UsingPto induce a measure on the set of minimizers of the Thomas-Fermi functional concludes the proof of TheoremA.6.
A.7 Appendix: Weyl asymptotics for the Dirichlet Pauli
and hence
lim sup
~→0
P
jej(~, b,Λ)
Escl(~, b,Λ) ≤lim sup
~→0
P
jej(~, b) Escl(~, b) = 1.
In fact, it is shown in the proof of [20, Theorem 3.1] that X
j
ej(~, b)≥Escl(~, b) +o b
~2 + 1
~3
,
implying one of the bounds in Remark A.50.
We will use the coherent states from Section A.4 to construct an appropriate trial state for the corresponding upper bound on the sum of eigenvalues, using the construction in [20] as a guide line. Let f ∈Cc∞(R3) with support contained in the unit ball B(0,1), and denote by Λ~ ={x ∈Λ|d(x, ∂Λ) >2~1/2}. We furthermore introduce
K~,b={(u, p, j, s)∈R3×R×N0× {±1} |p2+~b(2j+ 1 +s)≤V−(u)}, and letM~,b be the characteristic function ofK~,b∩ {(u, p, j, s)|u∈Λ~}. We define the trial state γ~ by
γ~ = b (2π~)2
X
s=±1
X
j≥0
Z
R3
Z
R
M~,b(u, p, j, s)Pu,p,j~,b Psdpdu. (A.109) Note by (A.74) for anyψ∈L2(R3;C2) that
b 2π~
ψ, Pu,p,j~,b Psψ
= 2π Z
R2
|Π(2)j ⊗ F3
f~(· −u)ψ(·, s)
(y,~−1p)|2dy, wheresuppf~ =~1/2suppf ⊆B(0,~1/2). Hence, ifu∈Λ~ andd(suppψ(·, s),Λ~)≥
~1/2, then we have hψ, Pu,p,j~,b Psψi= 0, because in this casef~(· −u)ψ(·, s) = 0. It follows that γ~ indeed is a suitable trial state for the Dirichlet problem onΛ, so by the variational principle,
X
j≥1
ej(~, b,Λ)≤Tr[H(~, b,Λ)γ~]
= b
(2π~)2 X
s=±1
X
j≥0
Z
R3
Z
R
M~,b(u, p, j, s) Tr[H(~, b,Λ)Pu,p,j~,b Ps] dpdu.
We proceed to estimate the terms on the right hand side individually. For the po- tential energy term, note that by definition of the coherent states,
Tr
V+Pu,p,j~,b
= Tr V
1
+2Pu,p,j~,b V
1
+2
= Z
R2
V
1
+2fx,u,p,j~,b
2 2dx
= Z
R2
Z
R3
V+(y)|ϕ~x,j,b(y⊥)|2|f~(y−u)|2dydx=V+∗ |f~|2(u),
with the same equality also holding for V−. For the kinetic energy term we have, using the IMS localization formula (A.80),
Tr[(−i~∇+bA)2Pu,p,j~,b ] = Z
R2
(−i~∇+bA)fx,u,p,j~,b
2 2dx
= Z
R2
ϕ~,bx,j⊗ei
p(·)
~ , f~(· −u)(−i~∇+bA)2f~(· −u)ϕ~,bx,j⊗ei
p(·)
~ dx
= Z
R2
ϕ~,bx,j⊗ei
p(·)
~ ,|f~(· −u)|2(−i~∇+bA)2ϕ~,bx,j⊗ei
p(·)
~ +~2
ϕ~x,j,b ⊗eip(·)~ ,|∇f~(· −u)|2ϕ~x,j,b ⊗eip(·)~ dx
= (p2+~b(2j+ 1)) Z
R2
fx,u,p,j~,b
2
2dx+~2 Z
R3
|∇f~|2
=p2+~b(2j+ 1) +~ Z
R3
|∇f|2, implying that
Tr[(σ·(−i~∇+bA))2Pu,p,j~,b Ps] = Tr[((−i~∇+bA)21C2+~bσ3)Pu,p,j~,b Ps]
=p2+~b(2j+ 1 +s) +~ Z
R3
|∇f|2. Collecting the terms and using (A.43) and (A.45), we have shown
X
j≥1
ej(~, b,Λ)≤ b (2π~)2
X
j≥0 s=±1
Z
Λ~×R
(p2+~b(2j+ 1 +s))1K~,bdudp
+ b
(2π~)2 X
j≥0 s=±1
Z
Λ~×R
V ∗ |f~|2(u) +~ Z
|∇f|2
1K~,bdudp
= 1
~3 Z
Λ~
V−(u)P~0b(V−(u))−P~b(V−(u)) du + 1
~3 Z
Λ~
V ∗ |f~|2(u) +~ Z
|∇f|2
P~0b(V−(u)) du.
