By Sylvia Serfaty

The subject of this ebook is structures of issues in Coulomb interplay, specifically, the classical Coulomb gasoline, and vortices within the Ginzburg–Landau version of superconductivity. The classical Coulomb and Log gases are classical statistical mechanics versions, that have visible vital advancements within the mathematical literature because of their reference to random matrices and approximation conception. At low temperature, those structures are anticipated to “cristallize” to so-called Fekete units, which express microscopically a lattice structure.

The Ginzburg–Landau version, however, describes superconductors. In superconducting fabrics subjected to an exterior magnetic box, densely packed aspect vortices emerge, forming excellent triangular lattice styles, so-called Abrikosov lattices.

This ebook describes those platforms and explores the similarity among them. It offers the mathematical instruments built to research the interplay among the Coulomb debris or the vortices, on the microscopic scale, and describes a “renormalized power” governing the purpose styles. this can be believed to degree the illness of some extent configuration, and to be minimized via the Abrikosov lattice in size 2.

The booklet offers a self-contained presentation of effects at the suggest box restrict of the Coulomb fuel approach, without or with temperature, and of the derivation of the renormalized power. It additionally presents a streamlined presentation of the same research that may be played for the Ginzburg–Landau version, together with a assessment of the vortex-specific instruments and the derivation of the severe fields, the mean-field restrict and the renormalized strength.

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4 The mean field limit where Cd is a constant depending only on the dimension d . 44) for ’ n large enough. 1/ when Á ! x y/dxdy < C1. 1/: ´ ´ Also Vd n ! C1 1 Hn . 45) are supported Ä I. 1/ and letting Á ! 0 finally gives us the -lim sup inequality, which concludes the proof. 19. x y/ dx dy < 1. This shows that the result still holds for all such interaction kernels. 20. To prove the -liminf relation, we have only used that V is l. s. c. and bounded below. To prove the -limsup relation, we have assumed that V is continuous for convenience.

2. The coincidence set for a one-dimensional obstacle problem. 49) has two options at each point: to touch the obstacle or not (and typically uses both possibilities). x/ q: e:g is closed and called the coincidence set or the contact set. ” The obstacle problem thus belongs to the class of so-called free-boundary problems, cf. [Fri]. Trying to compute the Euler–Lagrange associated to this problem by perturbing h by a small function, one is led to two possibilities depending on whether h D or h > .

10. 1. Since we assumed for simplicity that V is continuous and finite, it suffices to assume (A2) to have (A1) and (A3). With all the preceding, we may conclude with the following result, which goes back to [Cho]. 2 (Convergence of minimizers and minima of Hn ). Assume that V is continuous and satisfies (A2). x1 ; : : : ; xn /gn is a minimizer of Hn . Then, 1X ıxi ! x1 ; : : : ; xn / D I. 48) Proof. C1 n12 min Hn is bounded above (by I. 21. 5 Linking the equilibrium measure with the obstacle problem 29 P follows that, up to a subsequence, we have n1 niD1 ıxi !