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Subsections
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
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:
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.
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

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
![\begin{displaymath}
\Prob(V)\sim\exp\left[ - \frac{1}{2kT}\left(\frac{\partial^2 A}{\partial
V^2}\right)_{T}\,(\Delta V)^2\right]\end{displaymath}](img60.gif)
- Isothermic compressibility:
- We define

This is an intensive variable (why?).
Then

and
![\begin{displaymath}
\Prob(V)=\frac{1}{\sqrt{2\pi kTV \alpha_T}}\exp\left[-\frac{(\Delta
V)^2}{2kT V\alpha_T} \right]\,dV\end{displaymath}](img63.gif)
This is called Gauss Distribution
Average values:

and

Once again,

Once again maximal term method: at
the width becomes zero!
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Up: Maximal Work, Minimal Work
Previous: Maximal and Minimal Work
© 1997
Boris Veytsman
and Michael Kotelyanskii
Fri Sep 12 00:09:21 EDT 1997