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2.2 Hard Rods in a Vertical Tube - Shake Hard!

Consider a system of hard rods of mass $m=1$ and length $\sigma$ in a gravitational field with $g=1$, such that $u(z)=\mu-z$. Given some $z$-dependent density $\rho(z) $ the excess free energy (EFE) is
\begin{displaymath}
{\rm\cal{F}}_{ex} [\rho ] =
- \beta^{-1}\int dz   \rho(z) ...
...{\rm with} \;\;\;
t(z) \equiv \int_{z-\sigma}^{z} dy  \rho(y)
\end{displaymath} (2.7)

The Gibbs potential is
\begin{displaymath}
\Omega_{\phi}[\rho] \equiv {\rm\cal{F}}_{id} [\rho ]
+{\rm\cal{F}}_{ex} [\rho ] - \int dz   \rho(z) u(z)
\end{displaymath} (2.8)

with the free energy of the ideal (1-dimensional) gas,
\begin{displaymath}
{\rm\cal{F}}_{id} [\rho ]=
- \beta^{-1}\int \!dz   \rho(z) \ln [\lambda \rho(z) -1]
\end{displaymath} (2.9)

where $\lambda$ denotes the de Broglie length.

To minimize $\Omega_{\phi}$ we need Putting all this together we find for the equilibrium profile $\rho(z) $ the implicit relation
\begin{displaymath}
\beta u(z) = \ln \frac{\lambda \rho(z)}{1-t(z)}
+ \int_{z}^{z+\sigma}dy   \frac{\rho(y)}{1-t(y)},
\end{displaymath} (2.14)

which may be solved by iteration.

Applet HardRodsDFT: Start




An MD simulation of the same system produces the following profile:

Applet HardRodsMD: Start




Percus and Robledo-Varea have pointed out that the above expression for ${\rm\cal {F}}_{ex}[\rho ]$ may be rewritten as
\begin{displaymath}
{\rm\cal{F}}_{ex} [\rho ] = \frac{1}{2}
\int dz \;
\left[\r...
...))
\equiv
\int dz \;
\rho_{\sigma}(z)
\;
\psi(\rho_{\tau}(z))
\end{displaymath} (2.15)

with
\begin{displaymath}
\psi_{ex}(\rho) \equiv - kT \ln (1-\rho \sigma)
\end{displaymath} (2.16)

and
\begin{displaymath}
\rho_{\tau}(z) \equiv \int_{-\sigma/2}^{\sigma/2}
dy \; \rho(z+y)
\end{displaymath} (2.17)

Obviously the $\rho_{\tau}$ thus defined is a density that has been averaged over one particle diameter, while $\rho_{\sigma}$ is an average over the particle surface.

This suggests a general strategy for approximating ${\rm\cal {F}}_{ex}[\rho ]$, even when dealing with higher dimensions:

This and other Weighted Density Approximations are discussed in section 3.3. Before that, two simpler strategies, the Local Density Approximation (3.1) and the Square Gradient Approximation (3.2) will be sketched shortly.
next up previous
Next: 3. Approximations to Up: 2. Exact Expressions for Previous: 2.1 Functional Integration
Franz J. Vesely Oct 2001
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001