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

Journal of Mathematical Physics

Perturbation theory of a nonideal Bose gas. II

Indira Ramarao, Franz Mohling

Published Dec 1, 1974• 1 citation
Publisher

Abstract

With the pair Hamiltonian model as the starting point, perturbation theory calculations are performed for the radial distribution function and the structure factor through second order in the two-body interaction potential for a many-boson system below the Bose-Einstein transition temperature. For the special case of a dilute hard-sphere Bose gas at T=0° K, the structure factor is calculated in the low-momentum limit keeping terms through second order in the hard-sphere gas parameter (n a3)1/2, where n is the gas density and a is the hard-sphere diameter. The resulting expression for the structure factor is shown to satisfy the Feynman-Bijl relation explicitly.

DOI
10.1063/1.1666602

Related papers

Perturbation Theory of a Nonideal Bose Gas. I

Indira Ramarao, Franz Mohling

With the pair Hamiltonian model as the starting point, perturbation theory calculations are performed for the average energy and the momentum distribution through second order in the two-body interaction potential. A phenomenological theory of helium II is proposed. The chief underlying feature of this phenomenological theory, which is shown to have a firm basis in the microscopic theory, is the presence of a double quasiparticle spectrum. Phenomenological expressions for the average energy and other quantities of interest are also proposed. Simple prescriptions are obtained for calculating the quasiparticle energies and the chemical potential from the average energy. Via these prescriptions, the quasiparticle energies and the chemical potential are calculated through second order in the interaction potential at T ≠ 0 °K, and the results are compared with earlier work in the literature.

Journal of Mathematical Physics•Aug 1, 1971•3 citations

LOW-TEMPERATURE BEHAVIOUR OF A BOSE GAS WITH HARD-SPHERE INTERACTION

Raj K. Pathria, Adarsh Singh

Using the energy-momentum relation derived by Brueckner and Sawada, the low-temperature thermodynamical properties of a Bose gas with hard-sphere interaction are investigated. The two cases, (1) when this energy spectrum is similar to that observed in liquid helium II and (2) when it becomes similar to that derived by Lee, Huang, and Yang, are investigated separately. In the first case (where the spectrum exhibits a nonmonotonic character), the analytical derivation of the expressions for the various thermodynamical properties is rendered possible by dividing the momentum range into three suitable intervals such that within each interval the epsilon -p relation can be approximated to a simple algebraic one. The numerical magnitudes of the various parameters characterizing the algebraic relations are calculated for the case of a gas with density equal to that of liquid helium(II) and for three plausible values of the hard-sphere diameter. The calculated results agree qualitatively with those obtained from the experimental study of this liquid. For the second case, expressions are derived for the various macroscopic properties of the assembly in terms of suitable mathematical functions and then are written down in the form of series expansions for the two particular cases of interest. (auth)

•Sep 26, 1960•0 citations

Low-Temperature Behavior of a Dilute Bose System of Hard Spheres. I. Equilibrium Properties

T. D. Lee, Chongming Yang

By a generalization of the method used in a previous paper the distribution of energy levels of a dilute Bose system of hard spheres is found. These energy levels are then used to compute the statistical properties for the system. A phase transition is found, and the transition point is calculated to the lowest order of $a$, the diameter of the hard spheres. Furthermore, the thermodynamical functions of the system in both the gas phase and the degenerate phase are obtained, provided $(\frac{a}{\ensuremath{\lambda}})\ensuremath{\ll}1$ and $\ensuremath{\rho}{a}^{3}\ensuremath{\ll}1$, where $\ensuremath{\lambda}$ is the thermal wavelength and $\ensuremath{\rho}$ is the particle density.

Physical Review•Dec 1, 1958•193 citations

Evaluation of the Pair Distribution Function of a Hard-Sphere Bose Gas at Zero Temperature

A. Isihara, Donald W. Jepsen

An explicit evaluation of the pair distribution function of a hard-sphere Bose gas at zero temperature is made. The result correct to first order in a hard-sphere diameter and based on a chain-diagram approximation is plotted, and the corresponding numerical values are given. Approximate analytical expressions for the distribution function at various distances are also reported.

