×

Non-cooperative equilibria of Fermi systems with long range interactions. (English) Zbl 1307.82012

Mem. Am. Math. Soc. 1052, v, 155 p. (2013).
The present monograph is devoted to the systematic study of fermionic quantum models with long-range interactions. The set of translation invariant femionic quantum models with long-range interactions is defined, and it is shown to form a Banach space. The main results concern an explicit study of the structure of generalized equilibrium states for any model belonging to the Banach space. For each model from the Banach space, the provided methods and investigations allow to study and find the correlation functions of the equilibrium states. It is also shown that the themodynamics of each considered long-range interactions model is governed by a non-cooperative equilibrium of a zero-sum game.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
35Q82 PDEs in connection with statistical mechanics
91A10 Noncooperative games
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. · Zbl 0867.46001
[2] Erik M. Alfsen, Compact convex sets and boundary integrals, Springer-Verlag, New York-Heidelberg, 1971. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 57. · Zbl 0209.42601
[3] Robert R. Phelps, Lectures on Choquet’s theorem, 2nd ed., Lecture Notes in Mathematics, vol. 1757, Springer-Verlag, Berlin, 2001. · Zbl 0997.46005
[4] Robert B. Israel, Convexity in the theory of lattice gases, Princeton University Press, Princeton, N.J., 1979. Princeton Series in Physics; With an introduction by Arthur S. Wightman. · Zbl 0399.46055
[5] O. BRATTELLI AND D.W. ROBINSON, Operator Algebras and Quantum Statistical Mechanics, Vol. II, 2nd ed. New York: Springer-Verlag, 1996
[6] G. A. Raggio and R. F. Werner, Quantum statistical mechanics of general mean field systems, Helv. Phys. Acta 62 (1989), no. 8, 980-1003. · Zbl 0938.82501
[7] G. A. Raggio and R. F. Werner, The Gibbs variational principle for inhomogeneous mean-field systems, Helv. Phys. Acta 64 (1991), no. 5, 633-667. · Zbl 0938.82500
[8] Huzihiro Araki and Hajime Moriya, Equilibrium statistical mechanics of fermion lattice systems, Rev. Math. Phys. 15 (2003), no. 2, 93-198. · Zbl 1105.82003 · doi:10.1142/S0129055X03001606
[9] J.-B. Bru and W. de Siqueira Pedra, Effect of a locally repulsive interaction on \(s\)-wave superconductors, Rev. Math. Phys. 22 (2010), no. 3, 233-303. · Zbl 1192.82086 · doi:10.1142/S0129055X10003953
[10] J.-B. Bru, W. de Siqueira Pedra, and A.-S. Dömel, A microscopic two-band model for the electron-hole asymmetry in high-\(T_c\) superconductors and reentering behavior, J. Math. Phys. 52 (2011), no. 7, 073301, 28. · Zbl 1317.82060 · doi:10.1063/1.3600202
[11] D. LEBOEUF ET AL., Electron pockets in the Fermi surface of hole-doped high-Tc superconductors. Nature 450, 533-536 (2007)
[12] C. PFLEIDERER AND R. HACKL, Schizophrenic electrons. Nature 450,492-493 (2007)
[13] J. Ginibre, On the asymptotic exactness of the Bogoliubov approximation for many boson systems, Comm. Math. Phys. 8 (1968), 26-51. · Zbl 0155.32701
[14] J.-B. BRU AND W. DE SIQUEIRA PEDRA, Microscopic Foundations of the Meissner Effect - Thermodynamic Aspects, mp_arc 12-42 (2012) · Zbl 1281.82033
[15] Erling Størmer, Symmetric states of infinite tensor products of \(C^{\ast } \)-algebras, J. Functional Analysis 3 (1969), 48-68. · Zbl 0167.43403
[16] N. N. Jr. Bogolubov, On model dynamical systems in statistical mechanics, Physica 32 (1966), 933-944.
[17] N.N. BOGOLIUBOV JR., J.G. BRANKOV, V.A. ZAGREBNOV, A.M. KURBATOV AND N.S. TONCHEV, Metod approksimiruyushchego gamil’toniana v statisticheskoi fizike (The Approximating Hamiltonian Method in Statistical Physics). Sofia: Izdat. Bulgar. Akad. Nauk (Publ. House Bulg. Acad. Sci.), 1981 · Zbl 0469.70021
[18] N. N. Bogolyubov Jr., I. G. Brankov, V. A. Zagrebnov, A. M. Kurbatov, and N. S. Tonchev, Some classes of exactly solvable model problems of quantum statistical mechanics: the method of the approximating Hamiltonian, Uspekhi Mat. Nauk 39 (1984), no. 6(240), 3-45 (Russian).
