Baxterization, dynamical systems, and the symmetries of integrability

We resolve the `baxterization' problem with the help of the automorphism group of the Yang-Baxter (resp. star-triangle, tetrahedron, \dots) equations. This infinite group of symmetries is realized as a non-linear (birational) Coxeter group acting on matrices, and exists as such, {\em beyond the narrow context of strict integrability}. It yields among other things an unexpected elliptic parametrization of the non-integrable sixteen-vertex model. It provides us with a class of discrete dynamical systems, and we address some related problems, such as characterizing the complexity of iterations.Comment: 25 pages, Latex file (epsf style). WARNING: Postscript figures are BIG (600kB compressed, 4.3MB uncompressed). If necessary request hardcopy to [email protected] and give your postal mail addres

On the Symmetries of Integrability

We show that the Yang-Baxter equations for two dimensional models admit as a group of symmetry the infinite discrete group $A_2^{(1)}$. The existence of this symmetry explains the presence of a spectral parameter in the solutions of the equations. We show that similarly, for three-dimensional vertex models and the associated tetrahedron equations, there also exists an infinite discrete group of symmetry. Although generalizing naturally the previous one, it is a much bigger hyperbolic Coxeter group. We indicate how this symmetry can help to resolve the Yang-Baxter equations and their higher-dimensional generalizations and initiate the study of three-dimensional vertex models. These symmetries are naturally represented as birational projective transformations. They may preserve non trivial algebraic varieties, and lead to proper parametrizations of the models, be they integrable or not. We mention the relation existing between spin models and the Bose-Messner algebras of algebraic combinatorics. Our results also yield the generalization of the condition $q^n=1$ so often mentioned in the theory of quantum groups, when no $q$ parameter is available.Comment: 23 page

Thermal noise properties of two aging materials

In this lecture we review several aspects of the thermal noise properties in two aging materials: a polymer and a colloidal glass. The measurements have been performed after a quench for the polymer and during the transition from a fluid-like to a solid-like state for the gel. Two kind of noise has been measured: the electrical noise and the mechanical noise. For both materials we have observed that the electric noise is characterized by a strong intermittency, which induces a large violation of the Fluctuation Dissipation Theorem (FDT) during the aging time, and may persist for several hours at low frequency. The statistics of these intermittent signals and their dependance on the quench speed for the polymer or on sample concentration for the gel are studied. The results are in a qualitative agreement with recent models of aging, that predict an intermittent dynamics. For the mechanical noise the results are unclear. In the polymer the mechanical thermal noise is still intermittent whereas for the gel the violation of FDT, if it exists, is extremely small.Comment: to be published in the Proceedings of the XIX Sitges Conference on ''Jammming, Yielding and Irreversible Deformation in Condensed Matter'', M.-C.Miguel and M. Rubi eds.,Springer Verlag, Berli

Agroecological transitions in scientific research programs in Brazil and France.

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Symmetrical Temperature-Chaos Effect with Positive and Negative Temperature Shifts in a Spin Glass

The aging in a Heisenberg-like spin glass Ag(11 at% Mn) is investigated by measurements of the zero field cooled magnetic relaxation at a constant temperature after small temperature shifts $|\Delta T/T_g| < 0.012$. A crossover from fully accumulative to non-accumulative aging is observed, and by converting time scales to length scales using the logarithmic growth law of the droplet model, we find a quantitative evidence that positive and negative temperature shifts cause an equivalent restart of aging (rejuvenation) in terms of dynamical length scales. This result supports the existence of a unique overlap length between a pair of equilibrium states in the spin glass system.Comment: 4 page

Langevin simulations of the out-of-equilibrium dynamics of the vortex glass in high-temperature superconductors

We study the relaxation dynamics of flux lines in dirty high-temperature superconductors using numerical simulations of a London-Langevin model of the interacting vortex lines. By analysing the equilibrium dynamics in the vortex liquid phase we find a dynamic crossover to a glassy non-equilibrium regime. We then focus on the out-of-equilibrium dynamics of the vortex glass phase using tools that are common in the study of other glassy systems. By monitoring the two-times roughness and dynamic wandering we identify and characterize finite-size effects that are similar, though more complex, than the ones found in the stationary roughness of clean interface dynamics. The two-times density-density correlation and mean-squared-displacement correlation age and their temporal scaling follows a multiplicative law similar to the one found at criticality. The linear responses also age and the comparison with their associated correlations shows that the equilibrium fluctuation-dissipation relation is modified in a simple manner that allows for the identification of an effective temperature characterizing the dynamics of the slow modes. The effective temperature is closely related to the vortex liquid-vortex glass crossover temperature. Interestingly enough, our study demonstrates that the glassy dynamics in the vortex glass is basically identical to the one of a single elastic line in a disordered environment (with the same type of scaling though with different parameters). Possible extensions and the experimental relevance of these results are also discussed.Comment: 22 pages, 29 figure

Construction of Integrals of Higher-Order Mappings

We find that certain higher-order mappings arise as reductions of the integrable discrete A-type KP (AKP) and B-type KP (BKP) equations. We find conservation laws for the AKP and BKP equations, then we use these conservation laws to derive integrals of the associated reduced maps.Comment: appear to Journal of the Physical Society of Japa