** Next:** Inhomogeneous Systems
**Up:** Fluctuations in Inhomogeneous Systems.
** Previous:** Fluctuations in Inhomogeneous Systems.

We already know the theory of fluctuations of one variable . Just a reminder...

Professor:Do you know this?

Students:No.

Professor:Did you know this last semester?

Students:No.

Professor:Were you taught this last semester?

Students:Yes.

Illustration to the Method of Successive Iterations

Suppose we have one variable *x*. Near equilibrium we construct the
thermodynamic potential *A*(*x*) for a *subsystem*: the free energy
of the subsystem at given *x*. The total free energy of the subsystem
*plus* the environment is minimal:
. We use Lagrange multiplier and
obtain

Probability of a fluctuation is determined by the entropy
of the *whole* system:

(1) |

We obtained:

Normalization: or LetAverages are:

Free energy:

Let us discuss symmetrical phase above critical point ((2) |

Suppose we have *many* variables *x*_{i}, *i*=1, 2,..., *n*. Let
the equilibrium value

(3) |

Let

ThenLet us make a linear transformation:

*x*_{i} = *t*_{ik} *y*_{k}

Mathematicians prove, that *symmetric* matrix can be
diagonalized: there exists matrix such as

The quadratic form (3) becomes

The variables are completely separated. Each one has independent Gaussian distribution with The variablesLet us calculate :

Averaging this: so- 1.
- Write down quadratic form
- 2.
- Switch to normal coordinates
- 3.
- Calculate averages
- 4.
- Switch back

© 1997

Tue Oct 28 22:10:23 EST 1997