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

arXiv:1211.5185 · Physical Review E

Classical representation of a quantum system at equilibrium: Applications

Sandipan Dutta, James W. Dufty

Published Mar 4, 2013• 29 citations
arXivHTMLPDF

Abstract

In the preceding paper, the structure and thermodynamics of a given quantum system was represented by a corresponding classical system having an effective temperature, local chemical potential, and pair potential. Here, that formal correspondence is implemented approximately for applications to two quantum systems. The first is the electron gas (jellium) over a range of temperatures and densities. The second is an investigation of quantum effects on shell structure for charges confined by a harmonic potential.

DOI
10.1103/physreve.87.032102

Related papers

Classical representation of a quantum system at equilibrium: Theory

James W. Dufty, Sandipan Dutta

A quantum system at equilibrium is represented by a corresponding classical system, chosen to reproduce thermodynamic and structural properties. The motivation is to allow application of classical strong-coupling theories and molecular dynamics simulation to quantum systems at strong coupling. The correspondence is made at the level of the grand-canonical ensembles for the two systems. An effective temperature, local chemical potential, and pair potential are introduced to define the corresponding classical system. These are determined formally by requiring the equivalence of the grand potentials and their functional derivatives. Practical inversions of these formal definitions are indicated via the integral equations for densities and pair correlation functions of classical liquid theory. Application to the ideal Fermi gas is demonstrated, and the weak-coupling form for the pair potential is given. In a companion paper two applications are described: the thermodynamics and structure of uniform jellium over a range of temperatures and densities and the shell structure of harmonically bound charges.

arXiv:1211.5177•Physical Review E•Mar 4, 2013•30 citations

Thermodynamic extension of density-functional theory. II. Finite-temperature ensemble spin-density functional theory

Robert Balawender, A. Holas

The formalism developed in the first paper of the series [arXiv:0901.1060] is applied to two thermodynamic systems: (i) of three global observables (the energy, the total electron number and the spin number), (ii) of one global observable (the internal electron energy) and two local (position-dependent) observables (the total electron density and the spin density). The two-component potential of the many-electron system of interest is constructed of a scalar external potential and a collinear magnetic field (coupled only with the spin operator). Various equilibrium characteristics of two systems are defined and investigated. Conditions for the equivalence between two systems (the same equilibrium density matrix demanded) are derived and thoroughly discussed. The applicability of the Hohenberg-Kohn theorem is extended to the thermodynamic spin-density functional theory. Obtained results provide a rigorous mathematical foundation for future derivation of the zero-temperature limit of this theory and determination of its chemical reactivity indices.

arXiv:0904.3990•Apr 25, 2009•5 citations

Chemical potentials and thermodynamic characteristics of ideal Bose- and Fermi-gases in the region of quantum degeneracy

Andrii Sotnikov, K. V. Sereda, Yu. V. Slyusarenko

Calculations of chemical potentials for ideal monatomic gases with Bose-Einstein and Fermi-Dirac statistics as functions of temperature, across the temperature region that is typical for the collective quantum degeneracy effect, are presented. Numerical calculations are performed without any additional approximations, and explicit dependences of the chemical potentials on temperature are constructed at a fixed density of gas particles. Approximate polynomial dependences of chemical potentials on temperature are obtained that allow for the results to be used in further studies without re-applying the involved numerical methods. The ease of using the obtained representations is demonstrated on examples of deformation of distribution for a population of energy states at low temperatures, and on the impact of quantum statistics (exchange interaction) on the equations of state for ideal gases and some of the thermodynamic properties thereof. The results of this study essentially unify two opposite limiting cases in an intermediate region that are used to describe the equilibrium states of ideal gases, which are well known from university courses on statistical physics, thus adding value from an educational point of view.

Low Temperature Physics•Jan 1, 2017•13 citations

Equilibrium conditions for coupled classical-quantum systems

M. Fannes, André F. Verbeure

Systems with classical as well as quantal degrees of freedom are described. This is done by means of a relative Hamiltonian and a derivation satisfying a compatibility relation. Various equivalent equilibrium conditions are studied.

Journal of Physics A Mathematical and General•Dec 1, 1987•1 citation

Theory of cold dipolar and toroidal gases

D. Baillie

We develop and apply meanfield methods to two systems of significant interest in cold atom physics. The first system, and the main subject of this thesis, is a gas of polarized dipoles, in which the particles have a long range anisotropic interaction. The second system is a toroidally trapped gas that interacts via short range interactions. \n \nDipolar gases: We develop accurate techniques for dipolar Bose and Fermi gases accounting for both direct and exchange interactions. We detail an efficient numerical implementation of the meanfield treatment. \n \nWe study the thermodynamic and the first and second order correlation properties of the system. We find that exchange interactions cause the correlation functions to be anisotropic in the low temperature regime. We also find that many uniform gas thermodynamic predictions, for which direct interaction effects vanish, are qualitatively unreliable for trapped systems, most notably for oblate traps. \n \nUsing Hartree-Fock theory and analytic approximations, we examine the magnetostrictive position and momentum space distortions that occur in harmonically confined dipolar Bose and Fermi gases. The Bose gas momentum distribution distorts in the opposite sense to that of the Fermi gas. By relating exchange effects to short range correlations between the particles we discuss the energetic origin of this difference. \n \nIntroducing local momentum distortion fields, we develop a practical Hartree-Fock theory in which exchange effects are described by a local effective potential. This theory is applicable at zero and finite temperature, and is in excellent agreement with full Hartree-Fock calculations. \n \nContact gases: We show how considerable simplification of numerical results and analytical derivations can be achieved in the local density approximation using a variety of density of states techniques. \n \nWe examine the process of isentropic loading of a Bose or Fermi gas from a harmonic trap into the scale invariant toroidal regime that exhibits a high degree of system invariance when increasing the radius of the toroid. For bosons a regime of cooling is identified and we find that the Fermi gas is subjected to irreducible heating during loading, caused by the loss of one degree of freedom for thermalization.

