The Mathematical Theory of Thermodynamic Limits Thomas--Fermi Type Models

by Catto, Isabelle; Le Bris, Claude; Lions, Pierre-Louis

• ISBN13: 9780198501619
• ISBN10: 0198501617
• Format: Hardcover
• Copyright: 1998-12-10
• Publisher: Clarendon Press

Summary

The thermodynamic limit is a mathematical technique which allows us to consider crystals (or other macroscopic objects) as infinitely sized periodically arranged molecules. This means that we can derive models in solid state physics from models in quantum chemistry. Based on this technique,the book presents established as well as new mathematical results for a large class of models in quantum chemistry.

Table of Contents

1 General Presentation	1	(33)
1.1 Introduction	1	(1)
1.2 The thermodynamic limit from the mathematical viewpoint	1	(5)
1.3 Physical background	6	(5)
1.3.1 The thermodynamic limit of microscopic systems	7	(1)
1.3.2 Some basics of solid state theory	8	(3)
1.4 Historical Survey	11	(5)
1.4.1 Statistical mechanics works	11	(3)
1.4.2 The Thomas-Fermi theory	14	(2)
1.5 The Thermodynamic Limit in the TFW Setting	16	(15)
1.5.1 Description of the models and statement of the main results	17	(11)
1.5.2 Corollaries	28	(3)
1.6 The thermodynamic limit in the Hartree-Fock setting	31	(3)
2 Convergence of the energy for the Thomas-Fermi-von Weizsacker model with Yukawa potential	34	(51)
2.1 Introduction	34	(4)
2.2 Preliminaries: a priori estimates	38	(14)
2.3 Lower bound for the energy	52	(5)
2.4 Upper bound for the energy	57	(12)
2.5 Proof of the main theorem	69	(1)
2.6 Recovering the Coulomb potential: compactness	70	(10)
2.7 Extensions	80	(5)
2.7.1 Smeared out nuclei	80	(2)
2.7.2 Replacing Rho(5/3) by some other function	82	(1)
2.7.3 Default and excess of charge	83	(1)
2.7.4 Other shapes of periodic cells	84	(1)
3 Convergence of the energy for the Thomas-Fermi-von Weizsacker model	85	(71)
3.1 Introduction	85	(6)
3.2 Preliminaries: a priori estimates	91	(13)
3.2.1 The case of smeared nuclei	91	(6)
3.2.2 The point nuclei case	97	(7)
3.3 Proof of compactness	104	(9)
3.3.1 Smeared out nuclei	105	(1)
3.3.2 Convexity argument	106	(1)
3.3.3 Scaling argument	107	(2)
3.3.4 General argument	109	(2)
3.3.5 An argument via the estimation of the outside charge	111	(1)
3.3.6 Compactness in the TF case	112	(1)
3.4 Lower bound for the energy	113	(14)
3.4.1 Comparison proof	113	(2)
3.4.2 Direct proof for the (Cb - m) program	115	(4)
3.4.3 Direct proof for the (Cb - Delta) program	119	(7)
3.4.4 Adaptation to the Yukawa case	126	(1)
3.5 Upper bound for the energy	127	(9)
3.6 Some extensions	136	(16)
3.6.1 Other sizes of elementary cubic cells	136	(5)
3.6.2 Other shapes of nucleus	141	(1)
3.6.3 The Thomas-Fermi-Dirac-von Weizsacker model	142	(10)
3.7 Appendix	152	(4)
4 Convergence of the density for the Thomas-Fermi-von Weizsacker model with Yukawa potential	156	(38)
4.1 Introduction	156	(4)
4.2 Preliminary convergence results on the density	160	(5)
4.2.1 Local convergence	160	(2)
4.2.2 Convergence of the XXX-transform	162	(1)
4.2.3 Periodicity up to a translation	162	(3)
4.3 Periodicity of the limit density	165	(29)
4.3.1 Bounds from above for solutions	166	(4)
4.3.2 Bounds from below for solutions	170	(6)
4.3.3 Uniqueness result without the convolution term	176	(4)
4.3.4 Uniqueness result for the full equation: conclusion	180	(14)
5 Convergence of the density for the Thomas-Fermi-von Weizsacker model	194	(34)
5.1 Introduction	194	(2)
5.2 Preliminary convergence results	196	(3)
5.2.1 Convergence of the XXX-transform	196	(1)
5.2.2 Convergence up to a translation	197	(2)
5.3 A uniqueness result for a system of PDE's	199	(20)
5.3.1 The smeared nuclei case	199	(7)
5.3.2 The point nuclei case	206	(11)
5.3.3 Corollaries and remarks	217	(2)
5.4 Convergence of the density	219	(3)
5.5 Uniform convergence on interior domains	222	(6)
6 Convergence of the energy via the convergence of the density	228	(44)
6.1 Introduction	228	(3)
6.2 A new proof of the convergence of the energy	231	(15)
6.2.1 Proof of Theorem 6.6	232	(8)
6.2.2 Adaptation to the TF case	240	(6)
6.3 A general result for existence and uniqueness	246	(8)
6.3.1 Proof of Theorem 6.10	247	(4)
6.3.2 When there is no uniqueness	251	(3)
6.4 Geometries which are not that non-periodic	254	(10)
6.4.1 General periodic lattices, impurities.	255	(3)
6.4.2 Towards quasicrystals: almost periodic measures	258	(6)
6.5 Non-neutral systems	264	(8)
6.5.1 Negative ions	265	(2)
6.5.2 Positive ions	267	(5)
References	272