arXiv subjects

Baruch Meerson

Publications and source records attributed to Baruch Meerson.

At least 91 records · Page 5Linked to original sources

arXiv2015

Mortality, Redundancy, and Diversity in Stochastic Search

We investigate a stochastic search process in one dimension under the competing roles of mortality, redundancy, and diversity of the searchers. This picture represents a toy model for the fertilization of an oocyte by sperm. A population of $N$ independent and mortal diffusing searchers all start at $x=L$ and attempt to reach the target at $x=0$. When mortality is irrelevant, the search time scales as $τ_D/\ln N$ for $\ln N\gg 1$, where $τ_D\sim L^2/D$ is the diffusive time scale. Conversely, when the mortality rate $μ$ of the searchers is sufficiently large, the search time scales as $\sqrt{τ_D/μ}$, independent of $N$. When searchers have distinct and high mortalities, a subpopulation with a non-trivial optimal diffusivity are most likely to reach the target. We also discuss the effect of chemotaxis on the search time and its fluctuations.

cond-mat.stat-mech↗

arXiv2015

Full absorption statistics of diffusing particles with exclusion

Suppose that an infinite lattice gas of constant density $n_0$, whose dynamics are described by the symmetric simple exclusion process, is brought in contact with a spherical absorber of radius $R$. Employing the macroscopic fluctuation theory and assuming the additivity principle, we evaluate the probability distribution ${\mathcal P}(N)$ that $N$ particles are absorbed during a long time $T$. The limit of $N=0$ corresponds to the survival problem, whereas $N\gg \bar{N}$ describes the opposite extreme. Here $\bar{N}=4πR D_0 n_0 T$ is the \emph{average} number of absorbed particles (in three dimensions), and $D_0$ is the gas diffusivity. For $n_0\ll 1$ the exclusion effects are negligible, and ${\mathcal P}(N)$ can be approximated, for not too large $N$, by the Poisson distribution with mean $\bar{N}$. For finite $n_0$, ${\mathcal P}(N)$ is non-Poissonian. We show that $-\ln{\mathcal P}(N) \simeq n_0 N^2/\bar{N}$ at $N\gg \bar{N}$. At sufficiently large $N$ and $n_0<1/2$ the most likely density profile of the gas, conditional on the absorption of $N$ particles, is non-monotonic in space. We also establish a close connection between this problem and that of statistics of current in finite open systems.

cond-mat.stat-mech↗

arXiv2015

The number statistics and optimal history of non-equilibrium steady states of mortal diffusing particles

Suppose that a point-like steady source at $x=0$ injects particles into a half-infinite line. The particles diffuse and die. At long times a non-equilibrium steady state sets in, and we assume that it involves many particles. If the particles are non-interacting, their total number $N$ in the steady state is Poisson-distributed with mean $\bar{N}$ predicted from a deterministic reaction-diffusion equation. Here we determine the most likely density history of this driven system conditional on observing a given $N$. We also consider two prototypical examples of \emph{interacting} diffusing particles: (i) a family of mortal diffusive lattice gases with constant diffusivity (as illustrated by the simple symmetric exclusion process with mortal particles), and (ii) random walkers that can annihilate in pairs. In both examples we calculate the variances of the (non-Poissonian) stationary distributions of $N$.

cond-mat.stat-mech↗

arXiv2014

Two-strain competition in quasi-neutral stochastic disease dynamics

We develop a new perturbation method for studying quasi-neutral competition in a broad class of stochastic competition models, and apply it to the analysis of fixation of competing strains in two epidemic models. The first model is a two-strain generalization of the stochastic Susceptible-Infected-Susceptible (SIS) model. Here we extend previous results due to Parsons and Quince (2007), Parsons et al (2008) and Lin, Kim and Doering (2012). The second model, a two-strain generalization of the stochastic Susceptible-Infected-Recovered (SIR) model with population turnover, has not been studied previously. In each of the two models, when the basic reproduction numbers of the two strains are identical, a system with an infinite population size approaches a point on the deterministic coexistence line (CL): a straight line of fixed points in the phase space of sub-population sizes. Shot noise drives one of the strain populations to fixation, and the other to extinction, on a time scale proportional to the total population size. Our perturbation method explicitly tracks the dynamics of the probability distribution of the sub-populations in the vicinity of the CL. We argue that, whereas the slow strain has a competitive advantage for mathematically "typical" initial conditions, it is the fast strain that is more likely to win in the important situation when a few infectives of both strains are introduced into a susceptible population.

