Physical Review

Ground-State Energy of a Hard-Sphere Gas

Freeman J. Dyson

Published Apr 1, 1957• 149 citations

Abstract

Rigorous lower and upper bounds are established for the energy of the ground state of a Bose gas with hard-sphere interaction between particles.Received 17 December 1956DOI:https://doi.org/10.1103/PhysRev.106.20©1957 American Physical Society

DOI: 10.1103/physrev.106.20

Related papers

Upper Bound for the Ground State Energy of a Dilute Bose Gas of Hard Spheres

Giulia Basti, Serena Cenatiempo, Alessandro Giuliani +3 more

Abstract We consider a gas of bosons interacting through a hard-sphere potential with radius $$\mathfrak {a}$$ a in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term $$4\pi \rho \mathfrak {a}$$ 4 π ρ a and shows that corrections are smaller than $$C \rho \mathfrak {a} (\rho {{\mathfrak {a}}}^3)^{1/2}$$ C ρ a ( ρ a 3 ) 1 / 2 , for a sufficiently large constant $$C > 0$$ C > 0 . In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order $$\rho \mathfrak {a}(\rho {{\mathfrak {a}}}^3)^{1/2}$$ ρ a ( ρ a 3 ) 1 / 2 , in agreement with the Lee–Huang–Yang prediction.

Archive for Rational Mechanics and AnalysisOct 21, 20249 citations

Upper bound for the ground state energy of a dilute Bose gas of hard spheres

Giulia Basti, Serena Cenatiempo, Alessandro Giuliani +3 more

arXiv:2212.04431Dec 8, 20223 citations

Ground state energy of a Bose gas in the Gross–Pitaevskii regime

Giulia Basti, Serena Cenatiempo, Alessandro Olgiati +2 more

arXiv:2202.10270Journal of Mathematical PhysicsApr 1, 202211 citations

Ground-State Energy of a One-Dimensional Many-Boson System

Mark Bolsterli

Physical ReviewFeb 5, 19695 citations

Dilute Hard "Sphere" Bose Gas in Dimensions 2, 4 and 5

Chaohui Yang

The ground state energy for a dilute hard "sphere" Bose gas in various dimensions is studied theoretically.

arXiv:0807.0938Jul 7, 20081 citation

Energy of Interacting Fermi Systems

C. DeDominicis, Paul C. Martin

Physical ReviewFeb 15, 195738 citations

Physical Review

Ground-State Energy of a Hard-Sphere Gas

Freeman J. Dyson

Published Apr 1, 1957• 149 citations

Publisher

Abstract

DOI: 10.1103/physrev.106.20

Related papers

Upper Bound for the Ground State Energy of a Dilute Bose Gas of Hard Spheres

Giulia Basti, Serena Cenatiempo, Alessandro Giuliani +3 more

Abstract We consider a gas of bosons interacting through a hard-sphere potential with radius $\mathfrak {a}$ a in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term $4\pi \rho \mathfrak {a}$ 4 π ρ a and shows that corrections are smaller than $C \rho \mathfrak {a} (\rho {{\mathfrak {a}}}^3)^{1/2}$ C ρ a ( ρ a 3 ) 1 / 2 , for a sufficiently large constant $C > 0$ C > 0 . In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order $\rho \mathfrak {a}(\rho {{\mathfrak {a}}}^3)^{1/2}$ ρ a ( ρ a 3 ) 1 / 2 , in agreement with the Lee–Huang–Yang prediction.Abstract We consider a gas of bosons interacting through a hard-sphere potential with radius $$\mathfrak {a}$$ a in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term $$4\pi \rho \mathfrak {a}$$ 4 π ρ a and shows that corrections are smaller than $$C \rho \mathfrak {a} (\rho {{\mathfrak {a}}}^3)^{1/2}$$ C ρ a ( ρ a 3 ) 1 / 2 , for a sufficiently large constant $$C > 0$$ C > 0 . In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order $$\rho \mathfrak {a}(\rho {{\mathfrak {a}}}^3)^{1/2}$$ ρ a ( ρ a 3 ) 1 / 2 , in agreement with the Lee–Huang–Yang prediction.

