3. 3.1. Equilibrium between a System & a Heat Reservoir Isolated composite system A(0) = ( System of interest A ) + ( Heat reservoir A ) Heat reservoir :   , T = const. Let r be the label of the microstates of A.  with Probability of A in state r is 

4. 3.2. A System in the Canonical Ensemble Consider an ensemble of N identical systems sharing a total energy E. Let nr = number of systems having energy Er ( r = 0,1,2,... ).  = average energy per system Number of distinct configurations for a given E is Equal a priori probabilities   (X) means sum includes only terms that satisfy constraint on X. { nr* } =most probable distribution

5. Method of Most Probable Values   To maximize lnW subjected to constraints is equivalent to minimize, without constraint , are Lagrange multipliers  

6. with   Let and set  Same as sec 3.1 E.g.

7. Method of Mean Values ~ means “depend on {r} ”. Let Thus Constraints: Note: r in { nr } is a dummy variable that runs from 0 to , including s. 

8. Method of Steepest Descent ( Saddle Point ) is difficult to evaluate due to the energy constraint. Its asymptotic value ( N   ) can be evaluate by the MSD. Define the generating function   Binomial theorem  U removes the energy constraint. where

9. NU = integers  = coefficient of zN U in power expansion of . This is the case if all Er , except the ground state E0= 0, are integer multiples of a basic unit. analytic for |z| < R  C : |z| < R ( For { r~ 1 }, sharp min at z = x0 ) Let 

10.   N >>1   Fo z real, has sharp min at x0   max along ( i y )-axis For z complex :  x0 is a saddle point of .

11.  MSD: On C, integrand has sharp max near x0 . Gaussian dies quickly

12.   

13. C.f.  so that With { r = 1 } :

14. Fluctuations   where

15. 

16. Relative fluctuation 

17. 3.3. Physical Significance of Various Statistical Quantities in the Canonical Ensemble

18. 3.4. Alternative Expressions for the Partition Function

19. 3.5. The Classical Systems

20. 3.6. Energy Fluctuations in the Canonical Ensemble: Correspondence with the Microcanonical Ensemble

21. 3.7. Two Theorems: the “Equipartition” & the “Virial”

22. 3.8. A System of Harmonic Oscillators

23. 3.9. The Statistics of Paramagnetism

24. 3.10. Thermodynamics of Magnetic Systems: Negative Temperatures