Subsections

# Fluctuations in Homogeneous Systems

## One Variable

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

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Illustration to the Method of Successive Iterations

### Legendre Transforms and Conjugated Variables

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

with conjugated variable X. G(x) is a Legendre transform of A(x).

### Probability of a Fluctuation

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

or, making Legendre transform:
 (1)
Expansion: G(x) has a minimum at . So

The physical meaning of : since G=A-Xx,

Sometimes is called generalized susceptibility. If x is volume (density fluctuations), is compressibility. If x is entropy, is specific heat, etc.

### Gaussian Distribution

We obtained:

Normalization:

or

Let x0=0 (change zero of the scale!) and

Averages are:

## Example: Landau Theory for Homogeneous System

Free energy:

Let us discuss symmetrical phase above critical point (H=0, a>0). Then

and
 (2)
In the critical point fluctuations diverge. Note 1/V dependence.

## Case of Many Variables

### Problem

Suppose we have many variables xi, i=1, 2,..., n. Let the equilibrium value

The expansion becomes
 (3)
with (Why?). This is a quadratic form.

Let

Then

### Linear Transformations

Let us make a linear transformation:

xi = tik yk

Then

with

In matrix form:

and

### Diagonalization

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

with real .

The variables are completely separated. Each one has independent Gaussian distribution with

The variables yi are called normal coordinates

Let us calculate :

Averaging this:

so

1.