•Jan 1, 2012•0 citations

Approach to Equilibrium in Quantum Systems

Pierre M. V. Résibois

Prigogine and his co-workers developed a genera; theory of approach to equilibrium in classical mechanics. A inethod is presented which indicates the possibility for obtaining similar results for quantum systems. This method makes use of a representation for the density matrix which stresses the formal analogy between the von Neumann equation and the classical Liouville equation. This representation was used in the particular cases of weakly-coupled systems and of quantum plasmas. (B.O.G.)

Physical Review Letters•Nov 1, 1960•6 citations
In the preceding paper, the structure and thermodynamics of a given quantum system was represented by a corresponding classical system having an effective temperature, local chemical potential, and pair potential. Here, that formal correspondence is implemented approximately for applications to two quantum systems. The first is the electron gas (jellium) over a range of temperatures and densities. The second is an investigation of quantum effects on shell structure for charges confined by a harmonic potential.
A quantum system at equilibrium is represented by a corresponding classical system, chosen to reproduce thermodynamic and structural properties. The motivation is to allow application of classical strong-coupling theories and molecular dynamics simulation to quantum systems at strong coupling. The correspondence is made at the level of the grand-canonical ensembles for the two systems. An effective temperature, local chemical potential, and pair potential are introduced to define the corresponding classical system. These are determined formally by requiring the equivalence of the grand potentials and their functional derivatives. Practical inversions of these formal definitions are indicated via the integral equations for densities and pair correlation functions of classical liquid theory. Application to the ideal Fermi gas is demonstrated, and the weak-coupling form for the pair potential is given. In a companion paper two applications are described: the thermodynamics and structure of uniform jellium over a range of temperatures and densities and the shell structure of harmonically bound charges.
The formalism developed in the first paper of the series [arXiv:0901.1060] is applied to two thermodynamic systems: (i) of three global observables (the energy, the total electron number and the spin number), (ii) of one global observable (the internal electron energy) and two local (position-dependent) observables (the total electron density and the spin density). The two-component potential of the many-electron system of interest is constructed of a scalar external potential and a collinear magnetic field (coupled only with the spin operator). Various equilibrium characteristics of two systems are defined and investigated. Conditions for the equivalence between two systems (the same equilibrium density matrix demanded) are derived and thoroughly discussed. The applicability of the Hohenberg-Kohn theorem is extended to the thermodynamic spin-density functional theory. Obtained results provide a rigorous mathematical foundation for future derivation of the zero-temperature limit of this theory and determination of its chemical reactivity indices.
Calculations of chemical potentials for ideal monatomic gases with Bose-Einstein and Fermi-Dirac statistics as functions of temperature, across the temperature region that is typical for the collective quantum degeneracy effect, are presented. Numerical calculations are performed without any additional approximations, and explicit dependences of the chemical potentials on temperature are constructed at a fixed density of gas particles. Approximate polynomial dependences of chemical potentials on temperature are obtained that allow for the results to be used in further studies without re-applying the involved numerical methods. The ease of using the obtained representations is demonstrated on examples of deformation of distribution for a population of energy states at low temperatures, and on the impact of quantum statistics (exchange interaction) on the equations of state for ideal gases and some of the thermodynamic properties thereof. The results of this study essentially unify two opposite limiting cases in an intermediate region that are used to describe the equilibrium states of ideal gases, which are well known from university courses on statistical physics, thus adding value from an educational point of view.
We develop and apply meanfield methods to two systems of significant interest in cold atom physics. The first system, and the main subject of this thesis, is a gas of polarized dipoles, in which the particles have a long range anisotropic interaction. The second system is a toroidally trapped gas that interacts via short range interactions. \n \nDipolar gases: We develop accurate techniques for dipolar Bose and Fermi gases accounting for both direct and exchange interactions. We detail an efficient numerical implementation of the meanfield treatment. \n \nWe study the thermodynamic and the first and second order correlation properties of the system. We find that exchange interactions cause the correlation functions to be anisotropic in the low temperature regime. We also find that many uniform gas thermodynamic predictions, for which direct interaction effects vanish, are qualitatively unreliable for trapped systems, most notably for oblate traps. \n \nUsing Hartree-Fock theory and analytic approximations, we examine the magnetostrictive position and momentum space distortions that occur in harmonically confined dipolar Bose and Fermi gases. The Bose gas momentum distribution distorts in the opposite sense to that of the Fermi gas. By relating exchange effects to short range correlations between the particles we discuss the energetic origin of this difference. \n \nIntroducing local momentum distortion fields, we develop a practical Hartree-Fock theory in which exchange effects are described by a local effective potential. This theory is applicable at zero and finite temperature, and is in excellent agreement with full Hartree-Fock calculations. \n \nContact gases: We show how considerable simplification of numerical results and analytical derivations can be achieved in the local density approximation using a variety of density of states techniques. \n \nWe examine the process of isentropic loading of a Bose or Fermi gas from a harmonic trap into the scale invariant toroidal regime that exhibits a high degree of system invariance when increasing the radius of the toroid. For bosons a regime of cooling is identified and we find that the Fermi gas is subjected to irreducible heating during loading, caused by the loss of one degree of freedom for thermalization.