Lattice density functional theory (LDFT) is a statistical theory used in
physics and
thermodynamics to model a variety of physical phenomena with simple
lattice equations.
Description
Lattice models with nearest-neighbor interactions have been used extensively to model a wide variety of systems and phenomena, including the lattice gas, binary liquid solutions, order-disorder
phase transitions,
ferromagnetism, and
antiferromagnetism.[1] Most calculations of
correlation functions for nonrandom configurations are based on statistical mechanical techniques, which lead to equations that usually need to be solved numerically.
In 1925,
Ising[2] gave an exact solution to the one-dimensional (1D) lattice problem. In 1944
Onsager[3] was able to get an exact solution to a two-dimensional (2D) lattice problem at the critical density. However, to date, no three-dimensional (3D) problem has had a solution that is both complete and exact.[4] Over the last ten years, Aranovich and Donohue have developed lattice density functional theory (LDFT) based on a generalization of the Ono-Kondo equations to three-dimensions, and used the theory to model a variety of physical phenomena.
The theory starts by constructing an expression for
free energy, A=U-TS, where
internal energy U and
entropy S can be calculated using
mean field approximation. The grand potential is then constructed as Ω=A-μΦ, where μ is a
Lagrange multiplier which equals to the
chemical potential, and Φ is a constraint given by the lattice.
It is then possible to minimize the grand potential with respect to the local density, which results in a mean-field expression for local chemical potential. The theory is completed by specifying the chemical potential for a second (possibly bulk) phase. In an equilibrium process, μI=μII.
^Chen, Y.; Aranovich, G. L.; Donohue, M. D. (2006-04-07). "Thermodynamics of symmetric dimers: Lattice density functional theory predictions and simulations". The Journal of Chemical Physics. 124 (13). AIP Publishing: 134502.
Bibcode:
2006JChPh.124m4502C.
doi:
10.1063/1.2185090.
ISSN0021-9606.
PMID16613456.
^Chen, Y.; Wetzel, T. E.; Aranovich, G. L.; Donohue, M. D. (2008). "Configurational probabilities for monomers, dimers and trimers in fluids". Physical Chemistry Chemical Physics. 10 (38). Royal Society of Chemistry (RSC): 5840–7.
Bibcode:
2008PCCP...10.5840C.
doi:
10.1039/b805241g.
ISSN1463-9076.
PMID18818836.
^Chen, Y.; Aranovich, G. L.; Donohue, M. D. (2007-10-07). "Configurational probabilities for symmetric dimers on a lattice: An analytical approximation with exact limits at low and high densities". The Journal of Chemical Physics. 127 (13). AIP Publishing: 134903.
Bibcode:
2007JChPh.127m4903C.
doi:
10.1063/1.2780159.
ISSN0021-9606.
PMID17919050.
B. Bakhti, "Development of lattice density functionals and applications to structure formation in condensed matter systems". PhD thesis, Universität Osnabrück, Germany.