Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number 03–22 MATRIX INEQUALITIES IN STATISTICAL MECHANICS

Abstract: Some matrix inequalities used in statistical mechanics are presented. A straightforward proof of the Thermodynamic Inequality is given and its equivalence to the Peierls–Bogoliubov inequality is shown. 1. Golden–Thompson Inequality One of the earlier inequalities involving traces of matrices applied to sta-tistical mechanics is the Golden–Thompson inequality. In 1965, Golden [8], Symanzik [17], and C. Thompson [18], independently proved that tr (eA+B) ≤ tr (eAeB) (1.1) holds when A and B are Hermitian matrices. ¿From (1.1) Thompson derived a convexity property that was used to obtain an upper bound for the partition function of an antiferromagnetic chain tr (e−H/Θ), where H, a Hermitian op-erator, is the Hamiltonian of the physical system, and Θ = kT where k is the Boltzmann constant and T is the absolute temperature. Golden [8] obtained lower bounds for the Helmholtz free-energy function for a system in statistical or thermodynamic equilibrium. The Helmholtz free-energy function is given by F = −Θ log tr (e−H/Θ). Indeed, for any partition of the Hamiltonian H = H1+H2, the exponential can be represented by the well-known Lie–Trotter formula (for a proof see, for example, [5] or [20]) e−H/Θ = lim n→∞(e −H1/nΘe−H2/nΘ)n. (1.2) Since the exponential of a Hermitian matrix is a positive definite matrix, and recalling the following inequality for positive definite matrices A and B (see, e.g., [8]) tr(AB)2 p+1 ≤ tr(A2B2)2p, p a non-negative integer, (1.3