arXiv subjects

Pascal Grange

Publications and source records attributed to Pascal Grange.

At least 19 recordsLinked to original sources

arXiv2025

First passage of a run-and-tumble particle with exponentially-distributed tumble duration in the presence of a drift

We consider a run-and-tumble particle on a finite interval $[a,b]$ with two absorbing end points. The particle has an internal velocity state that switches between three values $v,0,-v$ at exponential times, thus incorporating positive tumble times. Moreover, a constant drift is added to the run-and-tumble motion at all times. The combination of these two features constitutes the main novelty of our model. The densities of the first-passage time through $a$ (given the initial position and velocity states) satisfy certain forward Fokker--Planck equations, whose Laplace transforms induce evolution equations for the exit probabilities and mean first-passage times of the particle. We solve these equations explicitly for all possible initial states. We consider the limiting regimes of instantaneous tumble and/or the limit of large $b$ to confirm consistency with existing results in the literature. In particular, in the limit of a half-line (large $b$), the mean first-passage time conditioned on the exit through $a$ is an affine function of the initial position if the drift is positive, as in the case of instantaneous tumble.

cond-mat.stat-mech↗

arXiv2024

Mean first-passage time at the origin of a run-and-tumble particle with periodic forces

We consider a run-and-tumble particle on a half-line with an absorbing target at the origin. The particle has an internal velocity state that switches between two opposite values at Poisson-distributed times. The position of the particle evolves according to an overdamped Langevin dynamics with a spatially-periodic force field such that every point in a given period interval is accessible to the particle. The survival probability of the particle satisfies a backward Fokker--Planck equation, whose Laplace transform yields systems of equations for the moments of the first-passage time of the particle at the origin. The mean first-passage time has already been calculated assuming that the particle exits the system almost surely. We calculate the probability that the particle reaches the origin in a finite time, given its initial position and velocity. We obtain an integral condition on the force, under which the particle has a non-zero survival probability. The conditional average of the first-passage time at the origin (over the trajectories that reach the origin) is obtained in closed form. As an application, we consider a piecewise-constant force field that alternates periodically between two opposite values. In the limit where the period is short compared to the mean free path of the particle, the mean first-return time to the origin coincides with the value obtained in the case of an effective constant drift, which we calculate explicitly.

cond-mat.stat-mech↗

arXiv2023

Local resetting in non-conserving zero-range processes with extensive rates

A non-conserving zero-range process with extensive creation, annihilation and hopping rates is subjected to local resetting. The model is formulated on a large, fully-connected network of states. The states are equipped with a (bounded) fitness level: particles are added to each state at a rate proportional to the fitness level of the state. Moreover, particles are annihilated at a constant rate, and hop at a fixed rate to a uniformly-drawn state in the network. This model has been interpreted in terms of population dynamics: the fitness is the reproductive fitness in a haploid population, and the hopping process models mutation. It has also been interpreted as a model of network growth with a fixed set of nodes (in which particles occupying a state are interpreted as links pointing to this state). In the absence of resetting, the model is known to reach a steady state, which in a certain limit may exhibit a condensate at maximum fitness. If the model is subjected to global resetting by annihilating all particles at Poisson-distributed times, there is no condensation in the steady state. If the system is subjected to local resetting, the occupation numbers of each state are reset to zero at independent random times. These times are distributed according to a Poisson process whose rate (the resetting rate) depends on the fitness. We derive the evolution equation satisfied by the probability law of the occupation numbers. We calculate the average occupation numbers in the steady state. The existence of a condensate is found to depend on the local behavior of the resetting rate at maximum fitness: if the resetting rate vanishes at least linearly at high fitness, a condensate appears at maximum fitness in the limit where the sum of the annihilation and hopping rates is equal to the maximum fitness.

