We study a system of N interacting fermions at positive temperature in a confining potential. In the regime where the intensity of the interaction scales as 1/N and with an effective semiclassical parameter ℏ = N−1/d where d is the space dimension, we prove the convergence to the corresponding Thomas-Fermi model at positive temperature.
REFERENCES
1.
G. L.
Aki
, P. A.
Markowich
, and C.
Sparber
, “Classical limit for semirelativistic Hartree systems
,” J. Math. Phys.
49
, 102110
(2008
).2.
V.
Bach
, S.
Breteaux
, S.
Petrat
, P.
Pickl
, and T.
Tzaneteas
, “Kinetic energy estimates for the accuracy of the time-dependent Hartree-Fock approximation with Coulomb interaction
,” J. Math. Pures Appl.
105
, 1
–30
(2015
).3.
V.
Bach
, E. H.
Lieb
, and J. P.
Solovej
, “Generalized Hartree-Fock theory and the Hubbard model
,” J. Stat. Phys.
76
, 3
–89
(1994
).4.
C.
Bardos
, F.
Golse
, A. D.
Gottlieb
, and N. J.
Mauser
, “Mean field dynamics of fermions and the time-dependent Hartree-Fock equation
,” J. Math. Pures Appl. (9)
82
, 665
–683
(2003
).5.
M.
Barranco
and J.-R.
Buchler
, “Equation of state of hot, dense stellar matter: Finite temperature nuclear Thomas-Fermi approach
,” Phys. Rev. C
24
, 1191
–1202
(1981
).6.
N.
Benedikter
, V.
Jaksic
, M.
Porta
, C.
Saffirio
, and B.
Schlein
, “Mean-field evolution of fermionic mixed states
,” Commun. Pure Appl. Math.
69
, 2250
–2303
(2016
).7.
N.
Benedikter
, P. T.
Nam
, M.
Porta
, B.
Schlein
, and R.
Seiringer
, “Optimal upper bound for the correlation energy of a Fermi gas in the mean-field regime
,” Commun. Math. Phys.
(published online).8.
N.
Benedikter
, M.
Porta
, C.
Saffirio
, and B.
Schlein
, “From the Hartree dynamics to the Vlasov equation
,” Arch. Ration. Mech. Anal.
221
, 273
–334
(2016
).9.
N.
Benedikter
, M.
Porta
, and B.
Schlein
, “Mean-field dynamics of fermions with relativistic dispersion
,” J. Math. Phys.
55
, 021901
(2014
).10.
N.
Benedikter
, M.
Porta
, and B.
Schlein
, “Mean-field evolution of fermionic systems
,” Commun. Math. Phys.
331
, 1087
–1131
(2014
).11.
L. G.
Brown
and H.
Kosaki
, “Jensen’s inequality in semi-finite von Neumann algebras
,” J. Oper. Theory
23
, 3
–19
(1990
), available at https://www.jstor.org/stable/24714523.12.
K. A.
Brueckner
, J. R.
Buchler
, S.
Jorna
, and R. J.
Lombard
, “Statistical theory of nuclei
,” Phys. Rev.
171
, 1188
–1195
(1968
).13.
R. D.
Cowan
and J.
Ashkin
, “Extension of the Thomas-Fermi-Dirac statistical theory of the atom to finite temperatures
,” Phys. Rev.
105
, 144
–157
(1957
).14.
H. L.
Cycon
, R. G.
Froese
, W.
Kirsch
, and B.
Simon
, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry
, Texts and Monographs in Physics, Study edtuion (Springer-Verlag
, Berlin
, 1987
).15.
E.
Dietler
, S.
Rademacher
, and B.
Schlein
, “From Hartree dynamics to the relativistic Vlasov equation
,” J. Stat. Phys.
172
, 398
–433
(2018
).16.
A.
Elgart
, L.
Erdős
, B.
Schlein
, and H.-T.
Yau
, “Nonlinear Hartree equation as the mean field limit of weakly coupled fermions
,” J. Math. Pures Appl.
83
, 1241
–1273
(2004
).17.
R. P.
Feynman
, N.
Metropolis
, and E.
Teller
, “Equations of state of elements based on the generalized Fermi-Thomas theory
,” Phys. Rev.
75
, 1561
–1573
(1949
).18.
S.
Fournais
, M.
Lewin
, and J. P.
Solovej
, “The semi-classical limit of large fermionic systems
,” Calc. Var. Partial Differ. Equations
57
, 105
(2018
).19.
