1. 3.The Canonical Ensemble 1.Equilibrium between a System & a Heat Reservoir 2.A System in the Canonical Ensemble 3.Physical Significance of Various Statistical Quantities in the Canonical Ensemble 4.Alternative Expressions for the Partition Function 5.The Classical Systems 6.Energy Fluctuations in the Canonical Ensemble: Correspondence with the Microcanonical Ensemble 7.Two Theorems: the “Equipartition” & the “Virial” 8.A System of Harmonic Oscillators 9.The Statistics of Paramagnetism 10.Thermodynamics of Magnetic Systems: Negative Temperatures

2. Reasons for dropping the microcanonical ensemble: 1. Mathematical: Counting states of given E is difficult. 2. 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 P r ( E r ) can be calculated in 2 ways: 1. P r  # of compatible states in reservoir. 2. P r ~ 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  Classical mech (Gibbs –corrected ):

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

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

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

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

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

10.   N >>1   Fo z real, has sharp min at x 0  For z complex :  max along ( i y )-axis  x 0 is a saddle point of.

11.  MSD:On C, integrand has sharp max near x 0. Gaussian dies quickly

12.   

13. C.f.  With {  r = 1 } : so that (r)  r is a dummy variable

14. Fluctuations   where

15. 

16. Relative fluctuation 

17. 3.3. Physical Significance of Various Statistical Quantities in the Canonical Ensemble Canonical distribution : Helmholtz free energy A ( T, V, N ) :

18.  = Partition function ( Zustandssumme / sum over states ) A, & hence lnZ, must be extensive. Gibbs free energy G ( T, P, N ) : Prob 3.5

19. P    c.f. E r is indep of T ( Fixed { P r } = Fixed S )

20. S    T = 0, non-degenerate ground state  ( 3 rd law )  ( microcanonical ) Disorder   Unpredictability     S  Information theory (Shannon)

21. 3.4.Alternative Expressions for the Partition Function Non-degenerate systems: Degenerate systems: g r = degeneracy of E r  Thermodynamic limit ( N, V   )  continuum approx. :

22.  Z(  > 0 ) = Laplace transform of g(E) Inverse transform: If g diverges, then  > 0 is real such that all poles of Z are to the left of 

23. 3.5.The Classical Systems Quantum  Classical states =    d   where Gibbs’ prescription:

24. Ideal Gas ( In Cartesian coordinates, sum has 3N terms ) where 

25. 

26.   

27. Non-interacting (free) particles : ( from sec 1.4 )  ( Gibbs factor added by hand ) 

28. 1-particle DOS : ( same as before )  contour closes on the left contour closes on the right ( same as before ) Prob 3.15

29. 3.6. Energy Fluctuations in the Canonical Ensemble: Correspondence with the Microcanonical Ensemble Relative root-mean-square fluctuation in E :  Almost all systems in a canonical ensemble have energy U. ( Just like the microcanonical ensemble )

30. P(E)P(E) max P at E* satisfies :  or  c.f.  ( Every system in ensemble has same N & V ) i.e., Most probable E = mean E

31.    Everything, except E, are kept const.

32.  P(E) is a Gaussian with mean U and dispersion (rms) P(E/U ) is a Gaussian with mean 1 and dispersion (rms)  P(E)   (E  U ) as N  

33. Ideal Gas    for N >> 1

34. N = 10,  = 1 Mathematica

35. Z O(N)O(N)