Aging by near-extinctions in many-variable interacting populations

Models of many-species ecosystems, such as the Lotka-Volterra and replicator equations, suggest that these systems generically exhibit near-extinction processes, where population sizes go very close to zero for some time before rebounding, accompanied by a slowdown of the dynamics (aging). Here, we investigate the connection between near-extinction and aging by introducing an exactly solvable many-variable model, where the time derivative of each population size vanishes both at zero and some finite maximal size. We show that aging emerges generically when random interactions are taken between populations. Population sizes remain exponentially close (in time) to the absorbing values for extended periods of time, with rapid transitions between these two values. The mechanism for aging is different from the one at play in usual glassy systems: at long times, the system evolves in the vicinity of unstable fixed points rather than marginal ones

Many-species ecological fluctuations as a jump process from the brink of extinction

Many-species ecological communities can exhibit persistent fluctuations driven by species interactions. These dynamics feature many interesting properties, such as the emergence of long timescales and large fluctuations, that have remained poorly understood. We look at such dynamics, when species are supported by migration at a small rate. We find that the dynamics are characterized by a single long correlation timescale. We prove that the time and abundances can be rescaled to yield a well-defined limiting process when the migration rate is small but positive. The existence of this rescaled dynamics predicts scaling forms for both abundance distributions and timescales, which are verified exactly in scaling collapse of simulation results. In the rescaled process, a clear separation naturally emerges at any given time between rare and abundant species, allowing for a clear-cut definition of the number of coexisting species. Species move back and forth between the rare and abundant subsets. The dynamics of a species entering the abundant subset starts with rapid growth from rare, appearing as an instantaneous jump in rescaled time, followed by meandering abundances with an overall negative bias. The emergence of the long timescale is explained by another rescaling theory for earlier times. Finally, we prove that the number of abundant species is tuned to remain below and without saturating a well-known stability bound, maintaining the system away from marginality. This is traced back to the perturbing effect of the jump processes of incoming species on the abundant ones

A run-and-tumble particle around a spherical obstacle: steady-state distribution far-from-equilibrium

We study the steady-state distribution function of a run-and-tumble particle evolving around a repulsive hard spherical obstacle. We show that the well-documented activity-induced attraction translates into a delta peak accumulation at the surface of the obstacle accompanied with an algebraic divergence of the density profile close to the obstacle. We obtain the full form of the distribution function in the regime where the typical distance run by the particle between two consecutive tumbles is much larger than the size of the obstacle. This provides an expression for the low-density pair distribution function of a fluid of highly persistent hard-core run-and-tumble particles. This also provides an expression for the steady-state probability distribution of highly-ballistic active Brownian particles and active Ornstein-Ulhenbeck particles around hard spherical obstacles

Active hard-spheres in infinitely many dimensions

Few equilibrium --even less so nonequilibrium-- statistical-mechanical models with continuous degrees of freedom can be solved exactly. Classical hard-spheres in infinitely many space dimensions are a notable exception. We show that even without resorting to a Boltzmann distribution, dimensionality is a powerful organizing device to explore the stationary properties of active hard-spheres evolving far from equilibrium. In infinite dimensions, we compute exactly the stationary state properties that govern and characterize the collective behavior of active hard-spheres: the structure factor and the equation of state for the pressure. In turn, this allows us to account for motility-induced phase-separation. Finally, we determine the crowding density at which the effective propulsion of a particle vanishes.Comment: Main text : 6 pages, 2 figures. Supplemental material : 7 pages, 2 figure

Path integrals and stochastic calculus

Path integrals are a ubiquitous tool in theoretical physics. However, their use is sometimes hindered by the lack of control on various manipulations -- such as performing a change of the integration path -- one would like to carry out in the light-hearted fashion that physicists enjoy. Similar issues arise in the field of stochastic calculus, which we review to prepare the ground for a proper construction of path integrals. At the level of path integration, and in arbitrary space dimension, we not only report on existing Riemannian geometry-based approaches that render path integrals amenable to the standard rules of calculus, but also bring forth new routes, based on a fully time-discretized approach, that achieve the same goal. We illustrate these various definitions of path integration on simple examples such as the diffusion of a particle on a sphere.Comment: 96 pages, 4 figures. New title, expanded introduction and additional references. Version accepted in Advandes in Physic

Matière active en dimension infinie et calcul stochastique appliqué aux intégrales de chemin

