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Eric Brunet

Publications and source records attributed to Eric Brunet.

14 recordsLinked to original sources
arXiv2022

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. Their solutions show that scrambling propagates at the maximal speed set by the Fermi velocity. At early times, we find exponential growth at a rate set by the inelastic scattering. At late times, we find that scrambling is governed by shock-wave dynamics with traveling waves exhibiting a discontinuity at the boundary of the light cone. Notably, we find perfectly causal dynamics where the solutions do not spill outside of the light cone.

cond-mat.stat-mech↗
arXiv2020

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 $\mathbb{R}^d$ 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 $N \to \infty$ 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), the hydrodynamic limit of the system. We then show that for this model the so-called selection principle holds, i.e. that as $N \to \infty$ the equilibrium density of the particle system converges to the steady state solution of the free boundary problem.

math.PR↗
arXiv2018

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.

math.AP↗
arXiv2009

Statistics at the tip of a branching random walk and the delay of traveling waves

We study the limiting distribution of particles at the frontier of a branching random walk. The positions of these particles can be viewed as the lowest energies of a directed polymer in a random medium in the mean-field case. We show that the average distances between these leading particles can be computed as the delay of a traveling wave evolving according to the Fisher-KPP front equation. These average distances exhibit universal behaviors, different from those of the probability cascades studied recently in the context of mean field spin-glasses.

cond-mat.dis-nn↗
arXiv2007

The stochastic edge in adaptive evolution

In a recent article, Desai and Fisher (2007) proposed that the speed of adaptation in an asexual population is determined by the dynamics of the stochastic edge of the population, that is, by the emergence and subsequent establishment of rare mutants that exceed the fitness of all sequences currently present in the population. Desai and Fisher perform an elaborate stochastic calculation of the mean time $τ$ until a new class of mutants has been established, and interpret $1/τ$ as the speed of adaptation. As they note, however, their calculations are valid only for moderate speeds. This limitation arises from their method to determine $τ$: Desai and Fisher back-extrapolate the value of $τ$ from the best-fit class' exponential growth at infinite time. This approach is not valid when the population adapts rapidly, because in this case the best-fit class grows non-exponentially during the relevant time interval. Here, we substantially extend Desai and Fisher's analysis of the stochastic edge. We show that we can apply Desai and Fisher's method to high speeds by either exponentially back-extrapolating from finite time or using a non-exponential back-extrapolation. Our results are compatible with predictions made using a different analytical approach (Rouzine et al. 2003, 2007), and agree well with numerical simulations.

q-bio.PE↗
arXiv2007

The traveling wave approach to asexual evolution: Muller's ratchet and speed of adaptation

We use traveling-wave theory to derive expressions for the rate of accumulation of deleterious mutations under Muller's ratchet and the speed of adaptation under positive selection in asexual populations. Traveling-wave theory is a semi-deterministic description of an evolving population, where the bulk of the population is modeled using deterministic equations, but the class of the highest-fitness genotypes, whose evolution over time determines loss or gain of fitness in the population, is given proper stochastic treatment. We derive improved methods to model the highest-fitness class (the stochastic edge) for both Muller's ratchet and adaptive evolution, and calculate analytic correction terms that compensate for inaccuracies which arise when treating discrete fitness classes as a continuum. We show that traveling wave theory makes excellent predictions for the rate of mutation accumulation in the case of Muller's ratchet, and makes good predictions for the speed of adaptation in a very broad parameter range. We predict the adaptation rate to grow logarithmically in the population size until the population size is extremely large.

q-bio.PE↗
arXiv2004

On the Relation between One-Species Diffusion-Limited Coalescence and Annihilation in One Dimension

The close similarity between the hierarchies of multiple-point correlation functions for the diffusion-limited coalescence and annihilation processes has caused some recent confusion, raising doubts as to whether such hierarchies uniquely determine an infinite particle system. We elucidate the precise relations between the two processes, arriving at the conclusion that the hierarchy of correlation functions does provide a complete representation of a particle system on the line. We also introduce a new hierarchy of probability density functions, for finding particles at specified locations and none in between. This hierarchy is computable for coalescence, through the method of empty intervals, and is naturally suited for questions concerning the ordering of particles on the line.

cond-mat.stat-mech↗
arXiv2004

An exactly soluble noisy traveling wave equation appearing in the problem of directed polymers in a random medium

We calculate exactly the velocity and diffusion constant of a microscopic stochastic model of $N$ evolving particles which can be described by a noisy traveling wave equation with a noise of order $N^{-1/2}$. Our model can be viewed as the infinite range limit of a directed polymer in random medium with $N$ sites in the transverse direction. Despite some peculiarities of the traveling wave equations in the absence of noise, our exact solution allows us to test the validity of a simple cutoff approximation and to show that, in the weak noise limit, the position of the front can be completely described by the effect of the noise on the first particle.

cond-mat.dis-nn↗
arXiv2003

Fluctuations of the winding number of a directed polymer in a random medium

For a directed polymer in a random medium lying on an infinite cylinder, that is in 1+1 dimensions with finite width and periodic boundary conditions on the transverse direction, the winding number is simply the algebraic number of turns the polymer does around the cylinder. This paper presents exact expressions of the fluctuations of this winding number due to, first, the thermal noise of the system and, second, the different realizations of the disorder in the medium.

cond-mat.dis-nn↗
arXiv2000

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.

cond-mat↗
arXiv2000

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.

cond-mat.stat-mech↗
arXiv2000

Shift in the velocity of a front due to a cut-off

We consider the effect of a small cut-off epsilon on the velocity of a traveling wave in one dimension. Simulations done over more than ten orders of magnitude as well as a simple theoretical argument indicate that the effect of the cut-off epsilon is to select a single velocity which converges when epsilon tends to 0 to the one predicted by the marginal stability argument. For small epsilon, the shift in velocity has the form K(log epsilon)^(-2) and our prediction for the constant K agrees very well with the results of our simulations. A very similar logarithmic shift appears in more complicated situations, in particular in finite size effects of some microscopic stochastic systems. Our theoretical approach can also be extended to give a simple way of deriving the shift in position due to initial conditions in the Fisher-Kolmogorov or similar equations.

cond-mat↗
arXiv2000

Probability distribution of the free energy of a directed polymer in a random medium

We calculate exactly the first cumulants of the free energy of a directed polymer in a random medium for the geometry of a cylinder. By using the fact that the n-th moment of the partition function is given by the ground state energy of a quantum problem of n interacting particles on a ring of length L, we write an integral equation allowing to expand these moments in powers of the strength of the disorder gamma or in powers of n. For n small and n of order (L gamma)^(-1/2), the moments take a scaling form which allows to describe all the fluctuations of order 1/L of the free energy per unit length of the directed polymer. The distribution of these fluctuations is the same as the one found recently in the asymmetric exclusion process, indicating that it is characteristic of all the systems described by the Kardar-Parisi-Zhang equation in 1+1 dimensions.

cond-mat.stat-mech↗