cond-mat.stat-mech↗

arXiv2023

Renewal processes with a trap under stochastic resetting

Renewal processes are zero-dimensional processes defined by independent intervals of time between zero crossings of a random walker. We subject renewal processes them to stochastic resetting by setting the position of the random walker to the origin at Poisson-distributed time with rate $r$. We introduce an additional parameter, the probability $\beta$ of keeping the sign state of the system at resetting time. Moreover, we introduce a trap at the origin, which absorbs the process with a fixed probability at each zero crossing. We obtain the mean lifetime of the process in closed form. For time intervals drawn from a L\'evy stable distribution of parameter $\theta$, the mean lifetime is finite for every positive value of the resetting rate, but goes to infinity when $r$ goes to zero. If the sign-keeping probability $\beta$ is higher than a critical level $\beta_c(\theta)$ (and strictly lower than $1$), the mean lifetime exhibits two extrema as a function of the resetting rate. Moreover, it goes to zero as $r^{-1}$ when $r$ goes to infinity. On the other hand, there is a single minimum if $\beta$ is set to one.

cond-mat.stat-mech↗

arXiv2022

Voter model under stochastic resetting

The voter model is a toy model of consensus formation based on nearest-neighbor interactions. A voter sits at each vertex in a hypercubic lattice (of dimension $d$) and is in one of two possible opinion states. The opinion state of each voter flips randomly, at a rate proportional to the fraction of the nearest neighbors that disagree with the voter. If the voters are initially independent and undecided, the model is known to lead to a consensus if and only if $d\leq 2$. In this paper the model is subjected to stochastic resetting: the voters revert independently to their initial opinion according to a Poisson process of fixed intensity (the resetting rate). This resetting prescription induces kinetic equations for the average opinion state and for the two-point function of the model. For initial conditions consisting of undecided voters except for one decided voter at the origin, the one-point function evolves as the probability of presence of a diffusive random walker on the lattice, whose position is stochastically reset to the origin. The resetting prescription leads to a non-equilibrium steady state. For an initial state consisting of independent undecided voters, the density of domain walls in the steady state is expressed in closed form as a function of the resetting rate. This function is differentiable at zero if and only if $d\geq 5$.

cond-mat.stat-mech↗

arXiv2021

Winding number of a Brownian particle on a ring under stochastic resetting

We consider a random walker on a ring, subjected to resetting at Poisson-distributed times to the initial position (the walker takes the shortest path along the ring to the initial position at resetting times). In the case of a Brownian random walker the mean first-completion time of a turn is expressed in closed form as a function of the resetting rate. The value is shorter than in the ordinary process if the resetting rate is low enough. Moreover, the mean first-completion time of a turn can be minimised in the resetting rate. At large time the distribution of winding numbers does not reach a steady state, which is in contrast with the non-compact case of a Brownian particle under resetting on the real line. The mean total number of turns (and the variance of the net number of turns) grow linearly with time, with a proportionality constant equal to the inverse of the mean first-completion time of a turn.

cond-mat.stat-mech↗

arXiv2020

Aggregation with constant kernel under stochastic resetting

The model of binary aggregation with constant kernel is subjected to stochastic resetting: aggregates of any size explode into monomers at independent stochastic times. These resetting times are Poisson distributed, and the rate of the process is called the resetting rate. The master equation yields a Bernoulli-type equation in the generating function of the concentration of aggregates of any size, which can be solved exactly. This resetting prescription leads to a non-equilibrium steady state for the densities of aggregates, which is a function of the size of the aggregate, rescaled by a function of the resetting rate. The steady-state density of aggregates of a given size is maximised if the resetting rate is set to the quotient of the aggregation rate by the size of the aggregate (minus one).