To compare with the semi-classical expression for the sum of eigenvalues, we will need the bounds [20, Theorem3.1]
k1 b
~2y12 +k2 1
~3y32 ≤ 1
~3P~0b(y)≤K1 b
~2y12 +K2 1
~3y32,
valid for all y ≥ 0, where k1, k2, K1, K2 are positive constants. These bounds also imply
c1
b
~2y32 +c2
1
~3y52 ≤ 1
~3P~b(y)≤C1
b
~2y32 +C2
1
~3y52, and
˜ c b
~2 + 1
~3 ≤
Z 1
~3P~b(V−(u)) du≤Ce b
~2 + 1
~3
. (A.110)
We obtain for the leading term
− 1
~3 Z
Λ~
P~b(V−(u)) du
=− 1
~3 Z
Λ
P~b(V−(u)) du+ Z
Λ\Λ~
C1 b
~2V−(u)32 +C2 1
~3V−(u)52 du, and applying Hölder’s inequality for the subleading terms,
1
~3 Z
Λ~
P~0b(V−(u)) du≤C Z
Λ
b
~2V−(u)32 + 1
~3V−(u)52 du
≤ C
~3 V−
3 2 3 2
+C b
~2
|Λ|23Z
Ω
V−(u)32du 13
, and
1
~3 Z
Λ~
(V−(u) +V ∗ |f~|2(u))P~0b(V−(u)) du
≤C Z
suppV−
(V−+V ∗ |f~|2) b
~2V
1
−2 + 1
~3V
3
−2
du
≤C b
~2 Z
Λ
V
3 2
− du13Z
suppV−
V ∗ |f~|2−V
3 2 du23 + C
~3 Z
Λ
V
5
−2 du35Z
suppV−
V ∗ |f~|2−V
5 2 du25
=C b
~2 V−
1 2 3 2
V ∗ |f~|2−V
L32(suppV−)
+ C
~3 V−
3 2 5 2
V ∗ |f~|2−V
L52(suppV−). Collecting these bounds, we have
X
j≥1
ej(~, b,Λ)≤ − 1
~3 Z
Λ
P~b(V−(u)) du+C b
~2 + 1
~3
ε(~), (A.111) whereε(~) is independent ofb, andlim~→0ε(~) = 0. Finally, by the upper bound in (A.110), we conclude
lim inf
~→0
P
jej(~, b,Λ) Escl(~, b,Λ) ≥1 uniformly inb.
Lastly, we also include a proof of CorollaryA.9, elaborating on the Weyl asymp- totics when ~ and b satisfy the scaling relations (A.5). For this, it is very useful to know the error bound in Remark A.50.