cond-mat.stat-mech↗

arXiv2014

Survival of a static target in a gas of diffusing particles with exclusion

Let a lattice gas of constant density, described by the symmetric simple exclusion process, be brought in contact with a "target": a spherical absorber of radius $R$. Employing the macroscopic fluctuation theory (MFT), we evaluate the probability ${\mathcal P}(T)$ that no gas particle hits the target until a long but finite time $T$. We also find the most likely gas density history conditional on the non-hitting. The results depend on the dimension of space $d$ and on the rescaled parameter $\ell=R/\sqrt{D_0T}$, where $D_0$ is the gas diffusivity. For small $\ell$ and $d>2$, ${\mathcal P}(T)$ is determined by an exact stationary solution of the MFT equations that we find. For large $\ell$, and for any $\ell$ in one dimension, the relevant MFT solutions are non-stationary. In this case $\ln {\mathcal P}(T)$ scales differently with relevant parameters, and it also depends on whether the initial condition is random or deterministic. The latter effects also occur if the lattice gas is composed of non-interacting random walkers. Finally, we extend the formalism to a whole class of diffusive gases of interacting particles.

cond-mat.stat-mech↗

arXiv2014

Large fluctuations in diffusion-controlled absorption

Suppose that $N_0$ independently diffusing particles, each with diffusivity $D$, are initially released at $x=\ell>0$ on the semi-infinite interval $0\leq x<\infty$ with an absorber at $x=0$. We determine the probability ${\cal P}(N)$ that $N$ particles survive until time $t=T$. We also employ macroscopic fluctuation theory to find the most likely history of the system, conditional on there being exactly $N$ survivors at time $t=T$. Depending on the basic parameter $\ell/\sqrt{4DT}$, very different histories can contribute to the extreme cases of $N=N_0$ (all particles survive) and $N=0$ (no survivors). For large values of $\ell/\sqrt{4DT}$, the leading contribution to ${\cal P}(N=0)$ comes from an effective point-like quasiparticle that contains all the $N_0$ particles and moves ballistically toward the absorber until absorption occurs.

cond-mat.stat-mech↗

arXiv2014

Extreme Fluctuations of Current in the Symmetric Simple Exclusion Process: a Non-Stationary Setting

We use the macroscopic fluctuation theory (MFT) to evaluate the probability distribution P of extreme values of integrated current J at a specified time t=T in the symmetric simple exclusion process (SSEP) on an infinite line. As shown recently [Phys. Rev. E 89, 010101(R) (2014)], the SSEP belongs to the elliptic universality class. Here, for very large currents, the diffusion terms of the MFT equations can be neglected compared with the terms coming from the shot noise. Using the hodograph transformation and an additional change of variables, we reduce the "inviscid" MFT equations to Laplace's equation in an extended space. This opens the way to an exact solution. Here we solve the extreme-current problem for a flat deterministic initial density profile with an arbitrary density 0<n<1. The solution yields the most probable density history of the system conditional on the extreme current, and leads to a super-Gaussian extreme-current statistics, - ln P = F(n) J^3/T, in agreement with Derrida and Gerschenfeld [J. Stat. Phys. 137, 978 (2009)]. We calculate the function F(n) analytically. It is symmetric with respect to the half-filling density n=1/2, diverges as n approached 0 or 1, and exhibits a singularity F(n) |n-1/2| at the half-filling density n=1/2.

