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The finite-temperature free energy attains a minimum on the density domain

Statement

For the finite-temperature model of Fixed-particle finite-temperature density-matrix free energy, the free energy $F_{β}$ has a global minimizer on $D_{r}$ .

Facts & Assumptions

Given: $n \geq 2$ , $0 < r < n$ , $β > 0$ , and a real $C^{1}$ energy $E$ on a neighborhood of $D_{r}$ .

[F1]

The energy is continuous and the entropy trace is defined on $D_{r}$ (Fixed-particle finite-temperature density-matrix free energy).

[F2]

$D_{r}$ is nonempty and compact (The fixed-particle density-matrix domain is nonempty and compact).

[F3]

The endpoint-extended entropy trace is continuous (The finite-temperature trace entropy is continuous at occupation endpoints).

[F4]

A continuous real-valued function on a nonempty compact metric space attains its minimum (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

Proof

1.1F1F3givenalgebra

The energy $E$ is continuous by [F1], and the entropy trace is continuous by [F3], so their weighted sum $F_{β}$ is continuous on $D_{r}$ .

2.1F2F4step 1.1∎

The domain is nonempty and compact by [F2]. Apply [F4] to $F_{β}$ to obtain a minimizer in $D_{r}$ .

Depends on

Used by

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Dependency tree · two levels

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Sources

Daniel Arovas, Thermodynamics and Statistical Mechanics, §7.4 Variational Density Matrix Method, UC San Diego / Physics LibreTexts (standard reference, not scraped)