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Type: report × search.filters.applied.f.series: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik, Volume 1303, ISSN 0946-8633 ×

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    Lagrangian and Hamiltonian two-scale reduction
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Giannoulis, Johannes; Herrmann, Michael; Mielke, Alexander
    Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system. In the first part we develop a general approach to this problem by considering non-canonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices (or Hamiltonian PDEs) and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, (iii) an elementary model reduction that is based on a Principle of Consistent Expansions. In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long wave-length motion or describe the macroscopic modulation of an oscillatory microstructure.
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    Competing particle systems and the Ghirlanda-Guerra identities
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Arguin, Louis-Pierre
    We study point processes on the real line whose configurations $X$ can be ordered decreasingly and evolve by increments which are functions of correlated gaussian variables. The correlations are intrinsic to the points and quantified by a matrix $Q=q_ij$. Quasi-stationary systems are those for which the law of $(X,Q)$ is invariant under the evolution up to translation of $X$. It was conjectured by Aizenman and co-authors that the matrix $Q$ of robustly quasi-stationary systems must ex This was established recently, up to a natural decomposition of the system, whenever the set $S_Q$ of values assumed by $q_ij$ is finite. In this paper, we study the general case, where $S_Q$ may be infinite. Using the past increments of the evolution, we show that the law of robustly quasi-stationary systems must obey the Ghirlanda-Guerra identities, which first appear in the study of spin glass models. This provides strong evidence that the above conjecture also holds in the general case.