cond-mat.stat-mech↗

arXiv2013

Extreme Current Fluctuations in a Nonstationary Stochastic Heat Flow

We employ the Hamiltonian formalism of macroscopic fluctuation theory to study large deviations of integrated current in the Kipnis-Marchioro-Presutti (KMP) model of stochastic hear flow when starting from a step-like initial condition. The KMP model belongs to the hyperbolic universality class where diffusion remains relevant no matter how large the fluctuating current is. The extreme current statistics for the KMP model turns out to be sub-Gaussian, as distinguished from the super-Gaussian statistics found for the Symmetric Simple Exclusion Process and other models of the elliptic class. The most probable time history of the system, which dominates the extreme current statistics of the KMP model, involves two large-amplitude solitary pulses: of the energy density field and of the conjugate "momentum" field. The coupled pulses propagate with a constant speed, but their amplitudes slowly grow with time, as the energy density pulse collects most of the available energy on its way.

cond-mat.stat-mech↗

arXiv2013

Extreme Current Fluctuations in Lattice Gases: Beyond Nonequilibrium Steady States

We use the macroscopic fluctuation theory (MFT) to study large current fluctuations in non-stationary diffusive lattice gases. We identify two universality classes of these fluctuations which we call elliptic and hyperbolic. They emerge in the limit when the deterministic mass flux is small compared to the mass flux due to the shot noise. The two classes are determined by the sign of compressibility of \emph{effective fluid}, obtained by mapping the MFT into an inviscid hydrodynamics. An example of the elliptic class is the Symmetric Simple Exclusion Process where, for some initial conditions, we can solve the effective hydrodynamics exactly. This leads to a super-Gaussian extreme current statistics conjectured by Derrida and Gerschenfeld (2009) and yields the optimal path of the system. For models of the hyperbolic class the deterministic mass flux cannot be neglected, leading to a different extreme current statistics.

cond-mat.stat-mech↗

arXiv2013

Immigration-extinction dynamics of stochastic populations

How high should be the rate of immigration into a stochastic population in order to significantly reduce the probability of observing the population extinct? Is there any relation between the population size distributions with and without immigration? Under what conditions can one justify the simple patch occupancy models which ignore the population distribution and its dynamics in a patch, and treat a patch simply as either occupied or empty? We address these questions by exactly solving a simple stochastic model obtained by adding a steady immigration to a variant of the Verhulst model: a prototypical model of an isolated stochastic population.

q-bio.PE↗

arXiv2013

Impact of Colored Environmental Noise on the Extinction of a Long-Lived Stochastic Population: Role of the Allee Effect

We study the combined impact of a colored environmental noise and demographic noise on the extinction risk of a long-lived and well-mixed isolated stochastic population which exhibits the Allee effect. The environmental noise modulates the population birth and death rates. Assuming that the Allee effect is strong, and the environmental noise is positively correlated and Gaussian, we derive a Fokker-Planck equation for the joint probability distribution of the population sizes and environmental fluctuations. In WKB approximation this equation reduces to an effective two-dimensional Hamiltonian mechanics, where the most likely path to extinction and the most likely environmental fluctuation are encoded in an instanton-like trajectory in the phase space. The mean time to extinction (MTE) is related to the mechanical action along this trajectory. We obtain new analytic results for short-correlated, long-correlated and relatively weak environmental noise. The population-size dependence of the MTE changes from exponential for weak environmental noise to no dependence for strong noise, implying a greatly increased extinction risk. The theory is readily extendable to population switches between different metastable states, and to stochastic population explosion, due to a combined action of demographic and environmental noise.

q-bio.PE↗

arXiv2012

Survival of the Scarcer

We investigate extinction dynamics in the paradigmatic model of two competing species A and B that reproduce (A-->2A, B-->2B), self-regulate by annihilation (2A-->0, 2B-->0), and compete (A+B-->A, A+B-->B). For a finite system that is in the well-mixed limit, a quasi-stationary state arises which describes coexistence of the two species. Because of discrete noise, both species eventually become extinct in time that is exponentially long in the quasi-stationary population size. For a sizable range of asymmetries in the growth and competition rates, the paradoxical situation arises in which the numerically disadvantaged species according to the deterministic rate equations survives much longer.

