Coarse-grained spin density-functional theory: Infinite-volume limit via the hyperfinite

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Abstract

Coarse-grained spin density functional theory (SDFT) is a version of SDFT which works with number/spin densities specified to a limited resolution - averages over cells of a regular spatial partition - and external potentials constant on the cells. This coarse-grained setting facilitates a rigorous investigation of the mathematical foundations which goes well beyond what is currently possible in the conventional formulation. Problems of existence, uniqueness, and regularity of representing potentials in the coarse-grained SDFT setting are here studied using techniques of (Robinsonian) nonstandard analysis. Every density which is nowhere spin-saturated is V-representable, and the set of representing potentials is the functional derivative, in an appropriate generalized sense, of the Lieb internal energy functional. Quasi-continuity and closure properties of the set-valued representing potentials map are also established. The extent of possible non-uniqueness is similar to that found in non-rigorous studies of the conventional theory, namely non-uniqueness can occur for states of collinear magnetization which are eigenstates of Sz.

Original languageEnglish (US)
Article number062104
JournalJournal of Mathematical Physics
Volume54
Issue number6
DOIs
StatePublished - Jun 3 2013

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Density Functional
density functional theory
Nonuniqueness
Quasi-continuity
Nonstandard Analysis
Closure Properties
Cell
Collinear
Energy Functional
uniqueness
cells
internal energy
regularity
continuity
Magnetization
closures
partitions
eigenvectors
Existence and Uniqueness
Regularity

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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Coarse-grained spin density-functional theory : Infinite-volume limit via the hyperfinite. / Lammert, Paul E.

In: Journal of Mathematical Physics, Vol. 54, No. 6, 062104, 03.06.2013.

Research output: Contribution to journalArticle

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