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Subsections


Spectral Representation

Definitions

We applied Fourier transform for space-dependent problems--and we were successful. Let us do the same for time-dependent problems:  
 \begin{displaymath}
 x_\omega = \int_{-\infty}^{\infty} x(t)e^{i\omega t}\,dx\end{displaymath} (1)
Inverse transform:

\begin{displaymath}
x(t) = \int_{-\infty}^{\infty} x_\omega e^{-i\omega t}\,\frac{d\omega}{2\pi}\end{displaymath}

Correlation Function

The product of x's is

\begin{displaymath}
\left\langle x(t)x(t')\right\rangle=\left\langle \iint_{-\in...
 ...\omega't')}\,
 \frac{d\omega\,d\omega'}{(2\pi)^2}\right\rangle \end{displaymath}

Or, since only x's are random,

\begin{displaymath}
\left\langle x(t)x(t')\right\rangle=\iint_{-\infty}^{\infty}...
 ...{-i(\omega t +\omega't')}\,
 \frac{d\omega\,d\omega'}{(2\pi)^2}\end{displaymath}

Symmetry with respect to time: $\left\langle x(t)x(t')\right\rangle$ depends only on t-t'. This means that in the integral $\omega t
+\omega't'\to\omega(t-t')$. This means that $\omega'=-\omega$.
Conclusion:
$\left\langle x_{\omega}x_{\omega'}\right\rangle$ should be proportional to $\delta$-function $\delta(\omega+\omega')$

Definition:
Let  
 \begin{displaymath}
 \left\langle x_{\omega}x_{\omega'}\right\rangle=2\pi\varphi_\omega\delta(\omega+\omega')
 \end{displaymath} (2)
Then

\begin{displaymath}
\varphi(t)=\int_{-\infty}^{\infty} \varphi_\omega e^{-i\omega
 t}\,\frac{d\omega}{2\pi} \end{displaymath}

We see that $\varphi_\omega$ is indeed the Fourier transform of $\varphi(t)$. This is exactly the function measured in inelastic scattering, relaxation experiments (NMR, ESR, dielectric spectroscopy, shear rheometry,...). Inverse:

\begin{displaymath}
\varphi_\omega = \int_{-\infty}^{\infty} \varphi(t)e^{i\omega t}\,dx\end{displaymath}

If t=0, we obtain

\begin{displaymath}
\varphi(0)=\int_{-\infty}^{\infty} \varphi_\omega \,\frac{d\omega}{2\pi} \end{displaymath}

But $\varphi(0) = \left\langle x^2(0)\right\rangle=kT/\gamma$. We obtained:

\begin{displaymath}
\int_{-\infty}^{\infty}\varphi_\omega \,\frac{d\omega}{2\pi} =
 \frac{kT}{\gamma} \end{displaymath}

Relaxation in Fourier Space

We obtained for correlation function

\begin{displaymath}
\varphi(t)=\frac{kT}{\gamma}e^{-\lambda \left\lvert t\right\rvert}\end{displaymath}

Take Fourier transform:  
 \begin{displaymath}
 \varphi_{\omega} = \int_{\infty}^{\infty}\frac{kT}{\gamma}e...
 ...ght\rvert}\,dt = \frac{2\lambda kT}{\gamma(\lambda^2+\omega^2)}\end{displaymath} (3)
This is a Lorentzian!

Relaxation with one characteristic time is Lorentzian.

Many characteristic times--more complex formulae.

Example: Velocity Autocorrelation Function

Consider a large particle surrounded by small particles (a colloidal particle among molecules). Equation for average velocity:

\begin{displaymath}
\frac{d\left\langle \mathbf{v}\right\rangle}{dt}=-\lambda\left\langle \mathbf{v}\right\rangle\end{displaymath}

with friction coefficient $\lambda$. For a spherical particle with radius r and mass m in liquid with viscosity $\eta$ 
 \begin{displaymath}
 \lambda=6\pi\eta r/m\end{displaymath} (4)
The average square[*]

\begin{displaymath}
\left\langle \mathbf{v}^2(0)\right\rangle = \frac{3kT}{m}\end{displaymath}

We immediately obtain autocorrelation function: Equations (3) and (5) are special cases of fluctuation-dissipation theorem.

In particular, if $\omega=0$

\begin{displaymath}
\begin{gathered}
 \varphi_{\omega=0} = \frac{2 kT}{\gamma\la...
 ...\frac{3kT}{m\lambda} =
 \frac{kT}{2\pi\eta mr} 
 \end{gathered}\end{displaymath}

Example: Diffusion

Consider particles with concentration c. Diffusion equation with diffusion coefficient D:

\begin{displaymath}
\frac{\partial c(\mathbf{r},t)}{\partial t} = D\nabla^2 c(\mathbf{r},t)\end{displaymath}

Fourier transform in space:

\begin{displaymath}
c(\mathbf{r},t) = \sum_{\mathbf{k}} c_{\mathbf{k}}e^{i\mathbf{k}\mathbf{r}}\end{displaymath}

We obtain:

\begin{displaymath}
\frac{\partial c_{\mathbf{k}}(t)}{\partial t} = -D\mathbf{k}^2 c_{\mathbf{k}}(t)\end{displaymath}

Let us now consider equilibrium. In equilibrium the fluid is homogeneous. For $\mathbf{k}\ne0$:

\begin{displaymath}
\left\langle c_{\mathbf{k}}\right\rangle=0,\quad
 \left\lang...
 ...ne\mathbf{k}'\\  kT/\gamma,&\mathbf{k}=\mathbf{k}'
 \end{cases}\end{displaymath}

with $\gamma=\partial^2A/\partial c^2$.

We immediately obtain:

\begin{displaymath}
\left\langle c_{\mathbf{k}}(0)c_{\mathbf{k}}(t)\right\rangle=\frac{kT}{\gamma}e^{-D\mathbf{k}^2t}\end{displaymath}

Or, in Fourier space

\begin{displaymath}
S(\mathbf{k},w) = \frac{2D\mathbf{k}^2 kT}{\gamma(D^2\mathbf{k}^4+\omega^2)}\end{displaymath}

This is dynamic structure factor.


next up previous
Next: Random Force and Its Up: Time Correlation Functions. Random Previous: Time Correlation Functions. Random

© 1997 Boris Veytsman and Michael Kotelyanskii
Tue Oct 28 22:16:24 EST 1997