Existence or nonexistence of grand-potential minimizers

Determine whether states minimizing the positive-temperature reduced Hartree–Fock grand potential exist in the absence of mass-noncriticality and energy-noncriticality assumptions, or prove that such minimizers do not exist in the critical cases.

Background

The main results establish semiclassical convergence and convergence of the mean-field potentials along minimizing sequences for a broad class of entropy functions. The analysis deliberately does not assume that the system is noncritical with respect to particle number or energy.

Although the paper identifies a unique limiting mean-field potential along every minimizing sequence, it does not establish whether the minimizing sequence itself converges to an admissible state attaining the grand-potential infimum. Thus the attainment question remains unresolved in the critical setting.

References

A key objective of this paper is to prove Theorems 1--3 without assuming mass non-criticality or energy non-criticality. In this setting, we are unable to show the existence (or lack thereof) of states that minimize the grand potential but we do show that along a minimizing sequence $\kappa3 w \ast \den \gamma_n \to \murHF_\kappa$ for some unique $\murHF_{\kappa}$.

— A semiclassical limit of reduced Hartree-Fock theory at positive temperature  (2608.14436 - Shillingford, 14 Aug 2026) in Section “Overview of Finite Temperature Phenomena”