1. 4.The Grand Canonical Ensemble 1.Equilibrium between a System & a Particle-Energy Reservoir 2.A System in the Grand Canonical Ensemble 3.Physical Significance of Various Statistical Quantities 4.Examples 5.Density & Energy Fluctuations in the Grand Canonical Ensemble: Correspondence with Other Ensembles 6.Thermodynamic Phase Diagrams 7.Phase Equilibrium & the Clausius-Clapeyron Equation

2. 4.1.Equilibrium between a System & a Particle-Energy Reservoir System A immersed in particle-energy reservoir A. A in microstate with N r & E s  with Using  

3.  A, A in eqm  

4. 4.2.A System in the Grand Canonical Ensemble Consider ensemble of N identical systems sharingparticles & energy Let n r,s = # of systems with N r & E s, then Let W { n r,s } = # of ways to realize a given set of distribution { n r,s }.  Let { n r,s * } = most probable set of distribution, i.e.,  Method of Most Probable Values :

5. Method of Mean Values :  (X) means sum includes only terms that satisfy constraint on X. Saddle point method  For a given the parameters  &  are determined from Classical mech (Gibb –corrected ):

6. 4.3. Physical Significance of Various Statistical Quantities The q-potential is defined as   dE s caused by dV.

7.     Euler’s eq. 

8. Fugacity   Variable dependence : Grand partition function Note: Z is much easier to evaluate than Z, especially for quantum statistics and/or interacting systems.

9. Grand Potential Approach Let F be the thermodynamic potential related to Z. Grand potential Particle, heat reservoir  Suggestion from canonical ensemble :  

10.   

11. Grand Potential See Reichl, §2.F.5. Grand potential :   Caution : Prob 4.2

12. Using we have 

13. 4.4.Examples Classical Ideal Gas : N ! = Correction for Indistinguishableness Freely moving particles  

15.    n = 3/2 : nonrelativistic gas. n = 3 : relativistic gas. Find A & S as functions of (T,V,N) yourself.

16. Non-Interacting, Localized Particles Non-Interacting, Localized Particles (distinguishable particles : model for solid ) : Particles localized   for or

17.   

18. See §3.8 Quantum 1-D oscillators: Classical limit : Consider a substance in vapor-solid phase equilibrium inside a closed vessel.  i.e., Phase equilibrium 

19. For ideal gas vapor :  For a monatomic gas :  If  From §3.5 Einstein model : solid ~ 3-D oscillators of same  

20.  ( e  / kT added by hand to account for the difference between binding energies of the solid & gas phases. ) At phase equilibrium: Solid phase appears :  Pure vapor :  T c = characteristic T or Since f /  e  / kT increases with T, this means Mathematica

21. 4.5.Density & Energy Fluctuations in the Grand Canonical Ensemble: Correspondence with Other Ensembles with  see §3.6 In general

22. Particle density :  Particle volume :  Euler’ s equation : 1 st law :     T = isothermal compressibility

23. Relative root mean square of n ~ 0 in the thermodynamic limit for finite  T At phase transition : , = critical exponents d = dimension of system Experiment on liquid-vapor transition : root mean square of n critical opalescence  Grand canonical  canonical ensemble

24. Energy Fluctuations  Caution : N = N( P,T ) 

25.  §3.6 :  

26. 4.6.Thermodynamic Phase Diagrams Phase diagram:Thermodynamic functions are analytic within a single phase, non-analytic on phase boundaries. T t = 83.8 K P t = 68.9 kPa T C = 150.7 K P C = 4.86 MPa supercritical fluid A = Triple point C = Critical point Ar Co-existence lines : S-L L-V S-V Solid Liquid Vapor

27. Ar Triple point T t = 83.8 K P t = 68.9 kPa Critical point T C = 150.7 K P C = 4.86 MPa supercritical fluid Co-existence lines : S-L L-V S-V

28. supercritical fluid Co-existence lines : S-L L-V S-V supercritical fluid Solid Liquid Vapor

29. 4 He 4 He (BE stat) : Critical point T C = 5.19 K P C = 227 kPa T = 2.18 K P S = 2.5MPa Superfluid characteristics (BEC) : Viscosity = 0. Quantized flow. Propagating heat modes. Macroscopic quantum coherence. 3 He (FD stat) : Critical point T C = 3.35 K P C = 227 kPa P S = 30MPa Superfluid below 10 mK due to BCS p-wave pairing.

30. 4.7.Phase Equilibrium & the Clausius-Clapeyron Equation Gibbs free energy  =  ( P,T ) = chemical potential Consider vessel containing N molecules at constant T & P. Let there be 2 phases initially: vapor (A) & liquid (B).  For a given T & P : At equilibrium, G is a minimum  for spontaneous changes See Reichl §2.F

31. T, P fixed  for spontaneous changes  At coexistence so that N A can assume any value between 0 & N. Coexistence curve in P-T plane is given by where Actual N A assumed is determined by U ( via latent heat of vaporization ).   

32.  Clausius-Clapeyron eq. ( for 1 st order transitions ) Latent heat per particle. Prob. 4.11, 4.14-6. At triple point  Slopes are related since Prob. 4.17.