Next: Quiz Up: Maximal Work, Minimal Work Previous: Maximal and Minimal Work

Subsections

# Thermodynamic Stability

## General Condition

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

## Stability and Metastability

There are two kinds of minima--local and global

• 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:

## Concave and Convex Functions

A body at constant pressure and temperature. Equilibrium condition:

What is the equilibrium volume V for the given pressure?

It means that:
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''
We proved, that A is a convex function of volume.

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

we proved that

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

## Once again Legendre Transforms

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

To invert p(x), we need . But this follows from the convexity condition p'(x)=f''(x)>0!
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

## Fluctuations of Volume. Gauss Distribution

We know the average volume of a given body V0. What is the probability to obtain V?

We start from the formula:

We obtain:

Since

we are left with

Isothermic compressibility:
We define

This is an intensive variable (why?).
Then

and

This is called Gauss Distribution

Average values:

and

Once again,

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

Next: Quiz Up: Maximal Work, Minimal Work Previous: Maximal and Minimal Work

© 1997 Boris Veytsman and Michael Kotelyanskii
Fri Sep 12 00:09:21 EDT 1997