Inequality with Applications in Statistical Mechanics

1965/11/01 by Colin J. Thompson · 243 citations
Computer Science · Mathematics · #Matrix Theory and Algorithms #Mathematical Inequalities and Applications #Spectral Theory in Mathematical Physics #Convexity #Mathematics #Hilbert space #Self-adjoint operator #Hermitian matrix #Pure mathematics #Inequality #Partition (number theory) #Property (philosophy) #Statistical mechanics #Mathematical analysis #Combinatorics #Physics #Quantum mechanics

paper · doi:10.1063/1.1704727

published in Journal of Mathematical Physics 6(11), 1812-1813 (American Institute of Physics)

openalex publication_date 1965/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/02

Abstract

We prove for Hermitian matrices (or more generally for completely continuous self-adjoint linear operators in Hilbert space) A and B that Tr (eA+B) ≤ Tr (eAeB). The inequality is shown to be sharper than the convexity property (0 ≤ α ≤ 1) Tr (eαA+(1−α)B) ≤ [Tr (eA)]α[Tr (eB)]1−α, and its possible use for obtaining upper bounds for the partition function is discussed briefly.

Citations

• Remark on Weyl's Note “Inequalities Between the Two Kinds of Eigenvalues of a Linear Transformation”

Cited by

• Generalized Log-Majorization and Multivariate Trace Inequalities
• Bounds on the heat kernel of the Schrödinger operator in a random electromagnetic field
• Weak Quasi-Factorization for the Belavkin-Staszewski Relative Entropy
• Matrix concentration inequalities for dependent binary random variables
• Golden–Thompson inequality [wikipedia]