Physical Review•Jun 5, 1967•6 citations

Many-body Bose systems and the hard-sphere model: dynamic properties from the weak to the strong interaction regime

R. Rota, F. Tramonto, Davide Emilio Galli +1 more

We obtain ab-initio estimations of the dynamic structure factor, S ( q ,ω), of Bose gases at zero temperature. More precisely, we use the Genetic Inversion via Falsification of Theories (GIFT) algorithm to perform analytic continuations of imaginary time correlation functions computed via an exact Path Integral projector method. Using the hard-sphere potential to model the two-body interactions between the atoms, we compute S ( q ,ω) changing the gas parameter from the dilute regime ( na 3 = 10 −4 ) up to the density corresponding to superfluid 4 He at equilibrium ( na 3 = 0.2138). With increasing density, we observe the emergence of a broad multiphonon contribution accompanying the quasiparticle peak and a crossover of the dispersion of elementary excitations from a Bogoliubov-like spectrum to a phonon-maxon- roton curve. Apart from the low wave vector region, for na 3 = 0.2138 the energy-momentum dispersion relation and the static density response function, χ( q ), turns out to be in good agreement with the superfluid 4 He experimental data at equilibrium density.

Journal of Physics Conference Series•Aug 14, 2014•2 citations

Quantum Hard-Sphere Gas. II

Marshall Luban

Using a method due to the author which correctly reproduces the scattering of two hard spheres for all relative momenta and all partial waves, we study the properties of a low-density hard-sphere boson gas at $T=0$\ifmmode^\circ\else\textdegree\fi{}K. The system is described by a Hamiltonian $\mathcal{H}$ defined for all relative distances of the $N$ particles. The terms of $\mathcal{H}$ corresponding to the forward motion of a particle through the system lead to the notion of an effective mass ${m}^{*}$ for each particle in excess of its actual mass $m$, with $\frac{{m}^{*}}{m}$ given by a monotonically increasing function of $\ensuremath{\rho}{a}^{3}$, where $\ensuremath{\rho}$ and $a$ are the number of particles per unit volume and the hard-sphere diameter, respectively. The excitation spectrum is linear at low momenta, displays a roton-like behavior for momenta $p\ensuremath{\gtrsim}\frac{\ensuremath{\hbar}}{a}$, possesses further extrema at higher momenta, and asymptotically approaches $\frac{{p}^{2}}{2{m}^{*}}$. The pair distribution function vanishes in the low-density limit for particle separations $r\ensuremath{\le}a$.