cond-mat.stat-mech↗

arXiv2020

Run-and-tumble particles on a line with a fertile site

We propose a model of run-and-tumble particles (RTPs) on a line with a fertile site at the origin. After going through the fertile site, a run-and-tumble particle gives rise to new particles until it flips direction. The process of creation of new particles is modelled by a fertility function (of the distance to the fertile site), multiplied by a fertility rate. If the initial conditions correspond to a single RTP with even probability density, the system is parity-invariant. The equations of motion can be solved in the Laplace domain, in terms of the density of right-movers at the origin. At large time, this density is shown to grow exponentially, at a rate that depends only on the fertility function and fertility rate. Moreover, the total density of RTPs (divided by the density of right-movers at the origin), reaches a stationary state that does not depend on the initial conditions, and presents a local minimum at the fertile site.

cond-mat.stat-mech↗

arXiv2020

Susceptibility to disorder of the optimal resetting rate in the Larkin model of directed polymers

We consider the Larkin model of a directed polymer with Gaussian-distributed random forces, with the addition of a resetting process whereby the transverse position of the end-point of the polymer is reset to zero with constant rate $r$. We express the average over disorder of the mean time to absorption by an absorbing target at a fixed value of the transverse position. Thanks to the independence properties of the distribution of the random forces, this expression is analogous to the mean time to absorption for a diffusive particle under resetting, which possesses a single minimum at an optimal value $r^\ast$ of the resetting rate . Moreover, the mean time to absorption can be expanded as a power series of the amplitude of the disorder, around the value $r^\ast$ of the resetting rate. We obtain the susceptibility of the optimal resetting rate to disorder in closed form, and find it to be positive.

cond-mat.stat-mech↗

arXiv2019

Entropy barriers and accelerated relaxation under resetting

The zero-temperature limit of the backgammon model under resetting is studied. The model is a balls-in-boxes model whose relaxation dynamics is governed by the density of boxes containing just one particle. As these boxes become rare at large times, the model presents an entropy barrier. As a preliminary step, a related model with faster relaxation, known to be mapped to a symmetric random walk, is studied by mapping recent results on diffusion with resetting onto the balls-in-boxes problem. Diffusion with an absorbing target at the origin (and diffusion constant equal to one), stochastically reset to the unit position, is a continuum approximation to the dynamics of the balls-in-boxes model, with resetting to a configuration maximising the number of boxes containing just one ball. In the limit of a large system, the relaxation time of the balls-in-boxes model under resetting is finite. The backgammon model subject to a constant resetting rate is then studied using an adiabatic approximation.

cond-mat.stat-mech↗

arXiv2019

Non-conserving zero-range processes with extensive rates under resetting

We consider a non-conserving zero-range process with hopping rate proportional to the number of particles at each site. Particles are added to the system with a site-dependent creation rate, and vanish with a uniform annihilation rate. On a fully-connected lattice with a large number of sites, the mean-field geometry leads to a negative binomial law for the number of particles at each site, with parameters depending on the hopping, creation and annihilation rates. This model can be mapped to population dynamics (if the creation rates are reproductive fitnesses in a haploid population, and the hopping rate is the mutation rate). It can also be mapped to a Bianconi--Barab\'asi model of a growing network with random rewiring of links (if creation rates are the rates of acquisition of links by nodes, and the hopping rate is the rewiring rate). The steady state has recently been worked out and gives rise to occupation numbers that reproduce Kingman's house-of-cards model of selection and mutation. In this paper we solve the master equation using a functional method, which yields integral equations satisfied by the occupation numbers. The occupation numbers are shown to forget initial conditions at an exponential rate that decreases linearly with the fitness level. Moreover, they can be computed exactly in the Laplace domain, which allows to obtain the steady state of the system under resetting. The result modifies the house-of-cards result by simply adding a skewed version of the initial conditions, and by adding the resetting rate to the hopping rate.

cond-mat.stat-mech↗

arXiv2019

Steady states in a non-conserving zero-range process with extensive rates as a model for the balance of selection and mutation