[19] J.G. BRANKOV, D.M. DANCHEV AND N.S. TONCHEV, Theory of Critical Phenomena in Finite-size Systems: Scaling and Quantum Effects. Singapore-New Jersey-London-Hong Kong: World Scientific, 2000 · Zbl 0967.82002
[20] Ola Bratteli and Derek W. Robinson, Operator algebras and quantum statistical mechanics. 1, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, New York, 1987. \(C^\ast \)- and \(W^\ast \)-algebras, symmetry groups, decomposition of states. · Zbl 0905.46046
[21] M. Fannes, The entropy density of quasi free states, Comm. Math. Phys. 31 (1973), 279-290. · Zbl 1125.82309
[22] M. Fannes, A continuity property of the entropy density for spin lattice systems, Comm. Math. Phys. 31 (1973), 291-294. · Zbl 1125.82310
[23] Robert B. Israel, Generic triviality of phase diagrams in spaces of long-range interactions, Comm. Math. Phys. 106 (1986), no. 3, 459-466. · Zbl 0613.60096
[24] Fumio Hiai, Milán Mosonyi, Hiromichi Ohno, and Dénes Petz, Free energy density for mean field perturbation of states of a one-dimensional spin chain, Rev. Math. Phys. 20 (2008), no. 3, 335-365. · Zbl 1153.82002 · doi:10.1142/S0129055X08003298
[25] W. De Roeck, Christian Maes, Karel Netočný, and Luc Rey-Bellet, A note on the non-commutative Laplace-Varadhan integral lemma, Rev. Math. Phys. 22 (2010), no. 7, 839-858. · Zbl 1197.82034 · doi:10.1142/S0129055X10004089
[26] M. Fannes, J. V. Pulè, and A. Verbeure, On Bose condensation, Helv. Phys. Acta 55 (1982), no. 4, 391-399.
[27] Joseph V. Pulé, André F. Verbeure, and Valentin A. Zagrebnov, On nonhomogeneous Bose condensation, J. Math. Phys. 46 (2005), no. 8, 083301, 8. · Zbl 1110.82005 · doi:10.1063/1.1985025
[28] J.-B. Bru and V. A. Zagrebnov, On condensations in the Bogoliubov weakly imperfect Bose gas, J. Statist. Phys. 99 (2000), no. 5-6, 1297-1338. · Zbl 0968.82007 · doi:10.1023/A:1018692823463
[29] C. N. Yang, Concept of off-diagonal long-range order and the quantum phases of liquid He and of superconductors, Rev. Modern Phys. 34 (1962), 694-704.
[30] W. METZNER, C. CASTELLANI AND C. DI CASTRO, Fermi systems with strong forward scattering. Advances in Physics 47(3), 317-445 (1998)
[31] H. YAMASE AND W. METZNER, Competition of Fermi surface symmetry breaking and superconductivity. Phys. Rev. B 75 155117-1-6 (2007)
[32] F.D.M. HALDANE. Helv. Phys. Acta 65, 152 (1992); Proceedings of the International School of Physics ‘Enrico Fermi’, Course CXXI, edited by R. A. Broglia and J. R. Schrieffer (Amsterdam: North-Holland, 1994).
[33] P. Kopietz, L. Bartosch, and F. Schütz, Introduction to the functional renormalization group, Lecture Notes in Physics, vol. 798, Springer-Verlag, Berlin, 2010. · Zbl 1196.82001
[34] N. G. Duffield and J. V. Pulé, A new method for the thermodynamics of the BCS model, Comm. Math. Phys. 118 (1988), no. 3, 475-494. · Zbl 0658.60141
[35] L. N. COOPER, Bound Electron Pairs in a Degenerate Fermi Gas. Phys. Rev 104, 1189-1190 (1956) · Zbl 0074.23705
[36] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Microscopic theory of superconductivity, Phys. Rev. (2) 106 (1957), 162-164. · Zbl 0090.45401
[37] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev. (2) 108 (1957), 1175-1204. · Zbl 0090.45401
[38] J.-B. BRU AND W. DE SIQUEIRA PEDRA, Inhomogeneous Fermi or Quantum Spin Systems on Lattices - I. J. Math. Phys. 53 123301 (2012); http://dx.doi.org/10.1063/1.4763465, (25 pages). · Zbl 1278.82009
[39] J.-B. BRU AND W. DE SIQUEIRA PEDRA, Inhomogeneous Fermi or Quantum Spin Systems on Lattices - II. In preparation. · Zbl 1278.82009
[40] G. L. Sewell, Quantum theory of collective phenomena, Monographs on the Physics and Chemistry of Materials, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications.
