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

TL;DR

This paper characterizes the set of $v$-representable densities on a one-dimensional torus at finite temperature, extending previous results to include excited states and more general densities, ensuring maximal $v$-representability.

Contribution

It provides an explicit description of $v$-representable densities at finite temperature on a 1D torus, including excited states, using Sobolev spaces and convexity of functionals.

Findings

Broader set of potentials than $L^p$ spaces including distributions

Guarantees Gâteaux differentiability of the thermal universal functional

Shows the set of $v$-representable densities is maximal

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, G\^ateaux 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.