We consider a non-conserving zero-range process with hopping rate proportional to the number of particles at each site. Particles are added to the system with a site-dependent creation rate, and removed from the system with a uniform annihilation rate. On a fully-connected lattice with a large number of sites, the mean-field geometry leads to a negative binomial law for the number of particles at each site, with parameters depending on the hopping, creation and annihilation rates. This model of particles is mapped to a model of population dynamics: the site label is interpreted as a level of fitness, the site-dependent creation rate is interpreted as a selection function, and the hopping process is interpreted as the introduction of mutants. In the limit of large density, the fraction of the total population occupying each site approaches the limiting distribution in the house-of-cards model of selection-mutation, introduced by Kingman. A single site can be occupied by a macroscopic fraction of the particles if the mutation rate is below a critical value (which matches the critical value worked out in the house-of-cards model). This feature generalises to classes of selection functions that increase sufficiently fast at high fitness. The process can be mapped to a model of evolving networks, inspired by the Bianconi--Barab\'asi model, but involving a large and fixed set of nodes. Each node forms links at a rate biased by its fitness, moreover links are destroyed at a uniform rate, and redirected at a certain rate. If this redirection rate matches the mutation rate, the number of links pointing to nodes of a given fitness level is distributed as the numbers of particles in the non-conserving zero-range process. There is a finite critical redirection rate if the density of quenched fitnesses goes to zero sufficiently fast at high fitness.

cond-mat.stat-mech↗

arXiv2018

Topology of the mesoscale connectome of the mouse brain

The wiring diagram of the mouse brain has recently been mapped at a mesoscopic scale in the Allen Mouse Brain Connectivity Atlas. Axonal projections from brain regions were traced using green fluoresent proteins. The resulting data were registered to a common three-dimensional reference space. They yielded a matrix of connection strengths between 213 brain regions. Global features such as closed loops formed by connections of similar intensity can be inferred using tools from persistent homology. We map the wiring diagram of the mouse brain to a simplicial complex (filtered by connection strengths). We work out generators of the first homology group. Some regions, including nucleus accumbens, are connected to the entire brain by loops, whereas no region has non-zero connection strength to all brain regions. Thousands of loops go through the isocortex, the striatum and the thalamus. On the other hand, medulla is the only major brain compartment that contains more than 100 loops.

q-bio.NC↗

arXiv2017

Crossover in the log-gamma polymer from the replica coordinate Bethe Ansatz

The coordinate Bethe Ansatz solution of the log-gamma polymer is extended to boundary conditions with one fixed end and the other attached to one half of a one-dimensional lattice. The large-time limit is studied using a saddle-point approximation,and the cumulative distribution function of the rescaled free energy of a long polymer is expressed as a Fredholm determinant. Scaling limits of the kernel are identified, leading to a crossover from the GUE to the GOE Tracy--Widom distributions. The continuum limit reproduces the crossover from droplet to flat initial conditions of the Kardar--Parisi--Zhang equation.

cond-mat.dis-nn↗

arXiv2017

Log-gamma directed polymer with one free end via coordinate Bethe Ansatz

The discrete polymer model with random Boltzmann weights with homogeneous inverse gamma distribution, introduced by Sepp\"al\"ainen, is studied in the case of a polymer with one fixed and one free end. The model with two fixed ends has been integrated by Thiery and Le Doussal, using coordinate Bethe Ansatz techniques and an analytic-continuation prescription. The probability distribution of the free energy has been obtained through the replica method, even though the moments of the partition sum do not exist at all orders due to the fat tail in the distribution of Boltzmann weights. To extend this approach to the polymer with one free end, we argue that the contribution to the partition sums in the thermodynamic limit is localised on parity-invariant string states. This situation is analogous to the case of the continuum polymer with one free end, related to the Kardar--Parisi--Zhang equation with flat boundary conditions and solved by Le Doussal and Calabrese. The expansion of the generating function of the partition sum in terms of numbers of strings can also be transposed to the log-gamma polymer model, with the induced Fredholm determinant structure. We derive the large-time limit of the rescaled cumulative distribution function, and relate it to the GOE Tracy--Widom distribution. The derivation is conjectural in the sense that it assumes completeness of a family of string states (and expressions of their norms already used in the fixed-end problem) and extends heuristically the order of moments of the partition sum to the complex plane.