Proof of Corollary A.9. We start out by considering the case0< β <∞. We wish to apply dominated convergence to the sequence P~b(V−(x)). Note for eachj ≥1that limN→∞[2~bj−V−(x)]− = [2kβj−V−(x)]− for almost every x ∈R3. Note also for eachxthat the number of non-zero terms in the sum in the semi-classical expression (A.107) is bounded byV−(x)(2~b)−1, and for eachj≥1we have[2~bj−V−(x)]−≤ [2~b−V−(x)]−. Hence we get the integrable pointwise bound
∞
X
j=1
[2~bj−V−(x)]
3
−2 ≤ V−(x)
2~b [2~b−V−(x)]
3
−2 ≤ V−(x)52 2~b , so by dominated convergence,
Nlim→∞
Z
R3
~b
∞
X
j=1
[2~bj−V−(x)]
3
−2 dx= Z
R3
kβ
∞
X
j=1
[2kβj−V−(x)]
3
−2 dx.
Combining this with the error bound in RemarkA.50 and the definitions of~andb by (A.4), we obtain
N→∞lim 1 N
X
j
ej(~, b, V) = lim
N→∞
1
NEscl(~, b, V)
= lim
N→∞− 1
~3N Z
R3
~b 3π2
V−(x)32 + 2
∞
X
j=1
[2~bj−V−(x)]
3
−2
dx
=−(1 +β)−35 Z
R3
Pkβ(V−(x)) dx.
For the caseβ = 0, note that the sum in the semi-classical expression becomes a Riemann sum, i.e.
N→∞lim 2~b
∞
X
j=1
[2~bj−V−(x)]
3
−2 = lim
N→∞2~b
bV−(x)(2~b)−1c
X
j=1
[2~bj−V−(x)]
3
−2
=
Z V−(x) 0
[y−V−(x)]
3
−2 dy= 2
5V−(x)52, so we obtain the result in the same way by applying RemarkA.50 as above.
In case βN → ∞, the contribution from all the higher Landau levels to the pressureP~bgoes to zero pointwise asN goes to infinity, so we conclude by monotone convergence and Remark A.50.
References
[1] W. Braun and K. Hepp. The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles. Comm. Math. Phys., 56(2):101–113, 1977. ISSN 0010- 3616.
[2] E. Caglioti, P.-L. Lions, C. Marchioro, and M. Pulvirenti. A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Comm.
Math. Phys.,143(3):501–525, 1992. ISSN 0010-3616.
[3] B. de Finetti.Funzione caratteristica di un fenomeno aleatorio. Atti della R. Accademia Nazionale dei Lincei, 1931. Ser. 6, Memorie, Classe di Scienze Fisiche, Matematiche e Naturali.
[4] P. Diaconis and D. Freedman. Finite exchangeable sequences. Ann. Probab.,8(4):745–
764, 1980. ISSN 0091-1798.
[5] S. Fournais, M. Lewin, and J. P. Solovej. The semi-classical limit of large fermionic systems. Calc. Var. Partial Differ. Equ., pages 57–105, 2018. doi:10.1007/
s00526-018-1374-2.
[6] C. Hainzl and R. Seiringer.Bounds on One-Dimensional Exchange Energies with Appli- cation to Lowest Landau Band Quantum Mechanics. Letters in Mathematical Physics, 55(2):133–142, 2001. ISSN 1573-0530. doi:10.1023/A:1010951905548.
[7] C. Hainzl and R. Seiringer. A Discrete Density Matrix Theory for Atoms in Strong Magnetic Fields. Communications in Mathematical Physics, 217(1):229–248, 2001.
ISSN 1432-0916. doi:10.1007/s002200100373.
[8] B. Hauksson and J. Yngvason.Asymptotic Exactness of Magnetic Thomas–Fermi The- ory at Nonzero Temperature.Journal of Statistical Physics,116(1):523–546, 2004. ISSN 1572-9613. doi:10.1023/B:JOSS.0000037223.74597.4e.
[9] E. Hewitt and L. J. Savage. Symmetric measures on Cartesian products. Trans. Amer.
Math. Soc.,80:470–501, 1955. ISSN 0002-9947.
[10] R. L. Hudson and G. R. Moody. Locally normal symmetric states and an analogue of de Finetti’s theorem. Z. Wahrscheinlichkeitstheor. und Verw. Gebiete,33(4):343–351, 1975/76.
