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Inhomogeneous Systems


In inhomogeneous systems G depends on the value of x in all points $\mathbf{r}$. G is a functional $G\bigl\{x(\mathbf{r})\bigr\}$--it depends on a function $x(\mathbf{r})$.

Analogy with multi variable case:
we have variables $x(\mathbf{r}_1)$,$x(\mathbf{r}_2)$,..., and want to minimize G.
Introduce normal coordinates that diagonalize G
Use Fourier transform--it leads to normal coordinates in linear problems

Landau Hamiltonian and Normal Coordinates

For inhomogeneous system the free energy is

G= G_0 + \int\left[aM^2+dM^4-HM+g\left(\nabla

Assumptions and simplifications:
We are above critical point, so a>0 (for negative a we expand around M0)
We neglect the term dM4, This is the most important approximation!!!

Then we make Fourier transform:

M(\mathbf{r}) = \sum_{\mathbf{k}} M_{\mathbf{k}}e^{i\mathbf{r}\mathbf{k}}, \quad
 M_{-\mathbf{k}} = M_{\mathbf{k}}^{\ast}\end{displaymath}

Quadratic term:


Gradient term:

\nabla M(\mathbf{r}) = \nabla \sum_{\mathbf{k}} M_{\mathbf{k...


\bigl(\nabla M(\mathbf{r})\bigr)^2 = -
 ..._{\mathbf{k}'}e^{i\mathbf{k}\mathbf{r}+i\mathbf{k}'\mathbf{r}} \end{displaymath}

Term with external field:

HM(\mathbf{r}) = H\sum_{\mathbf{k}} M_{\mathbf{k}}e^{i\mathbf{r}\mathbf{k}}\end{displaymath}

Mathematical result

\int_V e^{i\mathbf{k}\mathbf{r}}\,d\mathbf{r}= 
 0,& \mathbf{k}\ne0\  V,& \mathbf{k}=0

Integrals with quadratic & gradient terms contain

\int e^{i\mathbf{k}\mathbf{r}+i\mathbf{k}'\mathbf{r}}\,d\mathbf{r}\end{displaymath}

This is non-zero only if $\mathbf{k}'=-\mathbf{k}$. Then

M_{\mathbf{k}}M_{\mathbf{k}'} = M_{\mathbf{k}}M_{-\mathbf{k}}= \left\lvert M_{\mathbf{k}}\right\rvert^2 \end{displaymath}

and we obtain

G=G_0+V\sum_{\mathbf{k}}(a+g\mathbf{k}^2)\left\lvert M_{\mathbf{k}}\right\rvert^2 -VHM_{\mathbf{0}}\end{displaymath}

We represented free energy as a sum over Fourier components--this means $M_{\mathbf{k}}$ are normal coordinates!

Fluctuations in Fourier Space. Scattering

We immediately obtain for fluctuations

I(\mathbf{k})=\left\langle \left\lvert M_{\mathbf{k}}\right\rvert^2\right\rangle = \frac{kT}{2V(a+gk^2)}\end{displaymath}

This is exactly the function measured in scattering experiments!

What happens near critical points? Here $a=\alpha(T-T_c)$ is small.

Short wavelength fluctuations do not depend on the temperature:

\left\langle \left\lvert M_{\mathbf{k}}\right\rvert^2\right\rangle = \frac{kT}{2Vgk^2},\quad gk^2\gg a

Long wavelength fluctuations grow:

\left\langle \left\lvert M_{\mathbf{k}}\right\rvert^2\right\rangle = \frac{kT}{2V\alpha(T-T_c)}, \quad gk^2\ll

The characteristic wavelength tends to infinity: $k_0 =
This is called critical opalescence

Correlation in Real Space. Ornstein-Zernicke Function

The function

I(\mathbf{r}) = \sum_{\mathbf{k}} I(\mathbf{k}) e^{i\mathbf{r}\mathbf{k}}\end{displaymath}

Is called correlation function. It is just[*]

I(\mathbf{r})= \left\langle M(\mathbf{0})M(\mathbf{r})\right\rangle\end{displaymath}

Integrating $I(\mathbf{k})$, we obtain:

I(r) = \frac{kT_c}{8\pi g r}e^{-r/\xi}\end{displaymath}

with correlation radius

\xi = \sqrt{g/\alpha(T-T_c)}\end{displaymath}

This function is called Ornstein-Zernicke function.

What happens at $T\to T_c$? Correlation radius $\xi\to\infty$

At $r\gg\xi$ we have exponential decay with distance
At $r\ll\xi$ correlations decay as 1/r and are independent of the temperature!

Above we obtained the formula

\left\langle M^2\right\rangle = \frac{kT}{Va}\end{displaymath}

This works in thermodynamic limit $V\to\infty$. In other words $V\gg\xi^3$. Near Tc this condition becomes very stringent!

Ginsburg Number

Landau theory works if fluctuations are small. Average fluctuation in the volume $\xi^3$ is

\left\langle M^2\right\rangle = \frac{kT_c}{\xi^3\left\lvert...
 ...ight\rvert} = \frac{kT\left\lvert a\right\rvert^{1/2}}{g^{3/2}}\end{displaymath}

Comparing this to the equilibrium value below critical point $M_0^2\sim \left\lvert a\right\rvert/2d$, we obtain that $\left\langle M^2\right\rangle\ll M_0^2$ if

\left\lvert a\right\rvert\gg \frac{(kT)^2d^2}{g^3}\end{displaymath}

or, since $a=\alpha(T-T_c)$ 
 \frac{\left\lvert T-T_c\right\rvert}{T_c}\gg \frac{k^2T_cd^2}{\alpha g^3}=\mathsf{Gi}\end{displaymath} (4)
$\mathsf{Gi}$ is Ginsburg number.

Landau theory works, if we are both:

Close to critical point, so $\left\lvert T-T_c\right\rvert/T_c\ll 1$ (we used expansion!)
Still not close enough, so $\left\lvert T-T_c\right\rvert/T_c\gg\mathsf{Gi}$.

Why does Landau theory work well for polymers? Coefficient g does not depend on molecular weight N, but $d\sim1/N$ and $\alpha\sim1/N$. We obtain $\mathsf{Gi}\sim1/N$, i.e. Ginsburg number is small.

next up previous
Next: Universality and Scaling Invariance Up: Fluctuations in Inhomogeneous Systems. Previous: Fluctuations in Homogeneous Systems

© 1997 Boris Veytsman and Michael Kotelyanskii
Tue Oct 28 22:10:23 EST 1997