Subsections

# Entropy and Probabilities

## Principle of Maximal Entropy

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??

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.

## Temperature

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

## Sign of Temperature

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