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.Ce travail de thèse se divise en deux parties distinctes. La première est consacrée aux systèmes actifs de particules autopropulsées. Nous commençons par étudier le cas d’une particule dans un potentiel. Nous analysons les déviations par rapport à la dynamique d’équilibre de celle d’une particule active propulsée par un processus d’Ornstein-Ulhenbeck (AOUP) à petit temps de persistance et en présence de bruit thermique ainsi que les propriétés stationnaires d’une particule autopropulsées autour d’un obstacle sphérique dans la limite de grand temps de persistance. Nous nous intéressons ensuite aux propriétés collectives de ce systèmes. D’un point de vue analytique, leur compréhension est pour l’instant entravée par leur diffculté intrinsèque qui combine celles des systèmes hors d’équilibre à celles des liquides fortement corrélés. Depuis le milieu des années 1980, nous savons que les fuides d’équilibres peuvent être étudiés analytiquement dans la limite où la dimension de l’espace ambiant devient infinie. Les gains mathématiques sont alors considérables : non seulement l’énergie libre peut être calculée exactement mais aussi les coeffcients de transport. Ces idées eurent ensuite une influence majeure dans la théorie de la transition vitreuse en champ moyen. L’objectif ici est d’utiliser la limite de grande dimension dans le cas actif. Nous étudions d’abord les équations de la théorie de champ moyen dynamique dans la limite diluée, ce qui nous permet de quantifier la relation entre le déplacement quadratique moyen et la vitesse effective d’autopropulsion. Pour étudier les propriétés des systèmes actifs au-delà de la limite diluée nous proposons ensuite un schéma approché de resommation de la hiérarchie de Born-Bogolioubov-Green-Kirkwood-Yvon des fonctions de corrélation. Celui-ci permet de rendre compte de nombreuses propriétés observées dans les systèmes ac-tifs de dimension finie, en particulier de la transition de phase induite par la motilité et de la décroissance linéaire de la vitesse effective d’autopropulsion des sphères dures actives avec la densité. Ces travaux nous conduisent à introduire le concept d’amplitude effective des interactions potentielles. Nous montrons alors que celle-ci s’annule à la même densité que la vitesse effective d’autopropulsion qui est aussi la densité de transition vitreuse dynamique d’un système colloïdal d’équilibre de structure équivalente. Ces résultats dressent un parallèle intéressant entre la transition vitreuse des systèmes d’équilibre et l’annulation de la vitesse effective d’autopropulsion des systèmes actifs qui est une propriété de la mesure stationnaire d’un système unique. La spécificité soulignée par cette resommation approchée est la présence d’interactions multicorps dans la mesure stationnaire. Contrairement au cas des liquides classiques d’équilibre, celle-ci ne peut en effet pas s’écrire sous la forme d’un produit sur les paires du système. L’importance de ces interactions multicorps dans le diagramme des phases des systèmes actifs a récemment été soulignée en dimension 3. Nous continuons d’explorer cette idée en dimension infinie en étudiant le diagramme des phases de l’approximation dite de bruit coloré unifié de la dynamique AOUP. Nous montrons que celui-ci présente deux régions de coexistence de phase que les interactions de paires seules ne peuvent expliquer. La deuxième partie de cette thèse porte sur des extensions du calcul stochastique dans les intégrales de chemin et généralise des résultats récemment établis dans le cas de processus unidimensionnels. Après avoir expliqué pourquoi il est en général impossible d’utiliser les règles du calcul stochastique pour changer de variable au sein des intégrales de chemin en temps continu nous montrons comment modifier ces dernières enconséquence. Enfin, nous proposons une discrétisation d’ordre supérieur étendant celle de Stratonovich et rendant utilisable le calcul différentiel au sein des intégrales de chemin

Discretized and covariant path integrals for stochastic processes

Path integrals are a ubiquitous tool in theoretical physics. However, their use is sometimes hindered by the lack of control on various manipulations - such as performing a change of the integration path - one would like to carry out in the light-hearted fashion that physicists enjoy. Similar issues arise in the field of stochastic calculus, which we review to prepare the ground for a proper construction of path integrals. At the level of path integration, and in arbitrary space dimension, we not only report on existing Riemannian geometry-based approaches that render path integrals amenable to the standard rules of calculus, but also bring forth new routes, based on a fully time-discretized approach, that achieve the same goal. We illustrate these various definitions of path integration on simple examples such as the diffusion of a particle on a sphere