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- General Condition
- Stability and Metastability
- Concave and Convex Functions
- Once again Legendre Transforms
- Fluctuations of Volume. Gauss Distribution

The system is stable if the *total* entropy has
*maximum*. Equivalently, is positive for any
deviation.

It means that:

- At ,
*A*should be minimal - At ,
*G*should be minimal - At ,
*H*should be minimal

- If the system is neither in a global, nor in a local minimum, it
is
*unstable* - If the system is in the global minimum, it is
*stable* - If the system is in the local minimum, it is
*metastable:*it can spend a lot of time there, but eventually it will leave for the global minimum. This is called*partial equilibrium.*

Glasses--many metastable minima:

A body at constant pressure and temperature. Equilibrium condition:

What is the equilibrium volume *V* for the given pressure?

- 1.
- First derivative should be zero:
This is
*mechanical equilibrium condition* - 2.
- Second derivative should be positive:
This is
*convexity*condition.

**Definition:**- A function
*f*(*x*) is*convex*if*f*''(*x*)>0 and*concave*if*f*''(*x*)<0:

**Mnemonics:**- Concave = ``no coffee''

If , we can make this point *V* a
minimum by changing *P*. If , we
cannot help! Since

**General Statement:**- Thermodynamic potentials are
*convex*functions of their extensive arguments.

We now can prove that Legendre Transform of thermodynamic potentials is always possible!

To invert**Theorem:**- Thermodynamic potentials are
*concave*functions of their*intensive*variables. **Proof:**- If
*g*(*p*) is the Legendre transformation of*f*(*x*), then*g*'(*p*)=-*x*and

We know the average volume of a given body *V _{0}*. What is the
probability to obtain

We start from the formula:

We obtain:Since

we are left with**Isothermic compressibility:**- We define
This is an
*intensive*variable*(why?)*.

Average values:

and Once again,Once again maximal term method: at the width becomes zero!

© 1997

Fri Sep 12 00:09:21 EDT 1997