Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

v-Representability on a one-dimensional torus at elevated temperatures

Sarina M. Sutter, Markus Penz, Michael Ruggenthaler, Robert van Leeuwen, Klaas J. H. Giesbertz

TL;DR

This work characterizes v-representable densities for thermal density-functional theory of $N$ fermions on the one-dimensional torus at finite temperature. By employing the $H^1$ topology and a constrained-search framework for the thermal universal functional $F^\beta_{DM}$, the authors prove convexity and Gâteaux differentiability on the strictly positive densities $\mathscr{X}_{>0}$, and establish a precise equivalence between v-representability and a non-empty subdifferential. They show that every density arising from a Gibbs state is strictly positive and that any v-representable density corresponds to a unique distributional potential in $\mathscr{X}^*$, thereby closing the circle between density positivity and representability. The results provide a rigorous foundation for thermal DFT on a 1D torus with general interactions and highlight the necessity of distributional potentials; they also discuss potential extensions to grand-canonical ensembles and higher dimensions.

Abstract

We extend a previous result [Sutter et al., J. Phys. A: Math. Theor. 57, 475202 (2024)] to give an explicit form of the set of $v$-representable densities on the one-dimensional torus with any fixed number of particles in contact with a heat bath at finite temperature. The particle interaction has to satisfy some mild assumptions but is kept entirely general otherwise. For densities, we consider the Sobolev space $H^1$ and exploit the convexity of the functionals. This leads to a broader set of potentials than the usual $L^p$ spaces and encompasses distributions. By including temperature and thus considering all excited states in the Gibbs ensemble, Gâteaux differentiability of the thermal universal functional is guaranteed. This yields $v$-representability and it is demonstrated that the given set of $v$-representable densities is even maximal.

v-Representability on a one-dimensional torus at elevated temperatures

TL;DR

This work characterizes v-representable densities for thermal density-functional theory of fermions on the one-dimensional torus at finite temperature. By employing the topology and a constrained-search framework for the thermal universal functional , the authors prove convexity and Gâteaux differentiability on the strictly positive densities , and establish a precise equivalence between v-representability and a non-empty subdifferential. They show that every density arising from a Gibbs state is strictly positive and that any v-representable density corresponds to a unique distributional potential in , thereby closing the circle between density positivity and representability. The results provide a rigorous foundation for thermal DFT on a 1D torus with general interactions and highlight the necessity of distributional potentials; they also discuss potential extensions to grand-canonical ensembles and higher dimensions.

Abstract

We extend a previous result [Sutter et al., J. Phys. A: Math. Theor. 57, 475202 (2024)] to give an explicit form of the set of -representable densities on the one-dimensional torus with any fixed number of particles in contact with a heat bath at finite temperature. The particle interaction has to satisfy some mild assumptions but is kept entirely general otherwise. For densities, we consider the Sobolev space and exploit the convexity of the functionals. This leads to a broader set of potentials than the usual spaces and encompasses distributions. By including temperature and thus considering all excited states in the Gibbs ensemble, Gâteaux differentiability of the thermal universal functional is guaranteed. This yields -representability and it is demonstrated that the given set of -representable densities is even maximal.

Paper Structure

This paper contains 10 sections, 16 theorems, 54 equations, 1 table.

Table of Contents

  1. Introduction
  2. One-dimensional torus setting
  3. Helmholtz free energy and Gibbs state
  4. Relation between $v$-representability and non-empty subdifferential
  5. Existence of the Minimum
  6. Closing the Circle
  7. Gâteaux differentiability
  8. Summary and outlook
  9. Proof of Proposition \ref{['prop: minimizer of Helmhotz functional/grand potential']}
  10. Proof of Proposition \ref{['prop: Gibbs state density in densset']}

Key Result

Theorem 1

Define a set of densities and a space of (distributional) one-body potentials on the one-dimensional torus $\mathbb{T}$, Then for every $\rho\in\mathscr{X}_{>0}$ there is a unique potential $v$ with equivalence class $[v]\in\mathscr{X}^*$, such that $\rho$ is the density of the Gibbs state (see Proposition prop: minimizer of Helmhotz functional/grand potential) corresponding to the self-adjoint H

Theorems & Definitions (28)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Remark 1
  • Proposition 4
  • Proposition 5
  • Lemma 6
  • proof
  • Theorem 7
  • Lemma 8: lami2023attainability, Lemma 4
  • ...and 18 more