Microscopic models of traveling wave equations

Reaction-diffusion problems are often described at a macroscopic scale by partial derivative equations of the type of the Fisher or Kolmogorov-Petrovsky-Piscounov equation. These equations have a continuous family of front solutions, each of them corresponding to a different velocity of the front. By simulating systems of size up to N=10^(16) particles at the microscopic scale, where particles react and diffuse according to some stochastic rules, we show that a single velocity is selected for the front. This velocity converges logarithmically to the solution of the F-KPP equation with minimal velocity when the number N of particles increases. A simple calculation of the effect introduced by the cutoff due to the microscopic scale allows one to understand the origin of the logarithmic correction.Comment: 11 pages, 3 figure

Ground state energy of a non-integer number of particles with delta attractive interactions

We show how to define and calculate the ground state energy of a system of quantum particles with delta attractive interactions when the number of particles n$is non-integer. The question is relevant to obtain the probability distribution of the free energy of a directed polymer in a random medium. When one expands the ground state energy in powers of the interaction, all the coefficients of the perturbation series are polynomials in n, allowing to define the perturbation theory for non-integer n. We develop a procedure to calculate all the cumulants of the free energy of the directed polymer and we give explicit, although complicated, expressions of the first three cumulants.Comment: 11 pages, no figur

Global existence for a free boundary problem of Fisher-KPP type

Motivated by the study of branching particle systems with selection, we establish global existence for the solution $(u,\mu)$ of the free boundary problem $$\begin{cases} \partial_t u =\partial^2_{x} u +u & \text{for $t>0$ and $x>\mu_t$,}\\ u(x,t)=1 &\text{for $t>0$ and $x \leq \mu_t$}, \\ \partial_x u(\mu_t,t)=0 & \text{for $t>0$}, \\ u(x,0)=v(x) &\text{for $x\in \mathbb{R}$}, \end{cases}$$ when the initial condition $v:\mathbb{R}\to[0,1]$ is non-increasing with $v(x) \to 0$ as $x\to \infty$ and $v(x)\to 1$ as $x\to -\infty$. We construct the solution as the limit of a sequence $(u_n)_{n\ge 1}$, where each $u_n$ is the solution of a Fisher-KPP equation with same initial condition, but with a different non-linear term. Recent results of De Masi \textit{et al.}~\cite{DeMasi2017a} show that this global solution can be identified with the hydrodynamic limit of the so-called $N$-BBM, {\it i.e.} a branching Brownian motion in which the population size is kept constant equal to $N$ by killing the leftmost particle at each branching event

Traveling discontinuity at the quantum butterfly front

We formulate a kinetic theory of quantum information scrambling in the context of a paradigmatic model of interacting electrons in the vicinity of a superconducting phase transition. We carefully derive a set of coupled partial differential equations that effectively govern the dynamics of information spreading in generic dimensions. They exhibit traveling wave solutions that are discontinuous at the boundary of the light cone, and have a perfectly causal structure where the solutions do not spill outside of the light cone.Comment: 31 pages, 7 figure

Brownian bees in the infinite swarm limit

The Brownian bees model is a branching particle system with spatial selection. It is a system of N particles which move as independent Brownian motions in Rd and independently branch at rate 1, and, crucially, at each branching event, the particle which is the furthest away from the origin is removed to keep the population size constant. In the present work we prove that, as N→∞, the behaviour of the particle system is well approximated by the solution of a free boundary problem (which is the subject of a companion paper (Trans. Amer. Math. Soc. 374 (2021) 6269–6329)), the hydrodynamic limit of the system. We then show that for this model the so-called selection principle holds; that is, that as N→∞, the equilibrium density of the particle system converges to the steady-state solution of the free boundary problem

Le management en situation de complexité et d’incertitude. Apport de la Recherche et Développement

La seconde révolution industrielle a vu se développer des méthodes de management adaptées aux contextes successivement traversés, d’abord la nécessité de produire en masse, puis d’améliorer la qualité et la satisfaction des clients, puis d’améliorer la performance opérationnelle en réponse à la globalisation des marchés. Initialement inspirées d’une pensée mécaniste rationnelle, ces approches se sont progressivement étoffées en y intégrant d’autres dimensions, la psychologie, la sociologie, l’analyse systémique. Dans les années 1990, les entreprises s’étaient profondément remises en cause, avaient mis en place des modes de fonctionnement raffinés, mais deux nouveaux paramètres ont bouleversé leur environnement : la complexité et l’incertitude. Les entreprises de la troisième révolution industrielle ont d’emblée intégré ces paramètres, et mis en place de nouveaux modes de management. Pour autant, ceux-ci peuvent ne pas répondre à des activités où l’erreur peut être mortelle, où le bug n’est pas acceptable. Entre souci de performance et de bien-être au travail, entre principe de précaution et principe d’expérimentation, la troisième révolution industrielle cherche encore ses approches, alors qu’une quatrième révolution est peut-être déjà là et annonce de nouveaux bouleversements. Dans cette perspective marquée par une complexité et une incertitude croissantes, la Recherche et Développement présente un intérêt particulier puisque sa vocation, précisément, consiste à affronter le complexe et l’incertain. Dans cet article, nous dégagerons les grands fondamentaux du processus R&D, et analyserons en quoi ils peuvent servir de benchmark au management contemporain, en lui apportant des sources d’inspiration

A low Complexity Wireless Gigabit Ethernet IFoF 60 GHz H/W Platform and Issues

6 pagesInternational audienceThis paper proposes a complete IFoF system architecture derived from simplified IEEE802.15.3c PHY layer proposal to successfully ensure near 1 Gbps on the air interface. The system architecture utilizes low complexity baseband processing modules. The byte/frame synchronization technique is designed to provide a high value of preamble detection probability and a very small value of the false detection probability. Conventional Reed-Solomon RS (255, 239) coding is used for Channel Forward Error Correction (FEC). Good communication link quality and Bit Error Rate (BER) results at 875 Mbps are achieved with directional antennas