1. Symmetry

2. Phase Continuity Phase density is conserved by Liouville’s theorem.  Distribution function D  Points as ensemble members Consider as a fluid governed by the continuity equation.  Total time derivative q p

3. Hamilton Applied As before with Liouville’s theorem, apply Hamilton’s equations. The continuity of phase space is expressed in terms of the distribution function and the Hamiltonian.  Poisson bracket

4. Poisson Bracket The Poisson bracket can be defined for any two functions of the variables q, p. Poisson brackets have special properties.  Bilinear  Antisymmetric  Jacobi identity S1S1 {A + B, C} ={A, C} + {B, C} {  A, B} =  {A, B} {A, B} =  {B, A} {A, {B, C}} + {B, {C, A}} + {C, {A, B}} = 0

5. Matrix Form The dynamic variables can be assigned to a single set.  q 1, q 2, …, q n, p 1, p 2, …, p n  z 1, z 2, …, z 2n Hamilton’s equations can be written in terms of z . The Poisson bracket is equivalent to the matrix form.

6. Dynamical Variable A dynamical variable F can be expanded in terms of the independent variables. This can be expressed in terms of the Poisson bracket. The Poisson bracket of F with H gives the dynamics of F.

7. Symmetry A mathematical object that remains invariant under a transformation exhibits symmetry.  Geometric objects  Algebraic objects  Functions AB DC DA CB n an integer

8. Invariance A coordinate transformation changes the Lagrangian. An invariant Lagrangian exhibits symmetry.  Infinitessimal coordinate transformations  Conserved quantities emerge  Energy conservation when time-independent Invariance also emerges from the Poisson bracket.

9. Angular Momentum Example The two dimensional harmonic oscillator can be put in normalized coordinates.  m = k = 1 Find the change in angular momentum l.  It’s conserved

10. Ensemble Average Consider an arbitrary function F and its time derivative.  No explicit time dependence The ensemble average is taken over phase space.  z  for set of coordinates The mean value may depend on time from the density.

11. Mean Derivative The constancy of the density is used to substitute.  Density must vanish at infinity  Integrate by parts  Apply Hamilton’s equations The derivative of the mean is equal to the mean of the derivative.

12. Thermodynamic Equilibrium Dynamic variables in equilibrium are constant in time. The probability density must have no explicit time dependence.  True for phase density Since total derivative is also zero, so is the Poisson bracket.  Phase density is a constant of the motion for the system

13. Virial Component A virial is the product of a generalized force and position.  Single component without summation given  Mean expressed with canonical ensemble

14. Contained System Assume particles in a container.  H large for large z  Exponential damping Evaluate with partial integration. Any coordinate times the gradient of the Hamiltonian in that coordinate gives a mean of kT.  Virial theorem