Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Ground-state-energy theorem and the virial theorem of a many-particle system in d dimensions

Naoki Iwamoto

Published Nov 1, 1984• 10 citations
Publisher

Abstract

It is shown by dimensional analysis that Pauli's ground-state-energy theorem, which expresses the ground-state energy in the form of a coupling-constant integration of the potential energy, and the virial theorem are equivalent for a many-particle system interacting with a homogeneous interparticle potential ($\ensuremath{\propto}{r}^{l}$) in arbitrary ($d$) dimensions. The finite-temperature case is also discussed.

DOI
10.1103/physreva.30.2597

Related papers

General Theorem for Interacting Many-Particle Systems

節夫 三沢

It is shown that the kinetic energy of any interacting many-particle system in its ground state is a monotone increasing function with respect to the absolute value of the coupling constant (or the interaction strength). A concise derivation of the virial theorem for simple power law potentials is also presented with particular emphasis on the case of an electron gas in a uniform background of positive charge.

Progress of Theoretical Physics•Jun 1, 1968•5 citations

Virial Theorem for Nonrelativistic Quantum Fields in D Spatial Dimensions

Chris L. Lin, Carlos R. Ordóñez

The virial theorem for nonrelativistic complex fields in D spatial dimensions and with arbitrary many-body potential is derived, using path-integral methods and scaling arguments recently developed to analyze quantum anomalies in low-dimensional systems. The potential appearance of a Jacobian J due to a change of variables in the path-integral expression for the partition function of the system is pointed out, although in order to make contact with the literature most of the analysis deals with the J=1 case. The virial theorem is recast into a form that displays the effect of microscopic scales on the thermodynamics of the system. From the point of view of this paper the case usually considered, J=1 , is not natural, and the generalization to the case J≠1 is briefly presented.

arXiv:1503.05843•Advances in High Energy Physics•Jan 1, 2015•8 citations

Exact and asymptotic local virial theorems for finite fermionic systems

M. Brack, Koch, A, M. V. N. Murthy +1 more

We investigate the particle and kinetic-energy densities for a system of N fermions confined in a potential V (r). In an earlier paper (Brack and Murthy 2003 J. Phys. A: Math. Gen. 36 1111), some exact and asymptotic relations involving the particle density and the kinetic-energy density locally, i.e. at any given point r, were derived for isotropic harmonic oscillators in arbitrary dimensions. In this paper, we show that these local virial theorems (LVTs) also hold exactly for linear potentials in arbitrary dimensions and for the one-dimensional box. We also investigate the validity of these LVTs when they are applied to arbitrary smooth potentials. We formulate generalized LVTs that are suggested by a semiclassical theory which relates the density oscillations to the closed non-periodic orbits of the classical system. We test the validity of these generalized theorems numerically for various local potentials. Although formally they are only valid asymptotically for large particle numbers N, we show that practically they are surprisingly accurate also for moderate values of N.

•Jan 1, 2010•6 citations

Exact and asymptotic local virial theorems for finite fermionic systems

M. Brack, A Koch, M V N Murthy +1 more

We investigate the particle and kinetic-energy densities for a system of $N$ fermions confined in a potential $V(\bfr)$. In an earlier paper [J. Phys. A: Math. Gen. {\bf 36}, 1111 (2003)], some exact and asymptotic relations involving the particle density and the kinetic-energy density locally, i.e. at any given point $\bfr$, were derived for isotropic harmonic oscillators in arbitrary dimensions. In this paper we show that these {\it local virial theorems} (LVT) also hold exactly for linear potentials in arbitrary dimensions and for the one-dimensional box. We also investigate the validity of these LVTs when they are applied to arbitrary smooth potentials. We formulate generalized LVTs that are supported by a semiclassical theory which relates the density oscillations to the closed non-periodic orbits of the classical system. We test the validity of these generalized theorems numerically for various local potentials. Although they formally are only valid asymptotically for large particle numbers $N$, we show that they practically are surprisingly accurate also for moderate values of $N$.

arXiv:0903.2172•Journal of Physics A Mathematical and Theoretical•May 27, 2010•6 citations

Virial theorem and generalization of the Fermi-liquid theory for the homogeneous degenerate electron system

Viktor Borisovich Bobrov, S. A. Trigger, Anatolij G. Zagorodny

Using the general thermodynamic equalities and the virial theorem, it is shown that the ground state energy of the homogeneous electron liquid is uniquely defined by the one-particle electron distribution function over momenta and spin quantum numbers. On this basis, taking into account the results of quantum field theory for the one-particle Green’s function, we state that the ground state energy of the homogeneous electron liquid is a functional of the one-particle distribution function.

