×

zbMATH — the first resource for mathematics

The thermodynamic limit of quantum Coulomb systems. II: Applications. (English) Zbl 1165.81042
In a preceding paper, the authors introduced conditions of convergence to thermodynamic limit for function defined on the bounded open sets of the tri-dimensional space. Here they apply this result to three typical quantum systems which are respectively the perturbed crystal, quantum nuclei and electrons in a periodic magnetic field, and movable classical nuclei. Many known results are so recovered and generalized, like those of Lieb and Lebowitz for instance.
[Part I, cf. the authors, Adv. Math. 221, No. 2, 454–487 (2009; Zbl 1165.81041).]

MSC:
81V70 Many-body theory; quantum Hall effect
82B30 Statistical thermodynamics
81T10 Model quantum field theories
94A17 Measures of information, entropy
PDF BibTeX Cite
Full Text: DOI arXiv
References:
[1] Ammari, Z., Scattering theory for a class of fermionic Pauli-Fierz models, J. funct. anal., 208, 302-359, (2004) · Zbl 1050.81069
[2] Bach, V.; Lieb, E.H.; Solovej, J.-P., Generalized Hartree-Fock theory and the Hubbard model, J. statist. phys., 76, 1-2, 3-89, (1994) · Zbl 0839.60095
[3] Baxter, J.R., Inequalities for potentials of particle systems, Illinois J. math., 24, 4, 645-652, (1980) · Zbl 0476.31002
[4] Cancès, É.; Deleurence, A.; Lewin, M., A new approach to the modelling of local defects in crystals: the reduced Hartree-Fock case, Comm. math. phys., 281, 129-177, (2008) · Zbl 1157.82042
[5] Catto, I.; Le Bris, C.; Lions, P.-L., On the thermodynamic limit for Hartree-Fock type models, Ann. inst. H. Poincaré anal. non linéaire, 18, 6, 687-760, (2001) · Zbl 0994.35115
[6] Conlon, J.; Lieb, E.H.; Yau, H.T., The \(N^{7 / 5}\) law for charged bosons, Comm. math. phys., 116, 417-448, (1988)
[7] Conlon, J.; Lieb, E.H.; Yau, H.T., The Coulomb gas at low temperature and low density, Comm. math. phys., 125, 153-180, (1989) · Zbl 0682.76066
[8] Daubechies, I.; Lieb, E.H., One-electron relativistic molecules with Coulomb interaction, Comm. math. phys., 90, 4, 497-510, (1983) · Zbl 0946.81522
[9] Dereziński, J.; Gérard, C., Asymptotic completeness in quantum field theory. massive Pauli-Fierz Hamiltonians, Rev. math. phys., 11, 4, 383-450, (1999) · Zbl 1044.81556
[10] Dyson, F.J.; Lenard, A., Stability of matter I, J. math. phys., 8, 423-434, (1967) · Zbl 0948.81665
[11] Dyson, F.J.; Lenard, A., Stability of matter II, J. math. phys., 9, 698-711, (1968) · Zbl 0948.81666
[12] Fefferman, C., The thermodynamic limit for a crystal, Comm. math. phys., 98, 289-311, (1985) · Zbl 0603.35079
[13] Fisher, M.E., The free energy of a macroscopic system, Arch. ration. mech. anal., 17, 377-410, (1964)
[14] Graf, G.M., Stability of matter through an electrostatic inequality, Helv. phys. acta, 70, 1-2, 72-79, (1997) · Zbl 0864.58067
[15] Graf, G.M.; Schenker, D., On the molecular limit of Coulomb gases, Comm. math. phys., 174, 1, 215-227, (1995) · Zbl 0837.58048
[16] Hainzl, C.; Lewin, M.; Solovej, J.P., The thermodynamic limit of quantum Coulomb systems: A new approach, () · Zbl 1156.81038
[17] C. Hainzl, M. Lewin, J.P. Solovej, The thermodynamic limit of quantum Coulomb systems. Part I. General theory, Adv. Math. (2009), doi:10.1016/j.aim.2008.12.010 (this issue) · Zbl 1165.81041
[18] Hughes, W., Thermodynamics for Coulomb systems: A problem at vanishing particle densities, J. statist. phys., 41, 5-6, 975-1013, (1985) · Zbl 0646.35066
[19] Lanford, O.E.; Robinson, D.W., Mean entropy of states in quantum-statistical mechanics, J. math. phys., 9, 7, 1120-1125, (1965) · Zbl 0174.28303
[20] Lewin, M., A mountain pass for reacting molecules, Ann. Henri Poincaré, 5, 477-521, (2004) · Zbl 1062.81158
[21] Li, P.; Yau, S.T., On the Schrödinger equation and the eigenvalue problem, Comm. math. phys., 88, 309-318, (1983) · Zbl 0554.35029
[22] Lieb, E.H., The stability of matter, Rev. modern phys., 48, 4, 553-569, (1976)
[23] Lieb, E.H., The stability of matter: from atoms to stars, Bull. amer. math. soc. (N.S.), 22, 1, 1-49, (1990) · Zbl 0698.35135
[24] Lieb, E.H., The stability of matter and quantum electrodynamics, (), Milan J. math., Jahresber. Deutsch. math.-verein., 106, 3, 93-110, (2004), a further modification appears in: · Zbl 1255.81002
[25] Lieb, E.H.; Lebowitz, J.L., The constitution of matter: existence of thermodynamics for systems composed of electrons and nuclei, Adv. math., 9, 316-398, (1972) · Zbl 1049.82501
[26] Lieb, E.H.; Loss, M., Analysis, Grad. stud. math., vol. 14, (2001), Amer. Math. Soc. · Zbl 0966.26002
[27] Lieb, E.H.; Ruskai, M.B., A fundamental property of quantum-mechanical entropy, Phys. rev. lett., 30, 434-436, (1973)
[28] Lieb, E.H.; Ruskai, M.B., Proof of the strong subadditivity of quantum-mechanical entropy. with an appendix by B. Simon, J. math. phys., 14, 1938-1941, (1973)
[29] Lieb, E.H.; Thirring, W., Bound on kinetic energy of fermions which proves stability of matter, Phys. rev. lett., 35, 687-689, (1975)
[30] Lieb, E.H.; Yau, H.-T., The stability and instability of relativistic matter, Comm. math. phys., 118, 177-213, (1988) · Zbl 0686.35099
[31] M. Loss, Stability of matter, review for the Young Researchers Symposium in Lisbon, 2003
[32] Onsager, L., Electrostatic interaction of molecules, J. phys. chem., 43, 189-196, (1939)
[33] Robinson, D.W.; Ruelle, D., Mean entropy of states in classical statistical mechanics, Comm. math. phys., 5, 288-300, (1967) · Zbl 0144.48205
[34] Ruelle, D., Statistical mechanics. rigorous results, (1999), Imperial College Press/World Sci. Publ. · Zbl 1016.82500
[35] Simon, B., Functional integration and quantum physics, Pure appl. math., vol. 86, (1979), Academic Press New York · Zbl 0434.28013
[36] Solovej, J.P., Proof of the ionization conjecture in a reduced Hartree-Fock model, Invent. math., 104, 2, 291-311, (1991) · Zbl 0732.35066
[37] Solovej, J.P., The energy of charged matter, (), 113-129 · Zbl 1221.81210
[38] Wehrl, A., General properties of entropy, Rev. modern phys., 50, 2, 221-260, (1978) · Zbl 0484.70014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.