On the concentration properties of Interacting particle processes

These lecture notes present some new concentration inequalities for Feynman-Kac particle processes. We analyze different types of stochastic particle models, including particle profile occupation measures, genealogical tree based evolution models, particle free energies, as well as backward Markov chain particle models. We illustrate these results with a series of topics related to computational physics and biology, stochastic optimization, signal processing and bayesian statistics, and many other probabilistic machine learning algorithms. Special emphasis is given to the stochastic modeling and the quantitative performance analysis of a series of advanced Monte Carlo methods, including particle filters, genetic type island models, Markov bridge models, interacting particle Markov chain Monte Carlo methodologies

Nonmonotonic Evolution of the Blocking Temperature in Dispersions of Superparamagnetic Nanoparticles

We use a Monte Carlo approach to simulate the influence of the dipolar interaction on assemblies of monodisperse superparamagnetic ${\gamma}-Fe_{2}O_{3}$ nanoparticles. We have identified a critical concentration c*, that marks the transition between two different regimes in the evolution of the blocking temperature ($T_{B}$) with interparticle interactions. At low concentrations (c < c*) magnetic particles behave as an ideal non-interacting system with a constant $T_{B}$. At concentrations c > c* the dipolar energy enhances the anisotropic energy barrier and $T_{B}$ increases with increasing c, so that a larger temperature is required to reach the superparamagnetic state. The fitting of our results with classical particle models and experiments supports the existence of two differentiated regimes. Our data could help to understand apparently contradictory results from the literature.Comment: 13 pages, 7 figure

Concentration inequalities for mean field particle models

This article is concerned with the fluctuations and the concentration properties of a general class of discrete generation and mean field particle interpretations of nonlinear measure valued processes. We combine an original stochastic perturbation analysis with a concentration analysis for triangular arrays of conditionally independent random sequences, which may be of independent interest. Under some additional stability properties of the limiting measure valued processes, uniform concentration properties, with respect to the time parameter, are also derived. The concentration inequalities presented here generalize the classical Hoeffding, Bernstein and Bennett inequalities for independent random sequences to interacting particle systems, yielding very new results for this class of models. We illustrate these results in the context of McKean-Vlasov-type diffusion models, McKean collision-type models of gases and of a class of Feynman-Kac distribution flows arising in stochastic engineering sciences and in molecular chemistry.Comment: Published in at http://dx.doi.org/10.1214/10-AAP716 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Kinetics and Mechanism of Metal Nanoparticle Growth via Optical Extinction Spectroscopy and Computational Modeling: The Curious Case of Colloidal Gold

An overarching computational framework unifying several optical theories to describe the temporal evolution of gold nanoparticles (GNPs) during a seeded growth process is presented. To achieve this, we used the inexpensive and widely available optical extinction spectroscopy, to obtain quantitative kinetic data. In situ spectra collected over a wide set of experimental conditions were regressed using the physical model, calculating light extinction by ensembles of GNPs during the growth process. This model provides temporal information on the size, shape, and concentration of the particles and any electromagnetic interactions between them. Consequently, we were able to describe the mechanism of GNP growth and divide the process into distinct genesis periods. We provide explanations for several longstanding mysteries, for example, the phenomena responsible for the purple-greyish hue during the early stages of GNP growth, the complex interactions between nucleation, growth, and aggregation events, and a clear distinction between agglomeration and electromagnetic interactions. The presented theoretical formalism has been developed in a generic fashion so that it can readily be adapted to other nanoparticulate formation scenarios such as the genesis of various metal nanoparticles.Comment: Main text and supplementary information (accompanying MATLAB codes available on the journal webpage

Correlation functions of higher-dimensional Luttinger liquids

Using higher-dimensional bosonization, we study correlation functions of fermions with singular forward scattering. Following Bares and Wen [Phys. Rev. B 48, 8636 (1993)], we consider density-density interactions in d dimensions that diverge for small momentum transfers as q^{- eta} with eta = 2 (d-1). In this case the single-particle Green's function shows Luttinger liquid behavior. We discuss the momentum distribution and the density of states and show that, in contrast to d=1, in higher dimensions the scaling behavior cannot be characterized by a single anomalous exponent. We also calculate the irreducible polarization for q close to 2 k_F and show that the leading singularities cancel. We discuss consequences for the effect of disorder on higher-dimensional Luttinger liquids.Comment: 7 RevTex pages, 2 figures, minor modifications, to appear in Phys. Rev. B (Feb. 1999

Nonasymptotic analysis of adaptive and annealed Feynman-Kac particle models

Sequential and quantum Monte Carlo methods, as well as genetic type search algorithms can be interpreted as a mean field and interacting particle approximations of Feynman-Kac models in distribution spaces. The performance of these population Monte Carlo algorithms is strongly related to the stability properties of nonlinear Feynman-Kac semigroups. In this paper, we analyze these models in terms of Dobrushin ergodic coefficients of the reference Markov transitions and the oscillations of the potential functions. Sufficient conditions for uniform concentration inequalities w.r.t. time are expressed explicitly in terms of these two quantities. We provide an original perturbation analysis that applies to annealed and adaptive Feynman-Kac models, yielding what seems to be the first results of this kind for these types of models. Special attention is devoted to the particular case of Boltzmann-Gibbs measures' sampling. In this context, we design an explicit way of tuning the number of Markov chain Monte Carlo iterations with temperature schedule. We also design an alternative interacting particle method based on an adaptive strategy to define the temperature increments. The theoretical analysis of the performance of this adaptive model is much more involved as both the potential functions and the reference Markov transitions now depend on the random evolution on the particle model. The nonasymptotic analysis of these complex adaptive models is an open research problem. We initiate this study with the concentration analysis of a simplified adaptive models based on reference Markov transitions that coincide with the limiting quantities, as the number of particles tends to infinity.Comment: Published at http://dx.doi.org/10.3150/14-BEJ680 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Cosmological implications of a Dark Matter self-interaction energy density

We investigate cosmological constraints on an energy density contribution of elastic dark matter self-interactions characterized by the mass of the exchange particle and coupling constant. Because of the expansion behaviour in a Robertson-Walker metric we investigate self-interacting dark matter that is warm in the case of thermal relics. The scaling behaviour of dark matter self-interaction energy density shows that it can be the dominant contribution (only) in the very early universe. Thus its impact on primordial nucleosynthesis is used to restrict the interaction strength, which we find to be at least as strong as the strong interaction. Furthermore we explore dark matter decoupling in a self-interaction dominated universe, which is done for the self-interacting warm dark matter as well as for collisionless cold dark matter in a two component scenario. We find that strong dark matter self-interactions do not contradict super-weak inelastic interactions between self-interacting dark matter and baryonic matter and that the natural scale of collisionless cold dark matter decoupling exceeds the weak scale and depends linearly on the particle mass. Finally structure formation analysis reveals a linear growing solution during self-interaction domination; however, only non-cosmological scales are enhanced.Comment: 14 pages, 14 figures; version published in Phys. Rev.