Classical density functional theory: The local density approximation

Author:

Jex Michal1ORCID,Lewin Mathieu2ORCID,Madsen Peter S.3ORCID

Affiliation:

1. Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 11519 Prague, Czech Republic

2. CNRS & CEREMADE, Université Paris-Dauphine, PSL University, 75016 Paris, France

3. Department of Mathematics, LMU Munich, Theresienstrasse 39, 80333 Munich, Germany

Abstract

We prove that the lowest free energy of a classical interacting system at temperature [Formula: see text] with a prescribed density profile [Formula: see text] can be approximated by the local free energy [Formula: see text], provided that [Formula: see text] varies slowly over sufficiently large length scales. A quantitative error on the difference is provided in terms of the gradient of the density. Here [Formula: see text] is the free energy per unit volume of an infinite homogeneous gas of the corresponding uniform density. The proof uses quantitative Ruelle bounds (estimates on the local number of particles in a large system), which are derived in an appendix.

Funder

HORIZON EUROPE European Research Council

Ministry of Education, Youth and Sport of the Czech Republic

Publisher

World Scientific Pub Co Pte Ltd

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