Progress of Theoretical and Experimental Physics•Feb 4, 2013•1 citation

Virial expansion coefficients in the harmonic approximation

J. R. Armstrong, Nikolaj Thomas Zinner, D. V. Fedorov +1 more

The virial expansion method is applied within a harmonic approximation to an interacting N-body system of identical fermions. We compute the canonical partition functions for two and three particles to get the two lowest orders in the expansion. The energy spectrum is carefully interpolated to reproduce ground-state properties at low temperature and the noninteracting high-temperature limit of constant virial coefficients. This resembles the smearing of shell effects in finite systems with increasing temperature. Numerical results are discussed for the second and third virial coefficients as functions of dimension, temperature, interaction, and transition temperature between low- and high-energy limits.

arXiv:1205.2574•Physical Review E•Aug 15, 2012•21 citations
It is shown by dimensional analysis that Pauli's ground-state-energy theorem, which expresses the ground-state energy in the form of a coupling-constant integration of the potential energy, and the virial theorem are equivalent for a many-particle system interacting with a homogeneous interparticle potential (\ensuremath∝rl\ensuremath{\propto}{r}^{l}\ensuremath∝rl) in arbitrary (ddd) dimensions. The finite-temperature case is also discussed.
The virial theorem for nonrelativistic complex fields in D spatial dimensions and with arbitrary many-body potential is derived, using path-integral methods and scaling arguments recently developed to analyze quantum anomalies in low-dimensional systems. The potential appearance of a Jacobian J due to a change of variables in the path-integral expression for the partition function of the system is pointed out, although in order to make contact with the literature most of the analysis deals with the J=1 case. The virial theorem is recast into a form that displays the effect of microscopic scales on the thermodynamics of the system. From the point of view of this paper the case usually considered, J=1 , is not natural, and the generalization to the case J≠1 is briefly presented.
We investigate the particle and kinetic-energy densities for a system of N fermions confined in a potential V (r). In an earlier paper (Brack and Murthy 2003 J. Phys. A: Math. Gen. 36 1111), some exact and asymptotic relations involving the particle density and the kinetic-energy density locally, i.e. at any given point r, were derived for isotropic harmonic oscillators in arbitrary dimensions. In this paper, we show that these local virial theorems (LVTs) also hold exactly for linear potentials in arbitrary dimensions and for the one-dimensional box. We also investigate the validity of these LVTs when they are applied to arbitrary smooth potentials. We formulate generalized LVTs that are suggested by a semiclassical theory which relates the density oscillations to the closed non-periodic orbits of the classical system. We test the validity of these generalized theorems numerically for various local potentials. Although formally they are only valid asymptotically for large particle numbers N, we show that practically they are surprisingly accurate also for moderate values of N.
We investigate the particle and kinetic-energy densities for a system of NNN fermions confined in a potential $V(\bfr)$. In an earlier paper [J. Phys. A: Math. Gen. {\bf 36}, 1111 (2003)], some exact and asymptotic relations involving the particle density and the kinetic-energy density locally, i.e. at any given point $\bfr$, were derived for isotropic harmonic oscillators in arbitrary dimensions. In this paper we show that these {\it local virial theorems} (LVT) also hold exactly for linear potentials in arbitrary dimensions and for the one-dimensional box. We also investigate the validity of these LVTs when they are applied to arbitrary smooth potentials. We formulate generalized LVTs that are supported by a semiclassical theory which relates the density oscillations to the closed non-periodic orbits of the classical system. We test the validity of these generalized theorems numerically for various local potentials. Although they formally are only valid asymptotically for large particle numbers NNN, we show that they practically are surprisingly accurate also for moderate values of NNN.
The virial expansion method is applied within a harmonic approximation to an interacting N-body system of identical fermions. We compute the canonical partition functions for two and three particles to get the two lowest orders in the expansion. The energy spectrum is carefully interpolated to reproduce ground-state properties at low temperature and the noninteracting high-temperature limit of constant virial coefficients. This resembles the smearing of shell effects in finite systems with increasing temperature. Numerical results are discussed for the second and third virial coefficients as functions of dimension, temperature, interaction, and transition temperature between low- and high-energy limits.