1. The Grand Canonical Ensemble

2. Topics
Equilibrium between a System & a Particle-Energy Reservoir
A System in the Grand Canonical Ensemble
Physical Significance of Various Statistical Quantities
Examples

3. Topics
Density & Energy Fluctuations in the Grand Canonical Ensemble: Correspondence with Other Ensembles
Thermodynamic Phase Diagrams
Phase Equilibrium & the Clausius-Clapeyron Equation

4. Assume that A2 , is an energy & particle reservoir for A1.
Consider 2 macroscopic systems A1, A2, interacting & in equilibrium. The combined system
A0 = A1 + A2 is isolated. A2 A1
Assume that A2 , is an energy & particle reservoir for A1.
The following discussion is analogous to that in our undergrad stat mech review which found the conditions for equilibrium, in terms of the number of accessible states (E,x), between A1 & A2 , were:

5. The conditions for equilibrium, in terms of the number of accessible states (E,x), between A1 & A2 , were: A2 A1
The 2 temperature parameters  are equal:
2. The 2 mean generalized forces X are equal:
X1 = X2, X = kBT[∂ln/∂x]

6.   -[/kBT] where   Chemical Potential. Here   [∂ln/∂N]
When A1 & A2 exchange particles, it can be shown
that another condition for equilibrium is that 1 = 2 A2 A1
Here   [∂ln/∂N]
It can also be shown that
  -[/kBT]
where   Chemical Potential.
So, when the 2 systems are in equilibrium, their chemical potentials are equal: 1 = 2

7. Equilibrium between a System & a Particle-Energy Reservoir
For a system A interacting with particle &
energy reservoir A:
Probability distribution:

8. For system A interacting with particle & energy
reservoir A:
Probability distribution:
Using 1st Law:

9. A, A in equilibrium 
So:
So: 

10. Mean Values
Classical Stat Mech (Gibbs –corrected ):

11. Grand Potential (Helmholtz Free Energy) 
Prob 4.2