Spin Glasses

This is a short review about recent methods and results, mostly for mean field spin glasses, based on interpolation and comparison schemes. In particular, the Parisi spontaneous replica symmetry breaking phenomenon is described in the frame of extended variational principles, Derrida-Ruelle probability cascades, and overlap locking.Comment: 25 page

Mathematical aspects of mean field spin glass theory

A comprehensive review will be given about the rich mathematical structure of mean field spin glass theory, mostly developed, until now, in the frame of the methods of theoretical physics, based on deep physical intuition and hints coming from numerical simulation. Central to our treatment is a very simple and yet powerful interpolation method, allowing to compare different probabilistic schemes, by using convexity and positivity arguments. In this way we can prove the existence of the thermodynamic limit for the free energy density of the system, a long standing open problem. Moreover, in the frame of a generalized variational principle, we can show the emergency of the Derrida-Ruelle random probability cascades, leading to the form of free energy given by the celebrated Parisi \textit {Ansatz}. All these results seem to be in full agreement with the mechanism of spontaneous replica symmetry breaking as developed by Giorgio Parisi.Comment: proceedings of the "4th European Congress of Mathematics", Stockholm, 2004. 17 page

Dynamical ensembles in stationary states

We propose as a generalization of an idea of Ruelle to describe turbulent fluid flow a chaotic hypothesis for reversible dissipative many particle systems in nonequilibrium stationary states in general. This implies an extension of the zeroth law of thermodynamics to non equilibrium states and it leads to the identification of a unique distribution \m describing the asymptotic properties of the time evolution of the system for initial data randomly chosen with respect to a uniform distribution on phase space. For conservative systems in thermal equilibrium the chaotic hypothesis implies the ergodic hypothesis. We outline a procedure to obtain the distribution \m: it leads to a new unifying point of view for the phase space behavior of dissipative and conservative systems. The chaotic hypothesis is confirmed in a non trivial, parameter--free, way by a recent computer experiment on the entropy production fluctuations in a shearing fluid far from equilibrium. Similar applications to other models are proposed, in particular to a model for the Kolmogorov--Obuchov theory for turbulent flow.Comment: 31 pages, 3 figures, compile with dvips (otherwise no pictures

Large deviations for non-uniformly expanding maps

We obtain large deviation results for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average decays to zero exponentially fast with the number of iterates involved. As easy by-products we deduce escape rates from subsets of the basins of physical measures for these types of maps. The rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states. The corrections added to the published version of this text appear in bold; see last section for a list of changesComment: 36 pages, 1 figure. After many PhD students and colleagues having pointed several errors in the statements and proofs, this is a correction to published article answering those comments. List of main changes in a new last sectio

Global bifurcations close to symmetry

Heteroclinic cycles involving two saddle-foci, where the saddle-foci share both invariant manifolds, occur persistently in some symmetric differential equations on the 3-dimensional sphere. We analyse the dynamics around this type of cycle in the case when trajectories near the two equilibria turn in the same direction around a 1-dimensional connection - the saddle-foci have the same chirality. When part of the symmetry is broken, the 2-dimensional invariant manifolds intersect transversely creating a heteroclinic network of Bykov cycles. We show that the proximity of symmetry creates heteroclinic tangencies that coexist with hyperbolic dynamics. There are n-pulse heteroclinic tangencies - trajectories that follow the original cycle n times around before they arrive at the other node. Each n-pulse heteroclinic tangency is accumulated by a sequence of (n+1)-pulse ones. This coexists with the suspension of horseshoes defined on an infinite set of disjoint strips, where the first return map is hyperbolic. We also show how, as the system approaches full symmetry, the suspended horseshoes are destroyed, creating regions with infinitely many attracting periodic solutions

Fluctuations of the Magnetization in Thin Films due to Conduction Electrons

A detailed analysis of damping and noise due to a {\it sd}-interaction in a thin ferromagnetic film sandwiched between two large normal metal layers is carried out. The magnetization is shown to obey in general a non-local equation of motion which differs from the the Gilbert equation and is extended to the non-adiabatic regime. To lowest order in the exchange interaction and in the limit where the Gilbert equation applies, we show that the damping term is enhanced due to interfacial effects but it also shows oscillations as a function of the film thickness. The noise calculation is however carried out to all orders in the exchange coupling constant. The ellipticity of the precession of the magnetization is taken into account. The damping is shown to have a Gilbert form only in the adiabatic limit while the relaxation time becomes strongly dependent on the geometry of the thin film. It is also shown that the induced noise characteristic of sd-exchange is inherently colored in character and depends on the symmetry of the Hamiltonian of the magnetization in the film. We show that the sd-noise can be represented in terms of an external stochastic field which is white only in the adiabatic regime. The temperature is also renormalized by the spin accumulation in the system. For large intra-atomic exchange interactions, the Gilbert-Brown equation is no longer valid