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Trajectory--a line in phase space. Many-body systems: a complex
entangled trajectory. *Assumption:* after long time the system
``forgets'' the initial conditions. *Probability of each state does not
depend on where we started!*

What can it depend upon? Energy, total momentum, total volume,...

**Microcanonical Ensemble:**- Closed system all states
have
*same*energy, momentum,... **Gibbs Postulate:**- All states in microcanonical ensemble are equivalent. They have the same probability to occur.

- 1.
- I take one system and observe it for a long time
*(time averaging)*:(1) - 2.
- I take many systems from my ensemble (i.e. with same energy) and
observe them all in one time (ensemble average):
(2) (3)

*Ergodicity hypothesis:* ensemble average is equal to time
average.

Is ergodic hypothesis alway correct?

Gibbs postulate: yes.

*But* how large should be in (1) for the
limit to work? What happens if it is a year? Twenty years? *The
mountains flowed down at Thy presence* ^{}, but we are mortals....

Examples: water is liquid at frequencies , but solid at . Window panes flow in hundred years...

**Ergodic systems:**- is small, and ergodicity works. Gases, liquids, crystals.
**Non-ergodic systems:**- is large, and we cannot wait for equilibrium. Glasses.

Computer simulations: is the simulated system ergodic? Do we study its
*equilibrium* or *transient* features?

Each system in the ensemble is a point in the phase space. How many points are there?

Answer:

- Discrete space--just count them!
- Continuous space: measure the volume! Number of states The cannot be obtained in classical statistics. Quantum answer:

Geometric interpretation: cell model

Entropy:

(4) |

Suppose we increase the volume of our system *V* and the number of
particles *N*.

**Thermodynamic limit ():**- , ,and . This limit is
*always*taken! **Intensive variables:**- A variable
*A*is intensive if Examples: temperature, pressure, density, concentration,... **Extensive variables:**- A variable
*A*is intensive, if Examples: number of particles, volume, energy,...

Entropy of two *non-interacting* subsystems:

The total number of states:

*S* = *S _{1}* +

Let be the probability to meet the given state in the ensemble. Ensemble averaging:

is the(5) |

**Question:**- What is the distribution function for the
*microcanonical*ensemble? **Answer:**- It is zero if , and satisfies (5). Therefore

© 1997

Wed Sep 3 22:59:36 EDT 1997