Archive for Rational Mechanics and AnalysisOct 21, 20249 citations

Upper bound for the ground state energy of a dilute Bose gas of hard spheres

Giulia Basti, Serena Cenatiempo, Alessandro Giuliani +3 more

We consider a gas of bosons interacting through a hard-sphere potential with radius $\frak{a}$ in the thermodynamic limit. We derive a simple upper bound for the ground state energy per particle at low density. Our bound captures the leading term $4πρ\frak{a}$ and shows that corrections are smaller than $C ρ\frak{a} (ρ\frak{a}^3)^{1/2}$ , for a sufficiently large constant $C > 0$ . In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy is, in fact, of the order $ρ\frak{a} (ρ\frak{a}^3)^{1/2}$ , in agreement with the Lee-Huang-Yang prediction.We consider a gas of bosons interacting through a hard-sphere potential with radius $\frak{a}$ in the thermodynamic limit. We derive a simple upper bound for the ground state energy per particle at low density. Our bound captures the leading term $4πρ\frak{a}$ and shows that corrections are smaller than $C ρ\frak{a} (ρ\frak{a}^3)^{1/2}$, for a sufficiently large constant $C > 0$. In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy is, in fact, of the order $ρ\frak{a} (ρ\frak{a}^3)^{1/2}$, in agreement with the Lee-Huang-Yang prediction.

arXiv:2212.04431Dec 8, 20223 citations

Ground state energy of a Bose gas in the Gross–Pitaevskii regime

Giulia Basti, Serena Cenatiempo, Alessandro Olgiati +2 more

We review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson’s upper bound, which provides the correct leading order asymptotics for hard spheres. Afterward, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross–Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross–Pitaevskii limit. Here, we only sketch the main ideas; details will appear elsewhere.We review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson’s upper bound, which provides the correct leading order asymptotics for hard spheres. Afterward, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross–Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross–Pitaevskii limit. Here, we only sketch the main ideas; details will appear elsewhere.

arXiv:2202.10270Journal of Mathematical PhysicsApr 1, 202211 citations

Ground-State Energy of a One-Dimensional Many-Boson System

Mark Bolsterli

Physical ReviewFeb 5, 19695 citations

Dilute Hard "Sphere" Bose Gas in Dimensions 2, 4 and 5

Chaohui Yang

The ground state energy for a dilute hard "sphere" Bose gas in various dimensions is studied theoretically.

arXiv:0807.0938Jul 7, 20081 citation

Energy of Interacting Fermi Systems

C. DeDominicis, Paul C. Martin

Physical ReviewFeb 15, 195738 citations

Abstract We consider a gas of bosons interacting through a hard-sphere potential with radius

\mathfrak {a}

a in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term

4\pi \rho \mathfrak {a}

4 π ρ a and shows that corrections are smaller than

C \rho \mathfrak {a} (\rho {{\mathfrak {a}}}^3)^{1/2}

C ρ a ( ρ a 3 ) 1 / 2 , for a sufficiently large constant

C > 0

C > 0 . In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order

\rho \mathfrak {a}(\rho {{\mathfrak {a}}}^3)^{1/2}

ρ a ( ρ a 3 ) 1 / 2 , in agreement with the Lee–Huang–Yang prediction.

We consider a gas of bosons interacting through a hard-sphere potential with radius

\frak{a}

in the thermodynamic limit. We derive a simple upper bound for the ground state energy per particle at low density. Our bound captures the leading term

4πρ\frak{a}

and shows that corrections are smaller than

C ρ\frak{a} (ρ\frak{a}^3)^{1/2}

, for a sufficiently large constant

C > 0

. In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy is, in fact, of the order

ρ\frak{a} (ρ\frak{a}^3)^{1/2}

, in agreement with the Lee-Huang-Yang prediction.