##
**The mathematical theory of thermodynamic limits: Thomas-Fermi type models.**
*(English)*
Zbl 0938.81001

Oxford Mathematical Monographs. Oxford: Clarendon Press. xiii, 277 p. (1998).

This monograph is devoted to the rigorous mathematical analysis of the thermodynamic limit of the Thomas-Fermi-von Weizsäcker model with a Yukawa (short-range) and a Coulomb (long-range) potential for the interaction between the particles.

The main problems arising in this context and being analyzed in this book are the following ones. Assume \(N\) electrons and \(N\) nuclei of unit charge; let \(E_N\) and \(\rho_N\) be the ground-state energy and the minimizing density of this system, respectively; and let us assume that \(N\) goes to infinity. The mentioned problems are: 1) does the energy per unit volume go to a finite limit? 2) does the minimizing density approach (in some sense to be discussed) a limit? 3) does the limit density have the same periodicity as the assumed periodicity of the nuclei? From the physical point of view, these problems arise at least in two general theories: the thermodynamic limit of microscopic models (essential for the derivation of the postulates of thermodynamics from statistical mechanics) and the quantum theory of solid state (trying to derive from first principles the existence and stability of crystals or quasicrystals). The book is devoted to mathematicians and mathematical physicists, and requires a good familiarity with the theory of elliptic partial differential equations and variational principles.

The starting point of this research is the analysis due to E. H. Lieb and B. Simon [Adv. Math. 23, 22-116 (1977; Zbl 0938.81568)] on the Thomas-Fermi model. Instead, the authors have focused their attention on the Thomas-Fermi-von Weizsäcker model, which allows binding of molecules. To avoid some of the difficulties arising from the long-range character of the Coulomb potential, they use the well-known short-range Yukawa potential and compare the results obtained with it with those obtained in the Coulomb potential. Furthermore, they consider different possibilities for the description of the nuclei: they are considered either as point particles, or are described by smeared-out distribution charges, either smooth or composed by delta functions at several points of the unit cell.

The text is structured in six chapters. The first one is a general presentation (mathematical, physical, historical) of the problems to be analyzed, and a discussion of the main results obtained in the book. Chapters 2 and 3 are devoted to the problem of the convergence of the energy per unit volume (with Yukawa and Coulomb potentials, respectively), and some extensions are considered to other situations than point nuclei, or to exponents of the density different from the usual 5/3 of Thomas-Fermi type models. Chapters 4 and 5 deal with the convergence of the minimizing density (again, with Yukawa and Coulomb potentials, respectively). Results for the Coulomb potential are obtained either as a suitable limit of the results for the Yukawa potential and through a direct proof for the energy potential itself. Finally, Chapter 6 deals with the problem of the convergence of the energy via the convergence of the density, giving a new perspective on the relation between both problems which allows, in particular, to see that the uniform convergence of the density on the interior domains implies the convergence of the energy per unit volume, or which provides techniques which may be applied to non-periodic geometries (quasicrystals) and to non-neutral systems.

This book is not closed in itself, not only because it acknowledges the tradition of research in this field, with a list of 58 relevant references, but also because it points to the future. Indeed, it is announced to be followed by a work (in preparation) on these topics in the Hartree and Hartree-Fock setting. Furthermore, besides the problems being solved, it lists some open problems which have been identified as interesting and useful in the present work, and their relation to questions arising in other physical and mathematical contests is shown. As a consequence, the book is not only valuable for its interesting rigorous results but also for the stimulus provided to research in this subtle difficult area of mathematical physics.

The main problems arising in this context and being analyzed in this book are the following ones. Assume \(N\) electrons and \(N\) nuclei of unit charge; let \(E_N\) and \(\rho_N\) be the ground-state energy and the minimizing density of this system, respectively; and let us assume that \(N\) goes to infinity. The mentioned problems are: 1) does the energy per unit volume go to a finite limit? 2) does the minimizing density approach (in some sense to be discussed) a limit? 3) does the limit density have the same periodicity as the assumed periodicity of the nuclei? From the physical point of view, these problems arise at least in two general theories: the thermodynamic limit of microscopic models (essential for the derivation of the postulates of thermodynamics from statistical mechanics) and the quantum theory of solid state (trying to derive from first principles the existence and stability of crystals or quasicrystals). The book is devoted to mathematicians and mathematical physicists, and requires a good familiarity with the theory of elliptic partial differential equations and variational principles.

The starting point of this research is the analysis due to E. H. Lieb and B. Simon [Adv. Math. 23, 22-116 (1977; Zbl 0938.81568)] on the Thomas-Fermi model. Instead, the authors have focused their attention on the Thomas-Fermi-von Weizsäcker model, which allows binding of molecules. To avoid some of the difficulties arising from the long-range character of the Coulomb potential, they use the well-known short-range Yukawa potential and compare the results obtained with it with those obtained in the Coulomb potential. Furthermore, they consider different possibilities for the description of the nuclei: they are considered either as point particles, or are described by smeared-out distribution charges, either smooth or composed by delta functions at several points of the unit cell.

The text is structured in six chapters. The first one is a general presentation (mathematical, physical, historical) of the problems to be analyzed, and a discussion of the main results obtained in the book. Chapters 2 and 3 are devoted to the problem of the convergence of the energy per unit volume (with Yukawa and Coulomb potentials, respectively), and some extensions are considered to other situations than point nuclei, or to exponents of the density different from the usual 5/3 of Thomas-Fermi type models. Chapters 4 and 5 deal with the convergence of the minimizing density (again, with Yukawa and Coulomb potentials, respectively). Results for the Coulomb potential are obtained either as a suitable limit of the results for the Yukawa potential and through a direct proof for the energy potential itself. Finally, Chapter 6 deals with the problem of the convergence of the energy via the convergence of the density, giving a new perspective on the relation between both problems which allows, in particular, to see that the uniform convergence of the density on the interior domains implies the convergence of the energy per unit volume, or which provides techniques which may be applied to non-periodic geometries (quasicrystals) and to non-neutral systems.

This book is not closed in itself, not only because it acknowledges the tradition of research in this field, with a list of 58 relevant references, but also because it points to the future. Indeed, it is announced to be followed by a work (in preparation) on these topics in the Hartree and Hartree-Fock setting. Furthermore, besides the problems being solved, it lists some open problems which have been identified as interesting and useful in the present work, and their relation to questions arising in other physical and mathematical contests is shown. As a consequence, the book is not only valuable for its interesting rigorous results but also for the stimulus provided to research in this subtle difficult area of mathematical physics.

Reviewer: D.Jou (Bellaterra)

### MSC:

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

82-02 | Research exposition (monographs, survey articles) pertaining to statistical mechanics |

82B10 | Quantum equilibrium statistical mechanics (general) |

81V70 | Many-body theory; quantum Hall effect |

35Q40 | PDEs in connection with quantum mechanics |

81V55 | Molecular physics |

35J99 | Elliptic equations and elliptic systems |

47N50 | Applications of operator theory in the physical sciences |