1. 2. Elements of Ensemble Theory
Phase Space of a Classical System
Liouville’s Theorem & Its Consequences
The Microcanonical Ensemble
Examples
Quantum States & the Phase Space

2. At equilibrium:
Each macrostate represents a huge number of microstates.
 Observed values ~ time averaged over microstates.
Ensemble theory:
Ensemble average = Time average

3. 2.1. Phase Space of a Classical System
Hamiltonian formulism of mechanics:
Each dynamical state of a system of N particles is specified by 3N position coordinates { qi }, and 3N momentum coordinates { pi },
i.e., by a point { qi , pi } in the 6N –D phase space of the system.
Time evolution of the system is described by a path in phase space satisfying
For a conservative system, its phase space trajectory is restricted to the hypersurface
For a system in thermal equilibrium with a heat reservoir, its phase space trajectory is restricted to the “hypershell”

4. Ensemble of a system:
Set of identical copies of a system that includes every distinct state that satisfies the given boundary conditions and gives rise to the given macrostate.
In the thermodynamical limit ( N, V   , N / V = finite ) ,
the phase points of the ensemble form a continuum.
The ensemble average of a dynamical function f is given by
 = density function
An ensemble is stationary if
  f  is t – indep.
Ensemble of a system in equilibrium must be stationary.

5. 2.2. Liouville’s Theorem & Its Consequences
Liouville’s theorem
where the Poisson bracket is defined as :
# of phase points are conserved 
The current density of phase points is a 6N-D vector

6. 
where
Hence, the Liouville eq. is just the equation of continuity of the phase points :
For a system in equilibrium 

7. Solution 1:
  region of acceptible microstates in phase space.
Microcanonical ensemble
Principle of equal a priori probability :
Solution 2: 
Canonical ensemble
Chap 3:

8. 2.3. The Microcanonical Ensemble
  Hypersurface
  Hypershell
 f   ensemble average of f
= time average of  f  for systems in eqm.
=  time average of f  2 av are indep
= long time average of f over 1 ensemble member
= f measured
Let 0  fundamental volume of one microstate. Then

9. 2.4. Examples Microcanonical Classical ideal gas:
Surface area of an n-D sphere (App.C) is  

10. Sec 1.4 : 
Volume of state per degree of freedom = h

11. Single Free Particle Confined to V 

12. Simple Harmonic Oscillator
Ellipse  

13. 2.5. Quantum States & the Phase Space
Phase space volume of 1 quantum state may be estimated from the uncertainty principle:
Among the first to suggest
are Tetrode, Sackur, & Bose.
Bose (black-body radiation) :
for photons
Cf. Rayleigh’s number of modes 
Caution: Above formulae consider only a single polarization component.