Subsections

• One Variable
  • Legendre Transforms and Conjugated Variables
  • Probability of a Fluctuation
  • Gaussian Distribution
  • Mean Values
• Example: Landau Theory for Homogeneous System
• Case of Many Variables
  • Problem
  • Matrix form
  • Linear Transformations
  • Diagonalization
  • Return to Original Coordinates
  • Recipe for Multi-Variable Problem

Fluctuations in Homogeneous Systems

One Variable

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?

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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: $A(x)+A_{\text{env}}(x_{\text{env}})\to\min$. We use Lagrange multiplier and obtain

$$G(x)=A(x)-Xx\to\min$$

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:

$$\Prob(x)\sim\exp\bigl(S(x)/k+S_{\text{env}}(x)/k\bigr)$$

or, making Legendre transform:  

$$\Prob(x)\sim\exp\bigl(-G(x)/kT\bigr)$$ (1)

Expansion: G(x) has a minimum at $x_0=\left\langle x\right\rangle$. So

$$G(x)=G(x_0)+\frac12\gamma(x-x_0)^2,\quad\gamma\gt$$

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

$$\gamma = \frac{\partial^2 A}{\partial x^2} = \frac{\partial X}{\partial x}$$

Sometimes $\gamma^{-1}$ is called generalized susceptibility. If x is volume (density fluctuations), $\gamma^{-1}$ is compressibility. If x is entropy, $\gamma^{-1}T$ is specific heat, etc.

Gaussian Distribution

We obtained:

$$\Prob(x)\sim\exp\bigl(-\gamma(x-x_0)^2/2kT\bigr)$$

Normalization:

$$\int_{-\infty}^{\infty}\Prob(x)\,dx = 1$$

or

$$\Prob(x)\,dx = \sqrt{\frac{\gamma}{2\pi kT}}\exp\left(-\gamma(x-x_0)^2/2kT\right)\, dx$$

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

$$\Prob(x)\,dx = \sqrt{\frac{\gamma}{2\pi kT}}\exp\left(-\gamma x^2/2kT\right)\, dx$$

Mean Values

Averages are:

$$\begin{aligned} \left\langle x\right\rangle &= \int_{-\inft... ...}^{\infty} \Prob(x)x^2\,dx = \frac{kT}{\gamma} \end{aligned}$$

Example: Landau Theory for Homogeneous System

Free energy:

$$A = A_0+V\left[aM^2+dM^4-HM\right]$$

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

$$\gamma = 2aV$$

and  

$$\left\langle M^2\right\rangle=\frac{kT}{\gamma}=\frac{kT}{2V\alpha(T-T_c)}$$ (2)

In the critical point fluctuations diverge. Note 1/V dependence.

Case of Many Variables

Problem

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

$$\left\langle x_i\right\rangle = 0$$

The expansion becomes  

$$G = G_0 + \frac12\sum_{i,j=1}^n \gamma_{ij} x_i x_j$$ (3)

with $\gamma_{ij}=\gamma_{ji}$ (Why?). This is a quadratic form.

Matrix form

Let

$$\mathsf{x}= \begin{Vmatrix} x_1\ x_2\ \hdotsfor{1}\ ... ...},\quad \Gamma = \begin{Vmatrix} \gamma_{ij} \end{Vmatrix}$$

Then

$$G = G_0 + \frac12\mathsf{x}^{ \dag }\Gamma\mathsf{x}$$

Linear Transformations

Let us make a linear transformation:

x_i = t_ik y_k

Then

$$G=G_0+\frac12\sum_{i,k=1}^n \gamma'_{ik}y_i y_k$$

with

$$\gamma'_{ik} = \sum_{l,m=1}^n t_{li} \gamma_{lm} t_{mk}$$

In matrix form:

$$\mathsf{x} = \mathsf{T}\mathsf{y}$$

and

$$G = G_0 + \frac12 \mathsf{y}^{ \dag }\Gamma'\mathsf{y},\quad \Gamma'=\mathsf{T}^{ \dag }\Gamma\mathsf{T}$$

Diagonalization

Mathematicians prove, that symmetric matrix $\Gamma$ can be diagonalized: there exists matrix $\mathsf{T}$ such as

$$\Gamma'=\mathsf{T}^{ \dag }\Gamma\mathsf{T} = \begin{Vmat... ...& 0\ \hdotsfor{4}\ 0 & 0 & \dots & \lambda_n \end{Vmatrix}$$

with real $\lambda_i$.

The quadratic form (3) becomes

$$G = G_0 + \frac12\sum_{i=1}^n \lambda_i y_i^2$$

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

$$\left\langle y_i^2\right\rangle = \frac{kT}{\lambda_i},\quad \left\langle y_i y_j\right\rangle = 0,\, i\ne j$$

The variables y_i are called normal coordinates

Return to Original Coordinates

Let us calculate $\left\langle x_i x_j\right\rangle$:

$$x_i x_j = \sum_{l,m=1}^n t_{il}y_l t_{jm}y_m$$

Averaging this:

$$\left\langle y_i y_j\right\rangle = \begin{cases} kT/\lambda_i, & i=j\ 0, & i\ne j \end{cases}$$

so

$$\left\langle x_i x_j\right\rangle = \sum_{l=1}^n \frac{kTt_{il}t_{lj}}{\lambda_l}$$

Recipe for Multi-Variable Problem

1.

Write down quadratic form

2.

Switch to normal coordinates

3.

Calculate averages

4.

Switch back