Configurational transition in a Fleming - Viot-type model and probabilistic interpretation of Laplacian eigenfunctions

Convergence of a particle approximation for the quasi-stationary distribution of a diffusion process: Uniform estimates in a compact soft case

We establish the convergences (with respect to the simulation time t; the number of particles N; the timestep γ) of a Moran/Fleming-Viot type particle scheme toward the quasi-stationary distribution of a diffusion on the d-dimensional torus, killed at a smooth rate. In these conditions, quantitative bounds are obtained that, for each parameter (t →∞, N →∞ or γ → 0) are independent from the two others.

Slow-fast stochastic diffusion dynamics and quasi-stationarity for diploid populations with varying size

We are interested in the long-time behavior of a diploid population with sexual reproduction and randomly varying population size, characterized by its genotype composition at one bi-allelic locus. The population is modeled by a 3-dimensional birth-and-death process with competition, weak cooperation and Mendelian reproduction. This stochastic process is indexed by a scaling parameter K that goes to infinity, following a large population assumption. When the individual birth and natural death rates are of order K, the sequence of stochastic processes indexed by K converges toward a new slow-fast dynamics with variable population size. We indeed prove the convergence toward 0 of a fast variable giving the deviation of the population from quasi Hardy-Weinberg equilibrium, while the sequence of slow variables giving the respective numbers of occurrences of each allele converges toward a 2-dimensional diffusion process that reaches (0,0) almost surely in finite time. The population size and the proportion of a given allele converge toward a Wright-Fisher diffusion with stochastically varying population size and diploid selection. We insist on differences between haploid and diploid populations due to population size stochastic variability. Using a non trivial change of variables, we study the absorption of this diffusion and its long time behavior conditioned on non-extinction. In particular we prove that this diffusion starting from any non-trivial state and conditioned on not hitting (0,0) admits a unique quasi-stationary distribution. We give numerical approximations of this quasi-stationary behavior in three biologically relevant cases: neutrality, overdominance, and separate niches.

Tiling a plane in a dynamical process and its applications to arrays of quantum dots, drums, and heat transfer

We present a reaction-diffusion system consisting of N components. The evolution of the system leads to the partition of the plane into cells, each occupied by only one component. For large N, the stationary state becomes a periodic array of hexagonal cells. We present a functional of the densities of the components, which decreases monotonically during the evolution and attains its minimal value in the stationary state. This value is equal to the sum of the first Laplacian eigenvalues for all cells. Thus, the resulting partition of the plane is determined by minimization of the sum of the eigenvalues, and not by the minimization of the total perimeter of the cells as in the famous honeycomb problem.

Pattern formation in nonextensive thermodynamics: selection criterion based on the Renyi entropy production

We analyze a system of two different types of Brownian particles confined in a cubic box with periodic boundary conditions. Particles of different types annihilate when they come into close contact. The annihilation rate is matched by the birth rate, thus the total number of each kind of particles is conserved. When in a stationary state, the system is divided by an interface into two subregions, each occupied by one type of particles. All possible stationary states correspond to the Laplacian eigenfunctions. We show that the system evolves towards those stationary distributions of particles which minimize the Renyi entropy production. In all cases, the Renyi entropy production decreases monotonically during the evolution despite the fact that the topology and geometry of the interface exhibit abrupt and violent changes.

Minimization of the Renyi entropy production in the space-partitioning process

The spontaneous division of space in Fleming-Viot processes is studied in terms of non-extensive thermodynamics. We analyze a system of n different types of Brownian particles confined in a box. Particles of different types annihilate each other when they come into close contact. Each process of annihilation is accompanied by a simultaneous nucleation of a particle of the same type, so that the number of particles of each component remains constant. The system eventually reaches a stationary state, in which the available space is divided into n separate subregions, each occupied by particles of one type. Within each subregion, the particle density distribution minimizes the Renyi entropy production. We show that the sum of these entropy productions in the stationary state is also minimized, i.e., the resulting boundaries between different components adopt a configuration which minimizes the total entropy production. The evolution of the system leads to decreasing of the total entropy production monotonically in time, irrespective of the initial conditions. In some circumstances, the stationary state is not unique-the entropy production may have several local minima for different configurations. In the case of a rectangular box, the existence and stability of different stationary states are studied as a function of the aspect ratio of the rectangle.

Minimization of the Renyi entropy production in the stationary states of the Brownian process with matched death and birth rates

We analyze the Fleming-Viot process. The system is confined in a box, whose boundaries act as a sink of Brownian particles. The death rate at the boundaries is matched by the branching (birth) rate in the system and thus the number of particles is kept constant. We show that such a process is described by the Renyi entropy whose production is minimized in the stationary state. The entropy production in this process is a monotonically decreasing function of time irrespective of the initial conditions. The first Laplacian eigenvalue is shown to be equal to the Renyi entropy production in the stationary state. As an example we simulate the process in a two-dimensional box.