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This is a set of theories in statistical mechanics that profit from the | |||
fact that 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 | the density. That is, the density (in its usual sense of particles | ||
per volume), which is a | per volume), which is a funtion of the position in inhomogeneous systems, | ||
uniquely defines the | uniquely defines the free energy. By minimizing this free energy one | ||
arrives at the true | arrives at the true free energy of the system and the equilibrium | ||
densify function. The situation | |||
parallels the better known electronic density functional theory, | parallels the better known electronic density functional theory, | ||
in which the energy of a quantum system is shown to be a functional | in which the energy of a quantum system is shown to be a functional | ||
of the electronic density ( | of the electronic density (theorems by Hohenberg, Kohn, Sham, and Mermin.) | ||
Starting from this fact, approximations are usually made in order | Starting from this fact, approximations are usually made in order | ||
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In a local density theory the | In a local density theory the | ||
in which the dependence is local, as exemplified by the (exact) | in which the dependence is local, as exemplified by the (exact) | ||
free energy of an ideal system: | |||
<math>F_{id}=kT\int dr \rho(r) [\log \rho(r) -1 -U(r)],</math> | |||
where <math>U(r)</math> is an external potential. It is an easy exercise | where <math>U(r)</math> is an external potential. It is an easy exercise | ||
to show that | to show that Boltzmann's barometric law follows from minimization. | ||
An example of a weighed density theory would be the | An example of a weighed density theory would be the | ||
(also exact) excess | (also exact) excess free energy for a system | ||
of | of 1D hard rods: | ||
<math>F_{ex}=-kT\int dz \rho(z) \log [1-t(z)],</math> | |||
where <math>t(z)=\int_{z-\sigma}^z dy \rho(y)</math>, | where <math>t(z)=\int_{z-\sigma}^z dy \rho(y)</math>, | ||
precisely an average of the density over the length of | precisely an average of the density over the length of | ||
the hard rods, <math>\sigma</math>. "Excess" means "over | the hard rods, <math>\sigma</math>. "Excess" means "over | ||
ideal", i.e., it is the total <math> | ideal", i.e., it is the total <math>F=F_{id}+F_{ex}</math> | ||
that is to be minimized. | that is to be minimized. | ||
==See also== | ==See also== | ||
*[[ | *[[Dynamical density-functional theory]] | ||
*[[Ebner-Saam-Stroud]] | *[[Ebner-Saam-Stroud]] | ||
*[[Fundamental-measure theory]] | *[[Fundamental-measure theory]] | ||
*[[Hohenberg-Kohn-Mermin theorems]] | *[[Hohenberg-Kohn-Mermin theorems]] | ||
*[[Kierlik and Rosinberg's weighted density approximation]] | |||
*[[Quantum density-functional theory]] | *[[Quantum density-functional theory]] | ||
*[[Ramakrishnan-Youssouff]] | *[[Ramakrishnan-Youssouff]] | ||
*[[Tarazona's weighted density approximation]] | |||
*[[Weighted density approximation]] | *[[Weighted density approximation]] | ||
==Interesting reading== | ==Interesting reading== | ||
#Robert Evans "Density Functionals in the Theory of Nonuniform Fluids", in "Fundamentals of Inhomogeneous Fluids" (ed. D. Henderson). Marcel Dekker. | |||
#[http://dx.doi.org/10.1146/annurev.pc.34.100183.003215 Robert G. Parr "Density Functional Theory", Annual Review of Physical Chemistry '''34''' pp. 631-656 (1983)] | |||
#[http://dx.doi.org/10.1103/PhysRevA.43.4355 C. Ebner, H. R. Krishnamurthy and Rahul Pandit "Density-functional theory for classical fluids and solids", Physical Review A '''43''' pp. 4355 - 4364 (1991)] | |||
[[category: Density-functional theory]] | [[category: Density-functional theory]] |