[11] M. K.-H. Kiessling. Statistical mechanics of classical particles with logarithmic inter- actions. Comm. Pure. Appl. Math.,46:27–56, 1993.
[12] M. Lewin, P. S. Madsen, and A. Triay. Semi-classical limit of large fermionic systems at positive temperature. ArXiv e-prints, 2019.
[13] 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. doi:10.1016/j.aim.2013.12.010.
[14] M. Lewin, P. T. Nam, and N. Rougerie. Remarks on the quantum de Finetti theorem for bosonic systems. Appl. Math. Res. Express (AMRX), 2015(1):48–63, 2015. doi:
10.1093/amrx/abu006.
[15] M. Lewin, P. T. Nam, and N. Rougerie. The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases. Trans. Amer. Math. Soc, 368(9):6131–6157, 2016. doi:10.1090/tran/6537.
[16] E. H. Lieb and M. Loss. Analysis, volume 14 of Graduate Studies in Mathematics.
American Mathematical Society, Providence, RI, 2nd edition, 2001. ISBN 0-8218-2783- 9.
[17] E. H. Lieb and B. Simon. The Hartree-Fock theory for Coulomb systems. Commun.
Math. Phys.,53(3):185–194, 1977. ISSN 0010-3616.
[18] E. H. Lieb and B. Simon. The Thomas-Fermi theory of atoms, molecules and solids.
Adv. Math.,23(1):22–116, 1977. ISSN 0001-8708.
[19] E. H. Lieb, J. P. Solovej, and J. Yngvason.Asymptotics of heavy atoms in high magnetic fields: I. Lowest Landau band regions. Comm. Pure Appl. Math.,47(4):513–591, 1994.
[20] E. H. Lieb, J. P. Solovej, and J. Yngvason.Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical regions. Comm. Math. Phys.,161(1):77–124, 1994.
[21] E. H. Lieb, J. P. Solovej, and J. Yngvason. Ground states of large quantum dots in magnetic fields. Phys. Rev. B,51:10646–10665, 1995. doi:10.1103/PhysRevB.51.10646.
[22] E. H. Lieb and W. E. Thirring. Gravitational collapse in quantum mechanics with relativistic kinetic energy. Ann. Physics,155(2):494–512, 1984. ISSN 0003-4916.
[23] E. H. Lieb and H.-T. Yau. The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys.,112(1):147–174, 1987. ISSN 0010-3616.
[24] P. Madsen. In preparation. Ph.D. thesis, Aarhus University, 2019.
[25] J. Messer and H. Spohn. Statistical mechanics of the isothermal Lane-Emden equation.
J. Statist. Phys.,29(3):561–578, 1982. ISSN 0022-4715. doi:10.1007/BF01342187.
[26] N. Rougerie. De Finetti theorems, mean-field limits and Bose-Einstein condensation.
ArXiv e-prints, 2015.
[27] R. Seiringer. On the maximal ionization of atoms in strong magnetic fields. Journal of Physics A: Mathematical and General,34(9):1943–1948, 2001. doi:10.1088/0305-4470/
34/9/311.
[28] H. Spohn. On the Vlasov hierarchy. Math. Methods Appl. Sci., 3(4):445–455, 1981.
ISSN 0170-4214. doi:10.1002/mma.1670030131.
[29] E. Størmer. Symmetric states of infinite tensor products ofC∗-algebras. J. Functional Analysis, 3:48–68, 1969.
Semi-classical limit of large fermionic systems at positive temperature
By M. Lewin, P. S. Madsen, and A. Triay
Available at arXiv:1902.00310.
Abstract
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 as1/N and with an effective semi-classical parameter ~ = N−1/d where d is the space dimension, we prove the convergence to the corresponding Thomas-Fermi model at positive temperature.
