Consider a system of hard rods of mass
and length in a gravitational field with ,
such that . Given some -dependent density
the excess free energy (EFE) is
(2.7)
The Gibbs potential is
(2.8)
with the free energy of the ideal (1-dimensional) gas,
(2.9)
where denotes the de Broglie length.
To minimize we need
:
(2.10)
(2.11)
:
(2.12)
(2.13)
Putting all this together we find for the equilibrium profile
the implicit relation
Percus and Robledo-Varea have pointed out that
the above expression for
may be rewritten as
(2.15)
with
(2.16)
and
(2.17)
Obviously the thus defined is a density that has been
averaged over one particle diameter,
while is an average over the particle surface.
This suggests a general strategy for approximating
, even when dealing with higher dimensions:
Define two coarse-grained densities and
analogous to the 1-dimensional case
Use some accepted approximation for the excess free energy density
at the density and insert it in the integral
(2.18)
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:3. Approximations toUp:2. Exact Expressions for Previous:2.1 Functional Integration Franz J. Vesely Oct 2001
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001