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Subsections
Two macroscopic states A and B. Suppose S(A)>S(B). What can we
say about probabilities?
All microscopic states in microcanonical ensemble are equal.
Probability is proportional to the number of states.

The state with greater entropy is more probable
Thermodynamic limit:
,
,

In thermodynamic limit we can see only the states with maximal
entropy .
- Closed systems:
-
- in equilibrium:

- out of equilibrium:

If we look at a closed system, it will be either in equilibrium, or
equilibrating.
- Consequence:
- In all processes in closed systems entropy never
decreases (Clausius-Boltzmann).
- Reversible processes:

- Irreversible processes:

Equations of mechanics are time reversible . How can
irreversibility appear in the picture?
Glass falls from a table an is broken. All molecular motions are
reversible. What about irreversibility??
- Answer:
- Entropy is not a property of a microscopic state, it is
a property of ensemble . Irreversibility is in our
macro scopic eye!
Broken glass--many states, non-broken glass--one state.
Happy families are all alike; every unhappy family is unhappy in its
own way
.
- Definition:
- Temperature is the inverse of the derivative of
entropy by energy:

- Theorem:
- In equilibrium all parts of a closed system have the same
temperature.
- Proof:
- Consider a reversible process in a closed system:
Change of entropy:

or

We discussed systems that do not move. What happens if it moves
with the velocity v?
Entropy depends only on the internal energy:

What happens if a system separates into two parts that move apart?

If T<0, entropy increases --the process is favorable. Systems
with negative temperatures would want to blast off!
In fully equilibrated systems T>0. But partially
equilibrated systems might have negative T for some degrees
of freedom.
Next: Canonical Ensemble
Up: Probabilities of Macroscopic States.
Previous: Probabilities of Macroscopic States.
© 1997
Boris Veytsman
and Michael Kotelyanskii
Thu Sep 4 21:28:23 EDT 1997