Estimating the principal components of correlation matrices from all their empirical eigenvectors

We consider the problem of estimating the principal components of a population correlation matrix from a limited number of measurement data. Using a combination of random matrix and information-theoretic tools, we show that all the eigenmodes of the sample correlation matrices are informative, and not only the top ones. We show how this information can be exploited when prior information about the principal component, such as whether it is localized or not, is available by mapping the estimation problem onto the search for the ground state of a spin-glass-like effective Hamiltonian encoding the prior. Results are illustrated numerically on the spiked covariance model.Comment: 6 pages, 6 figures, to appear in Europhysics Letter

Microscopic origin of self-similarity in granular blast waves

The self-similar expansion of a blast wave, well-studied in air, has peculiar counterparts in dense and dissipative media such as granular gases. Recent results have shown that, while the traditional Taylor-von Neumann-Sedov (TvNS) derivation is not applicable to such granular blasts, they can nevertheless be well understood via a combination of microscopic and hydrodynamic insights. In this article, we provide a detailed analysis of these methods associating Molecular Dynamics simulations and continuum equations, which successfully predict hydrodynamic profiles, scaling properties and the instability of the self-similar solution. We also present new results for the energy conserving case, including the particle-level analysis of the classic TvNS solution and its breakdown at higher densities.Comment: 47 pages, 9 figures Supplementary Materials: 2 appendices, 3 figure

Growing non-equilibrium length in granular fluids: from experiment to fluctuating hydrodynamics

Velocity correlations in a 2D granular fluid are studied in experiments and numerical simulations. The transverse component of the velocity structure factor reveals two well defined energy scales, associated with the external "bath temperature" $T_b$ and with the internal granular one, $T_g<T_b$, relevant at large and small wavelengths respectively. Experimental and numerical data are discussed within a fluctuating hydrodynamics model, which allows one to define and measure a non-equilibrium coherence length $\xi$, growing with density, that characterizes order in the velocity field.Comment: 5 pages, 4 figure

Irreversible dynamics of a massive intruder in dense granular fluids

A Generalized Langevin Equation with exponential memory is proposed for the dynamics of a massive intruder in a dense granular fluid. The model reproduces numerical correlation and response functions, violating the equilibrium Fluctuation Dissipation relations. The source of memory is identified in the coupling of the tracer velocity $V$ with a spontaneous local velocity field $U$ in the surrounding fluid. Such identification allows us to measure the intruder's fluctuating entropy production as a function of $V$ and $U$, obtaining a neat verification of the Fluctuation Relation.Comment: 5 pages, 3 figures accepted for publication in EP

The Ratchet effect in an ageing glass

We study the dynamics of an asymmetric intruder in a glass-former model. At equilibrium, the intruder diffuses with average zero velocity. After an abrupt quench to $T$ deeply under the mode-coupling temperature, a net average drift is observed, steady on a logarithmic time-scale. The phenomenon is well reproduced in an asymmetric version of the Sinai model. The subvelocity of the intruder grows with $T_{eff}/T$, where $T_{eff}$ is defined by the response-correlation ratio, corresponding to a general behavior of thermal ratchets when in contact with two thermal reservoirs.Comment: 10 pages, 4 figure