Physical Review•May 17, 1965•6 citations
With the pair Hamiltonian model as the starting point, perturbation theory calculations are performed for the radial distribution function and the structure factor through second order in the two-body interaction potential for a many-boson system below the Bose-Einstein transition temperature. For the special case of a dilute hard-sphere Bose gas at T=0° K, the structure factor is calculated in the low-momentum limit keeping terms through second order in the hard-sphere gas parameter (n a3)1/2, where n is the gas density and a is the hard-sphere diameter. The resulting expression for the structure factor is shown to satisfy the Feynman-Bijl relation explicitly.
With the pair Hamiltonian model as the starting point, perturbation theory calculations are performed for the average energy and the momentum distribution through second order in the two-body interaction potential. A phenomenological theory of helium II is proposed. The chief underlying feature of this phenomenological theory, which is shown to have a firm basis in the microscopic theory, is the presence of a double quasiparticle spectrum. Phenomenological expressions for the average energy and other quantities of interest are also proposed. Simple prescriptions are obtained for calculating the quasiparticle energies and the chemical potential from the average energy. Via these prescriptions, the quasiparticle energies and the chemical potential are calculated through second order in the interaction potential at T ≠ 0 °K, and the results are compared with earlier work in the literature.
Using the energy-momentum relation derived by Brueckner and Sawada, the low-temperature thermodynamical properties of a Bose gas with hard-sphere interaction are investigated. The two cases, (1) when this energy spectrum is similar to that observed in liquid helium II and (2) when it becomes similar to that derived by Lee, Huang, and Yang, are investigated separately. In the first case (where the spectrum exhibits a nonmonotonic character), the analytical derivation of the expressions for the various thermodynamical properties is rendered possible by dividing the momentum range into three suitable intervals such that within each interval the epsilon -p relation can be approximated to a simple algebraic one. The numerical magnitudes of the various parameters characterizing the algebraic relations are calculated for the case of a gas with density equal to that of liquid helium(II) and for three plausible values of the hard-sphere diameter. The calculated results agree qualitatively with those obtained from the experimental study of this liquid. For the second case, expressions are derived for the various macroscopic properties of the assembly in terms of suitable mathematical functions and then are written down in the form of series expansions for the two particular cases of interest. (auth)
By a generalization of the method used in a previous paper the distribution of energy levels of a dilute Bose system of hard spheres is found. These energy levels are then used to compute the statistical properties for the system. A phase transition is found, and the transition point is calculated to the lowest order of aaa, the diameter of the hard spheres. Furthermore, the thermodynamical functions of the system in both the gas phase and the degenerate phase are obtained, provided $(\frac{a}{\ensuremath{\lambda}})\ensuremath{\ll}1$ and $\ensuremath{\rho}{a}^{3}\ensuremath{\ll}1$, where $\ensuremath{\lambda}$ is the thermal wavelength and $\ensuremath{\rho}$ is the particle density.
We obtain ab-initio estimations of the dynamic structure factor, S ( q ,ω), of Bose gases at zero temperature. More precisely, we use the Genetic Inversion via Falsification of Theories (GIFT) algorithm to perform analytic continuations of imaginary time correlation functions computed via an exact Path Integral projector method. Using the hard-sphere potential to model the two-body interactions between the atoms, we compute S ( q ,ω) changing the gas parameter from the dilute regime ( na 3 = 10 −4 ) up to the density corresponding to superfluid 4 He at equilibrium ( na 3 = 0.2138). With increasing density, we observe the emergence of a broad multiphonon contribution accompanying the quasiparticle peak and a crossover of the dispersion of elementary excitations from a Bogoliubov-like spectrum to a phonon-maxon- roton curve. Apart from the low wave vector region, for na 3 = 0.2138 the energy-momentum dispersion relation and the static density response function, χ( q ), turns out to be in good agreement with the superfluid 4 He experimental data at equilibrium density.
Using a method due to the author which correctly reproduces the scattering of two hard spheres for all relative momenta and all partial waves, we study the properties of a low-density hard-sphere boson gas at T=0T=0T=0\ifmmode^\circ\else\textdegree\fi{}K. The system is described by a Hamiltonian H\mathcal{H}H defined for all relative distances of the NNN particles. The terms of H\mathcal{H}H corresponding to the forward motion of a particle through the system lead to the notion of an effective mass m∗{m}^{*}m∗ for each particle in excess of its actual mass mmm, with m∗m\frac{{m}^{*}}{m}mm∗​ given by a monotonically increasing function of $\ensuremath{\rho}{a}^{3}$, where $\ensuremath{\rho}$ and aaa are the number of particles per unit volume and the hard-sphere diameter, respectively. The excitation spectrum is linear at low momenta, displays a roton-like behavior for momenta $p\ensuremath{\gtrsim}\frac{\ensuremath{\hbar}}{a}$, possesses further extrema at higher momenta, and asymptotically approaches p22m∗\frac{{p}^{2}}{2{m}^{*}}2m∗p2​. The pair distribution function vanishes in the low-density limit for particle separations $r\ensuremath{\le}a$.