B.1 Introduction
In this article we study mean-field-type limits for a system of N fermions at tem- perature T > 0 in a fixed confining potential. We assume that the interaction has an intensity of the order 1/N and that there is an effective semi-classical parameter
~ = N−1/d where d is the space dimension. In the limit N → ∞ we obtain the nonlinear Thomas-Fermi problem at the same temperature T >0. This paper is an extension of a recent work [18] by Fournais, Solovej and the first author where the case T = 0 was solved.
Physically, the Thomas-Fermi model is a rather crude approximation of quantum many-body systems in normal conditions, and it has to be refined in order to obtain a quantitative description of their equilibrium properties. However, certain physical systems in extreme conditions are rather well described by Thomas-Fermi theory.
79
It then becomes important to take into account the effect of the temperature. For instance, the positive-temperature Thomas-Fermi model has been thoroughly studied for very heavy atoms [17, 20, 30, 13, 53]. It has also played an important role in astrophysics, where the very high pressure encountered in the core of neutron stars and white dwarfs makes it valuable for all kinds of particles [48,47,12,5]. Finally, the Thomas-Fermi model is also useful for ultracold dilute atomic Fermi gases, but the interaction often becomes negligible due to the Pauli principle, except in the presence of spin or of several interacting species [21].
In the regime considered in this paper, a mean-field scaling is coupled to a semi- classical limit. This creates some mathematical difficulties. Before [18], this limit has been rigorously considered at T = 0 for atoms by Lieb and Simon in [41, 40] and for pseudo-relativistic stars by Lieb, Thirring and Yau in [44,45]. Upper and lower bounds on the next order correction have recently been derived in [26,7], for particles evolving on the torus. The positive temperature Thomas-Fermi model was derived for confined gravitational systems in [29,28,50,49,51] and for atoms in [53]. There are several mathematical works on the time-dependent setting [52, 62, 4,16, 1,19, 10,9,6,2,54,8,22,23,15], in which the Schrödinger dynamics has been proved to converge to the time-dependent Vlasov equation in the limit N → ∞. Finally, the first two terms in the expansion of the (free) energy of a Fermi gas with spin in the limit ρ→0 was provided in [39] at T = 0 and in [60] at T >0.
The mean-field limit at positive temperature for fermions is completely different from the bosonic case. It was proved in [32] that in the similar mean-field regime for bosons, the leading order is the same atT >0as whenT = 0. Only the next (Bogoli- ubov) correction depends on T [37]. In order to observe an effect of the temperature at the leading order of the bosonic free energy, one should takeT ∼N, a completely different limit where nonlinear Gibbs measures arise [24,33,35,36,34,57]. Without statistics (boltzons), the temperature does affect the leading order of the energy [31], and the same happens for fermions, as we will demonstrate.
Our method for studying the Fermi gas in the coupled mean-field/semi-classical limit relies on techniques previously introduced in [18]. Assuming that the interaction is positive-type (wb ≥ 0), the lower bound follows from using coherent states and inequalities on the entropy. We discuss later in RemarkB.6a conjectured inequality on the entropy of large fermionic systems which would imply the result for any interaction potential, not necessarily of positive-type. The upper bound is slightly more tedious. The idea is to construct a trial state with locally constant density in small boxes of side length much larger than ~, and to use the equivalence between the canonical and grand-canonical ensembles for the free Fermi gas. Finally, the convergence of states requires the tools recently introduced in [18] based on the classical de Finetti theorem for fermions.
The article is organized as follows. In the next section we introduce both theN- particle quantum Hamiltonian and the positive-temperature Thomas-Fermi theory which is obtained in the limit. We then state our main theorems, Theorem B.2 and
TheoremB.8. As an intermediate result for the upper bound, we show in Section B.3 how to approximate a classical density by anN body quantum state. In SectionB.4, we use this trial state and some known results about the free Fermi gas at positive temperature to prove our main result in the non-interacting case. The interacting case is dealt with in SectionB.5. Finally, in Section B.6we study the Gibbs state and the minimizers of the Thomas-Fermi functional at positive temperature (TheoremB.1).