# Density-functional theory

This is a set of theories in statistical mechanics that profit from the fact that the free energy of a system can be cast as a functional of the density. That is, the density (in its usual sense of particles per volume), which is a funtion of the position in inhomogeneous systems, uniquely defines the free energy. By minimizing this free energy one arrives at the true free energy of the system and the equilibrium densify function. The situation parallels the better known electronic density functional theory, in which the energy of a quantum system is shown to be a functional of the electronic density (theorems by Hohenberg, Kohn, Sham, and Mermin.)

Starting from this fact, approximations are usually made in order
to approach the true functional of a given system. An important
division is made between *local* and *weighed* theories.
In a local density theory the
in which the dependence is local, as exemplified by the (exact)
free energy of an ideal system:

where is an external potential. It is an easy exercise to show that Boltzmann's barometric law follows from minimization.

An example of a weighed density theory would be the (also exact) excess free energy for a system of 1D hard rods:

where , precisely an average of the density over the length of the hard rods, . "Excess" means "over ideal", i.e., it is the total that is to be minimized.

- Ebner-Saam-Stroud
- Fundamental-measure theory
- Hohenberg-Kohn-Mermin theorems
- Quantum density-functional theory
- Ramakrishnan-Youssouff
- Weighted density approximation
- Tarazona
- Dynamical density-functional theory

## Interesting reading

- Robert Evans "Density Functionals in the Theory of Nonuniform Fluids", in "Fundamentals of Inhomogeneous Fluids" (ed. D. Henderson). Marcel Dekker.
- Robert G. Parr "Density Functional Theory", Annual Review of Physical Chemistry
**34**pp. 631-656 (1983) - C. Ebner, H. R. Krishnamurthy and Rahul Pandit "Density-functional theory for classical fluids and solids", Physical Review A
**43**pp. 4355 - 4364 (1991)