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Physical Review Letters

Inequalities for the Free Energy of Certain Many-Body Systems

Peter H. Kleban

Published Mar 24, 1969• 3 citations
Publisher

Abstract

A new application of the Bogoliubov inequality to the compressibility of a quantum or classical nonrelativistic one-component many-body system is derived. For a (two-particle) potential $\frac{\ensuremath{\lambda}}{{r}^{n}}$, the inequality can be integrated to give simple, power-law upper and lower bounds on the ground-state energy as a function of the density. Analogous results are also valid at finite temperatures and for Lennard-Jones potentials at high densities.

DOI
10.1103/physrevlett.22.587

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A new application of the Bogoliubov inequality to the compressibility of a quantum or classical nonrelativistic one-component many-body system is derived. For a (two-particle) potential \ensuremathλrn\frac{\ensuremath{\lambda}}{{r}^{n}}rn\ensuremathλ​, the inequality can be integrated to give simple, power-law upper and lower bounds on the ground-state energy as a function of the density. Analogous results are also valid at finite temperatures and for Lennard-Jones potentials at high densities.
Variation principles for obtaining lower bounds by means of density matrices are applied to a number of many-body problems. This is done by means of restrictions derived from the study of exacly solvable models. Two classes of problems are considered: (a) obtaining a lower bound to the ground-state energy of a system of particles, and (b) obtaining a lower bound to the Helmholtz free energy. For the first case an exact lower bound is obtained for the ground-state energy of the particle-conserving Bogoliubov Hamiltonian. Of particular significance in the nonzero temperature case is a rigorous lower bound to the free energy of the Ising model with small external field at low temperatures.
We generalise the two-sided Bogoliubov inequality for classical particles from [L. Delle Site et al., J.Stat.Mech.Th.Exp. 083201 (2017)] to systems of quantum particles. As in the classical set-up, the inequality leads to upper and lower bounds for the free energy difference associated with the partitioning of a large system into smaller, independent subsystems. From a thermodynamic modelling point of view, the free energy difference determines the finite size correction needed to consistently treat a small system as a representation of a large system. Applications of the bounds to quantify finite size effects are ubiquitous in physics, chemistry, material science, or biology, to name just a few; in particular it is relevant for molecular dynamics simulations in which a small portion of a system is usually taken as representative of the idealized large system.
We present an inequality that gives a lower bound on the expectation value of certain two-body interaction potentials in a general state on Fock space in terms of the corresponding expectation value for thermal equilibrium states of non-interacting systems and the difference in the free energy. This bound can be viewed as a rigorous version of first order perturbation theory for many-body systems at positive temperature. As an application, we give a proof of the first two terms in a high density (and high temperature) expansion of the free energy of jellium with Coulomb interactions, both in the fermionic and bosonic case. For bosons, our method works above the transition temperature (for the non-interacting gas) for Bose-Einstein condensation. 1
Abstract We generalise the two-sided Bogoliubov inequality for classical particles (Delle Site et al. in J Stat Mech Theory Exp 083201, 2017 to systems of quantum particles. As in the classical set-up, the inequality leads to upper and lower bounds for the free energy difference associated with the partitioning of a large system into smaller, independent subsystems. From a thermodynamic modelling point of view, the free energy difference determines the finite size correction needed to consistently treat a small system as a representation of a large system. Applications of the bounds to quantify finite size effects are ubiquitous in physics, chemistry, material science, or biology, to name just a few; in particular, it is relevant for molecular dynamics simulations in which a small portion of a system is usually taken as representative of the idealized large system.
We present an inequality that gives a lower bound on the expectation value of certain two-body interaction potentials in a general state on Fock space in terms of the corresponding expectation value for thermal equilibrium states of non-interacting systems and the difference in the free energy. This bound can be viewed as a rigorous version of first-order perturbation theory for many-body systems at positive temperature. As an application, we give a proof of the first two terms in a high density (and high temperature) expansion of the free energy of jellium with Coulomb interactions, both in the fermionic and bosonic case. For bosons, our method works above the transition temperature (for the non-interacting gas) for Bose–Einstein condensation.