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Infinite dimensional active matter and stochastic calculus for path integration

Abstract : The forthcoming work is divided into two distinct parts. The first one deals with self-propelled particles systems. We start by studying the one particle in an external potential case. We derive the nonequilibrium properties of the Active Ornstein-Ulhenbeck Particle model at small persistence time in the presence of thermal noise and the stationary measure of a run-and-tumble particle around a hard spherical obstacle at large persistence time. We then focus on the collective behavior of such systems. From an analytical standpoint, not much is known given their high degree of complexity that combines those of out-of-equilibrium physics to those of strongly correlated liquids. Since the mid-eighties and Frisch’s & al. work, we have known that equilibrium fluids can be studied exactly in the limit where the dimension of the embedding space becomes infinite. The mathematical gains are then considerable: not only the free energy can be obtained analytically but also transport coefficients. These ideas later had a groundbreaking influence on the mean-field theory of the glass transition which is naturally expressed in infinite dimension. Here, the goal is to use the large dimension limit to gain theoretical insights into the behavior of active systems. The equations of the dynamical mean field theory are first studied in the dilute limit and we quantify the connections between the mean-square-displacement and the effective propulsion speed. To go beyond the dilute limit, we then propose an approximate resummation scheme of the Born-Bogolioubov-Green-Kirkwood-Yvon hierarchy of correlation functions. The latter allows us to account for various properties observed in finite dimensional systems, in particular for the Motility Induced Phase Separation and for the linear decrease with density of the effective self-propulsion speed of active hard spheres. We also introduce the concept of effective amplitude of potential interactions. We then show that this amplitude vanishes at the same density as the effective propulsion speed which is also that of the dynamical glass transition of an equilibrium colloidal system with equivalent structure. These results draw interesting links between the glass transition of equilibrium systems and the vanishing of the effective self-propulsion speed which is a stationary property of a unique active system. The specificity underlined by this approximate resummation is the presence of multibody interactions in the steady state measure. Unlike its equilibrium counterpart, it cannot be written as a product over the pairs in the system. The importance of these multibody interactions in the phase behavior of active systems was recently demonstrated in dimension 3. We keep exploring this idea by studying the phase diagram of the unified colored noise approximation of the AOUP dynamics. We show that it displays two regions of phase coexistence that the sole pair interactions are unable to account for. The second part of the manuscript deals with the extension of stochastic calculus to path integration and generalizes results recently obtained in the one-dimensional case. After explaining why it is in general impossible to use the rules of stochastic calculus to change variables within continuous time path integrals we show how to modify these rules consequently. We finally propose a higher-order discretization scheme extending that of Stratonovich and making the rules of standard differential calculus compatible with path integration.
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https://tel.archives-ouvertes.fr/tel-03504439
Contributor : Thibaut Arnoulx de Pirey Connect in order to contact the contributor
Submitted on : Wednesday, December 29, 2021 - 10:59:13 AM
Last modification on : Wednesday, January 5, 2022 - 3:50:13 AM
Long-term archiving on: : Wednesday, March 30, 2022 - 6:08:34 PM

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Thibaut Arnoulx de Pirey. Infinite dimensional active matter and stochastic calculus for path integration. Physics [physics]. Université de Paris / Université Paris Diderot (Paris 7), 2021. English. ⟨tel-03504439⟩

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