[41] Oscar E. Lanford III. and Derek W. Robinson, Statistical mechanics of quantum spin systems. III, Comm. Math. Phys. 9 (1968), 327-338. · Zbl 0172.27702
[42] J.-B. BRU AND W. DE SIQUEIRA PEDRA, Remarks on the \(\Gamma \)-regularization of Non-convex and Non-Semi-Continuous Functionals on Topological Vector Spaces. J. Convex Analysis 19(2), 467-483 (2012) · Zbl 1251.58004
[43] R. Haag, The mathematical structure of the Bardeen-Cooper- Schrieffer model., Nuovo Cimento (10) 25 (1962), 287-299 (English, with Italian summary). · Zbl 0113.46210
[44] W. Thirring and A. Wehrl, On the mathematical structure of the B.C.S.-model, Comm. Math. Phys. 4 (1967), 303-314. · Zbl 0163.23302
[45] G. EMCH, Algebraic Methods in Statistical Mechanics and Quantum Field Theory. New York: Wiley-Interscience, 1972 · Zbl 0235.46085
[46] S. MAZUR, Über konvexe Menge in linearen normierten Raumen. Studia. Math. 4, 70-84 (1933) · JFM 59.1074.01
[47] Barry Simon, The statistical mechanics of lattice gases. Vol. I, Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1993. · Zbl 0804.60093
[48] Valentin A. Zagrebnov and Jean-Bernard Bru, The Bogoliubov model of weakly imperfect Bose gas, Phys. Rep. 350 (2001), no. 5-6, 291-434. · Zbl 0971.82006 · doi:10.1016/S0370-1573(00)00132-0
[49] Jean-Pierre Aubin, Mathematical methods of game and economic theory, Reprint of the 1982 revised edition, Dover Publications, Inc., Mineola, NY, 2007. With a new preface by the author. · Zbl 1152.91005
[50] N. Bogolubov, On the theory of superfluidity, Acad. Sci. USSR. J. Phys. 11 (1947), 23-32.
[51] E.H. LIEB, R. SEIRINGER AND J. YNGVASON, Justification of \(c\)-Number Substitutions in Bosonic Hamiltonians. Phys. Rev. Lett. 94, 080401-1-4 (2005)
[52] A. SüTŐ, Equivalence of Bose-Einstein Condensation and Symmetry Breaking. Phys. Rev. Lett. 94, 080402-1-4 (2005)
[53] J.-B. Bru, Superstabilization of Bose systems. I. Thermodynamic study, J. Phys. A 35 (2002), no. 43, 8969-8994. · Zbl 1050.82002 · doi:10.1088/0305-4470/35/43/301
[54] J.-B. Bru, Superstabilization of Bose systems. II. Bose condensations and equivalence of ensembles, J. Phys. A 35 (2002), no. 43, 8995-9024. · Zbl 1050.82003 · doi:10.1088/0305-4470/35/43/302
[55] N.N. BOGOLIUBOV, On some problems of the theory of superconductivity, Physica 26, S1-S16 (1960)
[56] N.N. BOGOLIUBOV JR., A method for studying model Hamiltonians. Oxford: Pergamon, 1977
[57] M. Fannes, H. Spohn, and A. Verbeure, Equilibrium states for mean field models, J. Math. Phys. 21 (1980), no. 2, 355-358. · Zbl 0445.46049 · doi:10.1063/1.524422
[58] J. Lindenstrauss, G. Olsen, and Y. Sternfeld, The Poulsen simplex, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 1, vi, 91-114 (English, with French summary). · Zbl 0363.46006
[59] Winfried Schirotzek, Nonsmooth analysis, Universitext, Springer, Berlin, 2007. · Zbl 1120.49001
[60] Eberhard Zeidler, Nonlinear functional analysis and its applications. III, Springer-Verlag, New York, 1985. Variational methods and optimization; Translated from the German by Leo F. Boron. · Zbl 0583.47051
[61] Jean-Pierre Aubin, Optima and equilibria, 2nd ed., Graduate Texts in Mathematics, vol. 140, Springer-Verlag, Berlin, 1998. An introduction to nonlinear analysis; Translated from the French by Stephen Wilson. · Zbl 0930.91001
[62] John B. Conway, A course in functional analysis, Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1985. · Zbl 0558.46001
[63] Ebbe Thue Poulsen, A simplex with dense extreme points, Ann. Inst. Fourier. Grenoble 11 (1961), 83-87, XIV (English, with French summary). · Zbl 0104.08402
[64] Svatopluk Fučík, Jindřich Nečas, Jiří Souček, and Vladimír Souček, Spectral analysis of nonlinear operators, Lecture Notes in Mathematics, Vol. 346, Springer-Verlag, Berlin-New York, 1973. · Zbl 0268.47056
[65] Eberhard Zeidler, Nonlinear functional analysis and its applications. I, Springer-Verlag, New York, 1986. Fixed-point theorems; Translated from the German by Peter R. Wadsack. · Zbl 0583.47050
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.