cond-mat.dis-nn↗

arXiv2016

Quantum centipedes with strong global constraint

A centipede made of $N$ quantum walkers on a one-dimensional lattice is considered. The distance between two consecutive legs is either one or two lattice spacings, and a global constraint is imposed: the maximal distance between the first and last leg is $N+1$. This is the strongest global constraint compatible with walking. For an initial value of the wave function corresponding to a localized configuration at the origin, the probability law of the first leg of the centipede can be expressed in closed form in terms of Bessel functions. The dispersion relation and the group velocities are worked out exactly. Their maximal group velocity goes to zero when $N$ goes to infinity, which is in contrast with the behaviour of group velocities of quantum centipedes without global constraint, which were recently shown by Krapivsky, Luck and Mallick to give rise to ballistic spreading of extremal wave-front at non-zero velocity in the large-$N$ limit. The corresponding Hamiltonians are implemented numerically, based on a block structure of the space of configurations corresponding to compositions of the integer $N$. The growth of the maximal group velocity when the strong constraint is gradually relaxed is explored, and observed to be linear in the density of gaps allowed in the configurations. Heuristic arguments are presented to infer that the large-$N$ limit of the globally constrained model can yield finite group velocities provided the allowed number of gaps is a finite fraction of $N$.

cond-mat.stat-mech↗

arXiv2015

Computational neuroanatomy: mapping cell-type densities in the mouse brain, simulations from the Allen Brain Atlas

The Allen Brain Atlas (ABA) of the adult mouse consists of digitized expression profiles of thousands of genes in the mouse brain, co-registered to a common three-dimensional template (the Allen Reference Atlas). This brain-wide, genome-wide data set has triggered a renaissance in neuroanatomy. Its voxelized version (with cubic voxels of side 200 microns) can be analyzed on a desktop computer using MATLAB. On the other hand, brain cells exhibit a great phenotypic diversity (in terms of size, shape and electrophysiological activity), which has inspired the names of some well-studied cell types, such as granule cells and medium spiny neurons. However, no exhaustive taxonomy of brain cells is available. A genetic classification of brain cells is under way, and some cell types have been characterized by their transcriptome profiles. However, given a cell type characterized by its transcriptome, it is not clear where else in the brain similar cells can be found. The ABA can been used to solve this region-specificity problem in a data-driven way: rewriting the brain-wide expression profiles of all genes in the atlas as a sum of cell-type-specific transcriptome profiles is equivalent to solving a quadratic optimization problem at each voxel in the brain. However, the estimated brain-wide densities of 64 cell types published recently were based on one series of co-registered coronal in situ hybridization (ISH) images per gene, whereas the online ABA contains several image series per gene, including sagittal ones. In the presented work, we simulate the variability of cell-type densities in a Monte Carlo way by repeatedly drawing a random image series for each gene and solving optimization problems. This yields error bars on the region-specificity of cell types.

q-bio.NC↗

arXiv2015

Cell-type-specific computational neuroanatomy, simulations from the sagittal and coronal Allen Brain Atlas

The Allen Atlas of the adult mouse brain is a brain-wide, genome-wide data set that has been made available online, triggering a renaissance in neuroanatomy. In particular, it has been used to define brain regions in a computational, data-driven way, and to estimate the region-specificity of cell types characterized independently by their transcriptional activity. However, these results were based on one series of co-registered (coronal) ISH image series per gene, whereas the online ABA contains several image series per genes, including sagittal ones. Since the sagittal series cover mostly the left hemisphere, we can simulate the variability of results by repeatedly drawing a random image series for each gene and restricting the computation to the left hemisphere. This gives rise to an estimate of error bars on the results of computational neuroanatomy.

q-bio.NC↗