Abstract
We consider the grand canonical pressure for Coulombic matter with nuclear charges ∼Zin a magnetic field Band at nonzero temperature. We prove that its asymptotic limit as Z→∞ with B/Z 3→0 can be obtained by minimizing a Thomas–Fermi type pressure functional.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
The Stability of Matter: From Atoms to Stars, Selecta of E. H. Lieb, ed. by W. Thirring, (Springer-Verlag, Berlin etc., 2001).
E. H. Lieb and B. Simon, The Thomas-Fermi theory of atoms, molecules, and solids, Adv. in Math. 23:22–116 (1977).
E. H. Lieb, Thomas-Fermi and related theories of atoms and molecules, Rev. Mod. Phys. 53:603–641 (1981); Erratum, Rev. Mod. Phys. 54:311 (1982).
W. Thirring, A lower bound with the best possible constant for the Coulomb Hamiltonian, Comm. Math. Phys. 79:1–7 (1981).
I. Fushiki, E. H. Gudmundsson, and C. J. Pethick, Surface structure of neutron stars with high magnetic fields, Astrophys. J. 342:958–975 (1989).
I. Fushiki, E. H. Gudmundsson, C. J. Pethick, and J. Yngvason, Matter in a magnetic field in the Thomas-Fermi and related theories, Ann. Phys. 216:29–72 (1992).
ö. E. Rögnvaldsson, I. Fushiki, C. J. Pethick, E. H. Gudmundsson, and J. Yngvason, Thomas-Fermi calculations of atoms and matter in magnetic neutron stars: Effects of higher landau bands, Astrophys. J. 416:276 (1993).
J. Yngvason, Thomas-Fermi theory for matter in a magnetic field as a limit of quantum mechanics, Lett. Math. Phys. 22:107–117 (1991).
E. H. Lieb, J. P. Solovej, and J. Yngvason, Asymptotics of heavy atoms in high magnetic fields: II. semiclassical regions, Comm. Math. Phys. 161:77–124 (1994).
V. Ivrii, Asymptotics of the ground state energy of heavy molecules in the strong magnetic field. I, Russian J. Math. Phys. 4:29–74 (1996); II, Russian J. Math. Phys. 5:321-354 (1997).
L. Erdoős and J. P. Solovej, Semiclassical eigenvalue estimates for the Pauli operator with strong non-homogeneous magnetic fields: II. Leading order asymptotic estimates, Comm. Math. Phys. 188:599–656 (1997).
H. Narnhofer and W. Thirring, Asymptotic exactness of finite temperature Thomas-Fermi theory, Ann. Phys. 134:128–140 (1981).
W. Thirring, A Course in Mathematical Physics, Vol.4: Quantum Theory of Large Systems(Springer-Verlag, Wien, 1983).
J. Messer, Temperature Dependent Thomas-Fermi Theory, Springer Lecture Notes in Physics, Vol. 147 (Springer, Berlin etc., 1981).
A. Thórólfsson, ö. E. Rögnvaldsson, J. Yngvason, and E. H. Gudmundsson, Thomas-Fermi calculations of atoms and matter in magnetic neutron stars. II. Finite temperature effects, Astrophys. J. 502:847–857 (1998).
O. B. Firsov, Calculation of the interaction potential of atoms for small nuclear separation, Sov. Phys.-JETP 5:1192–1196 (1957). See also: R. Benguria, The von Weizsäcker and Exchange Corrections in Thomas-Fermi Theory, Ph.D. thesis (Princeton University, 1979) (unpublished).
E. H. Lieb and M. Loss, Analysis, second edition (Amer. Math. Soc., 2001).
L. Hörmander, The Analysis of Linear Partial Differential Operators II(Springer-Verlag, Berlin etc., 1983).
E. H. Lieb and W. E. Thirring, Bound for the kinetic energy of fermions which proves the stability of matter, Phys. Rev. Lett. 35:687–689 (1975).
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, 513–591 (1994); E. H. Lieb, J. P. Solovej, and J. Yngvason, Heavy atoms in the strong magnetic field of a neutron star, Phys. Rev. Lett. 69:749-752 (1992).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hauksson, B., Yngvason, J. Asymptotic Exactness of Magnetic Thomas—Fermi Theory at Nonzero Temperature. Journal of Statistical Physics 116, 523–546 (2004). https://doi.org/10.1023/B:JOSS.0000037223.74597.4e
Issue Date:
DOI: https://doi.org/10.1023/B:JOSS.0000037223.74597.4e