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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.

Thermodynamic limit: , ,

In thermodynamic limit we can see only the states with**Closed systems:**-
- in equilibrium:
- out of equilibrium:

**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!

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:

In *fully equilibrated systems* *T*>0. *But* partially
equilibrated systems *might* have negative *T* for some degrees
of freedom.

© 1997

Thu Sep 4 21:28:23 EDT 1997