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1.2 Functional Differentiation

It may be shown that the equilibrium density, as a function of $\vec{r}$, is given by
\begin{displaymath}
\rho_{eq}(\vec{r})= - \frac{\delta \Omega [u]}{\delta u(\vec{r})}
\end{displaymath} (1.5)

Proof? $\rightarrow$ 5

Similarly, for the density correlations we have
\begin{displaymath}
-kT \frac{\delta^{2} \Omega}{\delta u(\vec{1}) \delta u(\vec...
...= <\rho(\vec{1}) \rho(\vec{2})>-<\rho(\vec{1})><\rho(\vec{2})>
\end{displaymath} (1.6)

etc. (subscript $(..)_{\textstyle eq}$ omitted).

Reminder: $<\rho(\vec{1}) \rho(\vec{2})>$ is proportional to the pair correlation function $g(\vec{1},\vec{2})$.

Helmholtz Free Energy as a Density Functional:

Defining
\begin{displaymath}
{\rm\cal{F}}[\rho] \equiv \Omega [u] + \int d\vec{r} u(\vec{r}) \rho(\vec{r})
\end{displaymath} (1.7)

(functional Legendre transform) and
\begin{displaymath}
{\rm\cal{F}}_{ex}[\rho] \equiv
{\rm\cal{F}}[\rho]-{\rm\cal{F...
... \rho(\vec{r})
\left[ \ln (\rho(\vec{r}) \lambda^{3})-1\right]
\end{displaymath} (1.8)

we find a complementary hierarchy of correlation functions:
\begin{displaymath}
c^{(1)}(\vec{1})=
-\beta \frac{\delta {\rm\cal{F}}_{ex}[\rho]}{\delta \rho(\vec{1})}
\end{displaymath} (1.9)


\begin{displaymath}
c^{(2)}(\vec{1},\vec{2})=-\beta
\frac{\delta^{2}{\rm\cal{F}}_{ex}[\rho]}{\delta \rho(\vec{1})\delta\rho(\vec{2})}
\end{displaymath} (1.10)

etc., where $c^{(n)}$ is the $n$-th direct correlation function.

Reminder: $c^{(2)}$ is the direct pair correlation function which is related to the net pcf $h \equiv g-1$ via the Ornstein-Zernike equation.

For some homogeneous systems we know $c^{(2)}$ as a function of density. It is reasonable to take advantage of this knowledge in some manner (see below).

The direct one-particle pcf $c^{(1)}$ vanishes for the homogeneous ideal gas; for interacting fluids it defines the chemical excess potential, both for homogeneous and inhomogeneous systems.

Theorems of Hohenberg-Mermin-Kohn:

The following theorems are at the basis of density functional method:
next up previous
Next: 1.3 Basic Strategy of Up: 1. Basics Previous: 1.1 Functionals in Statistical
Franz J. Vesely Oct 2001
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001