J.
Fröhlich
and A.
Knowles
, “A microscopic derivation of the time-dependent Hartree-Fock equation with Coulomb two-body interaction
,” J. Stat. Phys.
145
, 23
–50
(2011
).20.
J. J.
Gilvarry
and G. H.
Peebles
, “Solutions of the temperature-perturbed Thomas-Fermi equation
,” Phys. Rev.
99
, 550
–552
(1955
).21.
S.
Giorgini
, L. P.
Pitaevsk
II, and S.
Stringari
, “Theory of ultracold atomic Fermi gases
,” Rev. Mod. Phys.
80
, 1215
–1274
(2008
).22.
F.
Golse
, C.
Mouhot
, and T.
Paul
, “On the mean field and classical limits of quantum mechanics
,” Commun. Math. Phys.
343
, 165
–205
(2016
).23.
F.
Golse
and T.
Paul
, “The Schrödinger equation in the mean-field and semiclassical regime
,” Arch. Ration. Mech. Anal.
223
, 57
–94
(2017
).24.
A. D.
Gottlieb
, “Examples of Bosonic de Finetti states over finite dimensional Hilbert spaces
,” J. Stat. Phys.
121
, 497
–509
(2005
).25.
A.
Grabsch
, S. N.
Majumdar
, G.
Schehr
, and C.
Texier
, “Fluctuations of observables for free fermions in a harmonic trap at finite temperature
,” SciPost Phys.
4
, 14
(2018
).26.
C.
Hainzl
, M.
Porta
, and F.
Rexze
, “On the correlation energy of the mean-field Fermi gas
,” e-print arXiv:1806.11411 (2018
).27.
B.
Hauksson
and J.
Yngvason
, “Asymptotic exactness of magnetic Thomas-Fermi theory at nonzero temperature
,” J. Stat. Phys.
116
, 523
–546
(2004
).28.
P.
Hertel
, H.
Narnhofer
, and W.
Thirring
, “Thermodynamic functions for fermions with gravostatic and electrostatic interactions
,” Commun. Math. Phys.
28
, 159
–176
(1972
).29.
P.
Hertel
and W.
Thirring
, “Free energy of gravitating fermions
,” Commun. Math. Phys.
24
, 22
–36
(1971
).30.
R.
Latter
, “Temperature behavior of the Thomas-Fermi statistical model for atoms
,” Phys. Rev.
99
, 1854
–1870
(1955
).31.
M.
Lewin
, P. T.
Nam
, and N.
Rougerie
, “Derivation of Hartree’s theory for generic mean-field Bose systems
,” Adv. Math.
254
, 570
–621
(2014
).32.
M.
Lewin
, P. T.
Nam
, and N.
Rougerie
, “Bose gases at positive temperature and non-linear Gibbs measures
,” in Proceedings of the International Congress of Mathematical Physics
, 2015
; e-print arXiv:1602.05166 (2016
).33.
M.
Lewin
, P. T.
Nam
, and N.
Rougerie
, “Derivation of nonlinear Gibbs measures from many-body quantum mechanics
,” J. l’École polytech. Math.
2
, 65
–115
(2015
).34.
M.
Lewin
, P. T.
Nam
, and N.
Rougerie
, “Classical field theory limit of 2D many-body quantum Gibbs states
,” e-print arXiv:1810.08370 (2018
).35.
M.
Lewin
, P. T.
Nam
, and N.
Rougerie
, “Gibbs measures based on 1D (an)harmonic oscillators as mean-field limits
,” J. Math. Phys.
59
, 041901
(2018
).36.
M.
Lewin
, P. T.
Nam
, and N.
Rougerie
, “The interacting 2D Bose gas and nonlinear Gibbs measures
,” in Gibbs Measures for Nonlinear Dispersive Equations
, edited by B. S.
Giuseppe Genovese
and V.
Sohinger
(Oberwolfach Mini-Workshop
, 2018
).37.
M.
Lewin
, P. T.
Nam
, S.
Serfaty
, and J. P.
Solovej
, “Bogoliubov spectrum of interacting Bose gases
,” Commun. Pure Appl. Math.
68
, 413
–471
(2015
).38.
E. H.
Lieb
and M.
Loss
, Analysis
, Graduate Studies in Mathematics Vol. 14, 2nd ed. (American Mathematical Society
, Providence, RI
, 2001
).39.
E. H.
Lieb
, R.
Seiringer
, and J. P.
