Maxime Clusel

A ‘granocentric’ model for random packing of jammed emulsions

Packing problems are ubiquitous, ranging from oil extraction through porous rocks to grain storage in silos and the compaction of pharmaceutical powders into tablets. At a given density, particulate systems pack into a mechanically stable and amorphous jammed state. Previous theoretical studies have explored a connection between this jammed state and the glass transition, the thermodynamics of jamming and geometric modelling of random packings. Nevertheless, a simple underlying mechanism for the random assembly of athermal particles, analogous to crystalline ordering, remains unknown. Here we use three-dimensional measurements of packings of polydisperse emulsion droplets to build a simple statistical model in which the complexity of the global packing is distilled into a local stochastic process. From the perspective of a single particle, the packing problem is reduced to the random formation of nearest neighbours, followed by a choice of contacts among them. The two key parameters in the model—the available space around a particle and the ratio of contacts to neighbours—are directly obtained from experiments. We demonstrate that this ‘granocentric’ view captures the properties of the polydisperse emulsion packing—ranging from the microscopic distributions of nearest neighbours and contacts, to local density fluctuations, to the global packing density. Application of our results to monodisperse and bidisperse systems produces quantitative agreement with previously measured trends in global density. Our model therefore reveals a general principle of organization for random packing and may provide the foundations for a theory of jammed matter.

Generalized extreme value statistics and sum of correlated variables

We show that generalized extreme value statistics—the statistics of the kth largest value among a large set of random variables—can be mapped onto a problem of random sums. This allows us to identify classes of non-identical and (generally) correlated random variables with a sum distributed according to one of the three (k-dependent) asymptotic distributions of extreme value statistics, namely the Gumbel, Frechet and Weibull distributions. These classes, as well as the limit distributions, are naturally extended to real values of k, thus providing a clear interpretation to the onset of Gumbel distributions with non-integer index k in the statistics of global observables. This is one of the very few known generalizations of the central limit theorem to non-independent random variables. Finally, in the context of a simple physical model, we relate the index k to the ratio of the correlation length to the system size, which remains finite in strongly correlated systems.

GLOBAL FLUCTUATIONS IN PHYSICAL SYSTEMS: A SUBTLE INTERPLAY BETWEEN SUM AND EXTREME VALUE STATISTICS

Fluctuations of global additive quantities, like total energy or magnetization for instance, can in principle be described by statistics of sums of (possibly correlated) random variables. Yet, it turns out that extreme values (the largest value among a set of random variables) may also play a role in the statistics of global quantities, in a direct or indirect way. This review discusses different connections that may appear between problems of sums and of extreme values of random variables, and emphasizes physical situations in which such connections are relevant. Along this line of thought, standard convergence theorems for sums and extreme values of independent and identically distributed random variables are recalled, and some rigorous results as well as more heuristic reasonings are presented for correlated or non-identically distributed random variables. More specifically, the role of extreme values within sums of broadly distributed variables is addressed, and a general mapping between extreme values and sums is presented, allowing us to identify a class of correlated random variables whose sum follows (generalized) extreme value distributions. Possible applications of this specific class of random variables are illustrated on the example of two simple physical models. A few extensions to other related classes of random variables sharing similar qualitative properties are also briefly discussed, in connection with the so-called BHP distribution.

The role of quantum measurement in stochastic thermodynamics

This article sets up a new formalism to investigate stochastic thermodynamics in the quantum regime, where stochasticity and irreversibility primarily come from quantum measurement. In the absence of any bath, we define a purely quantum component to heat exchange, that corresponds to energy fluctuations caused by quantum measurement. Energetic and entropic signatures of measurement-induced irreversibility are then explored for canonical experiments of quantum optics, and the energetic cost of counter-acting decoherence is studied on a simple state-stabilizing protocol. By placing quantum measurement in a central position, our formalism contributes to bridge a gap between experimental quantum optics and quantum thermodynamics, and opens new paths to characterize the energetic features of quantum processing.Rebuilding quantum thermodynamics on quantum measurementMeasuring a quantum system is an ultimately random operation. It induces a genuinely quantum time arrow, increasing the system’s entropy. But because it perturbs its state, quantum measurement also provides energy to the quantum system. These energetic quantum fluctuations play the same role as thermal fluctuations in thermodynamics, while being of quantum nature. Building on such “Quantum Heat”, a group of scientists from france provided a thermodynamic analyzis of canonical experiments of quantum optics. They show that quantum heat is a major concept to evaluate the performances of a basic protocol to counteract the decoherence of a quantum bit. The findings pave the way towards a new generation of quantum engines, powered by quantum measurement. They bring new tools to investigate the energetic cost of quantum protocols performed at ultra-low temperature, in the presence of decoherence.

Model for random packing of polydisperse frictionless spheres

We propose a statistical model for the random packing of frictionless polydisperse spheres in which the complexity of the global packing is distilled into a local stochastic process. We simplify the problem by considering the “granocentric” point of view of a single particle in the bulk, thereby reducing random packing to the assembly of nearest neighbours, followed by a random choice of contacts among them. The model is based on only two parameters, the available solid angle around each particle and the ratio of contacts to neighbors, which are both directly obtainable from experiments or simulations. As a result, the model analytically predicts the microscopic distributions of nearest neighbours and contacts, the local density fluctuations as well as the global density of the packing. We find that this granocentric view captures the essential properties of the polydisperse emulsion packing. This model suggests a general principle of organization for random packing and provides a statistical tool for quantifying the effect of the particle size distribution on the geometry of random packing in a variety of contexts of industrial relevance.

Alternative description of the 2D Blume–Capel model using Grassmann algebra

We use Grassmann algebra to study the phase transition in the two-dimensional ferromagnetic Blume-Capel model from a fermionic point of view. This model presents a phase diagram with a second order critical line which becomes first order through a tricritical point, and was used to model the phase transition in specific magnetic materials and liquid mixtures of He

Applications of extreme value statistics in physics

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Origin of the approximate universality of distributions in equilibrium correlated systems

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Unfolding proteins with an atomic force microscope: force-fluctuation-induced nonexponential kinetics.

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Second-order critical lines of spin-S Ising models in a splitting field using Grassmann techniques

. In particular, we are able to map the spin-1 system of the BC model onto an effective fermionic action from which we obtain the exact mass of the theory, the condition of vanishing mass defines the critical line. This effective action is actually an extension of the free fermion Ising action with an additional quartic interaction term. The effect of this term is merely to render the excitation spectrum of the fermions unstable at the tricritical point. The results are compared with recent numerical Monte-Carlo simulations.