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Coulomb gases and Ginzburg-Landau vortices. (English) Zbl 1335.82002
Zurich Lectures in Advanced Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-152-1/pbk; 978-3-03719-652-6/ebook). viii, 157 p. (2015).
The lecture notes are devoted to the mathematical study of vortices in the Ginzburg-Landau (GL) model of superconductivity and classical Coulomb gases. The book is divided into ten chapters. First, an introductory chapter presents briefly the two topics and the connection between them describing the GL model and the two-dimensional Coulomb gas. Chapter 2 studies the leading-order or mean-field behavior of the Coulomb gas Hamiltonian, $$H_n$$, giving a self-contained and general treatment. First, the Hamiltonian is analyzed only, via $$\Gamma$$-convergence (in the sense of De Giorgi), leading to the mean-field description of its minimizers. Then these results are applied to the statistical mechanics model associated to the Hamiltonian, i.e., to characterizing the states with nonzero temperature. The next-order behavior is studied in Chapter 3. With this aim, the configurations are blown-up by the inverse value of the typical distance between two points, so that the points are well-separated (typically with distance $$O(1)$$) and a way of expanding the Hamiltonian to the next order is found. As a result, the splitting lower bound is stated and in the splitting of the Hamiltonian, a lower-order term in the form of a function Hamiltonian, “the precursor to renormalized energy $$W$$”, appears at fixed number of points $$n$$. Chapter 4 finds the asymptotic limit of this lower-order term as $$n \to +\infty$$ (a limiting object appears, called renormalized energy). This energy is the total Coulomb interaction energy of an infinite configuration of points in the whole space in a constant neutralizing background. This chapter defines this limiting object itself and studies some of its properties. Chapter 5 carries out the passage to the limit $$n \to +\infty$$ in deriving $$W$$, based on the results of Chapter 3 in order to extract $$W$$ as a limiting energy. A lower bound in the large $$n$$ limit is expressed in terms of an average of $$W$$ with respect to a suitable measure that encodes all the possible blow-up profiles. It is accomplished using a general method formulated abstractly. Chapter 6 obtains the upper bound that optimally matches the lower bound obtained in Chapter 5. This upper bound relies on an important construction called the “screening” of a point configuration. As a consequence of matching the upper and lower bounds, an asymptotic expansion of min $$H_n$$ with the prefactor $$\min \widetilde{\mathcal W}$$ is obtained (where $$\widetilde{\mathcal W}$$ is defined from $$W$$ in Chapter 5), and the fact that minimizers of $$H_n$$ converge to minimizers of $$\widetilde{\mathcal W}$$. Moreover, it is shown that the Gibbs measure concentrates on minimizers of $$\widetilde{\mathcal W}$$ as the inverse temperature $$\beta \to +\infty$$. Chapter 7 presents some non-rigorous heuristics on the GL model that allows one to see how and when vortices are expected to form in minimizers. Chapter 8 presents two main mathematical tools for the GL model, namely (i) the “vortex balls construction” method, which allows one to get completely general lower bounds for the energy of a configuration in terms of its vortices, and (ii) the “Jacobian estimate”, which gives a quantitative estimate and meaning relating the vortices of an arbitrary configuration (or the Jacobian in the gauge-independent version) to its underlying vortices. Based on the tools presented in Chapter 8, Chapter 9 obtains a mean-field limit or leading-order behavior of minimizers (or ground states) of the GL functional (without temperature), highlighting the analogy with the Coulomb gas. Chapter 10 sketches the method, which allows one to derive the renormalization energy $$W$$ from the minimization of GL, at the next order, beyond the mean-field limit presented in Chapter 9. In total, the lecture notes are written on high physical and mathematical level. They state clear connection between the GL model of superconductivity and classical Coulomb gases. The book will be useful for graduate and PhD students, specializing in the scientific area, and also for scientists working in this and related areas of the physics of matter.

##### MSC:
 82-02 Research exposition (monographs, survey articles) pertaining to statistical mechanics 82B05 Classical equilibrium statistical mechanics (general) 82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics 82B26 Phase transitions (general) in equilibrium statistical mechanics 82D55 Statistical mechanics of superconductors 35Q82 PDEs in connection with statistical mechanics 35Q56 Ginzburg-Landau equations
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