Solovej
, “Ground-state energy of the low-density Fermi gas
,” Phys. Rev. A
71
, 053605
(2005
).40.
E. H.
Lieb
and B.
Simon
, “The Hartree-Fock theory for Coulomb systems
,” Commun. Math. Phys.
53
, 185
–194
(1977
).41.
E. H.
Lieb
and B.
Simon
, “The Thomas-Fermi theory of atoms, molecules and solids
,” Adv. Math.
23
, 22
–116
(1977
).42.
E. H.
Lieb
and W. E.
Thirring
, “Bound on kinetic energy of fermions which proves stability of matter
,” Phys. Rev. Lett.
35
, 687
–689
(1975
).43.
E. H.
Lieb
and W. E.
Thirring
, “Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities
,” in Studies in Mathematical Physics
(Princeton University Press
, 1976
), pp. 269
–303
.44.
E. H.
Lieb
and W. E.
Thirring
, “Gravitational collapse in quantum mechanics with relativistic kinetic energy
,” Ann. Phys.
155
, 494
–512
(1984
).45.
E. H.
Lieb
and H.-T.
Yau
, “The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics
,” Commun. Math. Phys.
112
, 147
–174
(1987
).46.
P. S.
Madsen
, Ph.D. thesis, Aarhus University
, 2019
.47.
N. H.
March
, “Equations of state of elements from the Thomas-Fermi theory II: Case of incomplete degeneracy
,” Proc. Phys. Soc.
68
, 1145
(1955
).48.
R. E.
Marshak
and H. A.
Bethe
, “The generalized Thomas-Fermi method as applied to stars
,” Astrophys. J.
91
, 239
(1940
).49.
J.
Messer
, “Nonmonotonicity of the mass distribution and existence of the gravitational phase transition
,” Phys. Lett. A
83
, 304
–306
(1981
).50.
J.
Messer
, “On the gravitational phase transition in the Thomas-Fermi model
,” J. Math. Phys.
22
, 2910
–2917
(1981
).51.
J.
Messer
, Temperature Dependent Thomas-Fermi theory
, Lecture Notes in Physics Vol. 147 (Springer-Verlag
, Berlin-New York
, 1981
).52.
H.
Narnhofer
and G.
Sewell
, “Vlasov hydrodynamics of a quantum mechanical model
,” Commun. Math. Phys.
79
, 9
–24
(1981
).53.
H.
Narnhofer
and W.
Thirring
, “Asymptotic exactness of finite temperature Thomas-Fermi theory
,” Ann. Phys.
134
, 128
–140
(1981
).54.
S.
Petrat
and P.
Pickl
, “A new method and a new scaling for deriving fermionic mean-field dynamics
,” Math. Phys. Anal. Geom.
19
, 3
(2016
).55.
M.
Reed
and B.
Simon
, Methods of Modern Mathematical Physics. I. Functional Analysis
(Academic Press
, 1972
).56.
D. W.
Robinson
, The Thermodynamic Pressure in Quantum Statistical Mechanics
, Lecture Notes in Physics Vol. 9 (Springer-Verlag
, Berlin, New York
, 1971
).57.
N.
Rougerie
, “De Finetti theorems, mean-field limits and Bose-Einstein condensation
,” e-print arXiv:1506.05263 (2015
).58.
D.
Ruelle
, Statistical Mechanics. Rigorous Results
(World Scientific
, Singapore
; Imperial College Press
, London
, 1999
).59.
K.
Schönhammer
, “Deviations from Wick’s theorem in the canonical ensemble
,” Phys. Rev. A
96
, 012102
(2017
).60.
R.
Seiringer
, “The thermodynamic pressure of a dilute Fermi gas
,” Commun. Math. Phys.
261
, 729
–757
(2006
).61.
B.
Simon
, “The classical limit of quantum partition functions
,” Commun. Math. Phys.
71
, 247
–276
(1980
).62.
H.
Spohn
, “On the Vlasov hierarchy
,” Math. Methods Appl. Sci.
3
, 445
–455
(1981
).63.
W. E.
Thirring
, Quantum Mathematical Physics: Atoms, Molecules and Large Systems
, 2nd ed. (Springer
, 2002
).64.
A.
Triay
, “Mean-field limits in quantum mechanics
,” Ph.D. thesis, University of Paris-Dauphine
, 2019
.65.
53 works under the condition that for some a > 0 for large p. Not all interaction potentials can, therefore, be covered.
© 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.