cond-mat.stat-mech↗

arXiv2012

Emergence of fluctuating traveling front solutions in macroscopic theory of noisy invasion fronts

The position of an invasion front, propagating into an unstable state, fluctuates because of the shot noise coming from the discreteness of reacting particles and stochastic character of the reactions and diffusion. A recent macroscopic theory [Meerson and Sasorov, Phys. Rev. E 84, 030101(R) (2011)] yields the probability of observing, during a long time, an unusually slow front. The theory is formulated as an effective classical Hamiltonian field theory which operates with the density field and the conjugate "momentum" field. Further, the theory assumes that the most probable density field history of an unusually slow front represents, up to small corrections, a traveling front solution of the Hamilton equations. Here we verify this assumption by solving the Hamilton equations numerically for models belonging to the directed percolation universality class.

cond-mat.stat-mech↗

arXiv2012

Void formation in diffusive lattice gases

What is the probability that a macroscopic void will spontaneously arise, at a specified time T, in an initially homogeneous gas? We address this question for diffusive lattice gases, and also determine the most probable density history leading to the void formation. We employ the macroscopic fluctuation theory by Bertini et al and consider both annealed and quenched averaging procedures (the initial condition is allowed to fluctuate in the annealed setting). We show that in the annealed case the void formation probability is given by the equilibrium Boltzmann-Gibbs formula, so the probability is independent of T (and also of the void shape, as only the volume matters). In the quenched case, which is intrinsically non-equilibrium, we evaluate the void formation probability analytically for non-interacting random walkers and probe it numerically for the simple symmetric exclusion process. For voids that are small compared with the diffusion length, the equilibrium result for the void formation probability is recovered. We also re-derive our main results for non-interacting random walkers from an exact microscopic analysis.

cond-mat.stat-mech↗

arXiv2012

Velocity fluctuations of noisy reaction fronts propagating into a metastable state: testing theory in stochastic simulations

The position of a reaction front, propagating into a metastable state, fluctuates because of the shot noise of reactions and diffusion. A recent theory [B. Meerson, P.V. Sasorov, and Y. Kaplan, Phys. Rev. E 84, 011147 (2011)] gave a closed analytic expression for the front diffusion coefficient in the weak noise limit. Here we test this theory in stochastic simulations involving reacting and diffusing particles on a one-dimensional lattice. We also investigate a small noise-induced systematic shift of the front velocity compared to the prediction from the spatially continuous deterministic reaction-diffusion equation.

cond-mat.stat-mech↗

arXiv2012

Fluctuations of Current in Non-Stationary Diffusive Lattice Gases

We employ the macroscopic fluctuation theory to study fluctuations of integrated current in one-dimensional lattice gases with a step-like initial density profile. We analytically determine the variance of the current fluctuations for a class of diffusive processes with a density-independent diffusion coefficient, but otherwise arbitrary. Our calculations rely on a perturbation theory around the noiseless hydrodynamic solution. We consider both quenched and annealed types of averaging (the initial condition is allowed to fluctuate in the latter situation). The general results for the variance are specialized to a few interesting models including the symmetric exclusion process and the Kipnis-Marchioro-Presutti model. We also probe large deviations of the current for the symmetric exclusion process. This is done by numerically solving the governing equations of the macroscopic fluctuation theory using an efficient iteration algorithm.

cond-mat.stat-mech↗

arXiv2012

Minimizing the population extinction risk by migration

Many populations in nature are fragmented: they consist of local populations occupying separate patches. A local population is prone to extinction due to the shot noise of birth and death processes. A migrating population from another patch can dramatically delay the extinction. What is the optimal migration rate that minimizes the extinction risk of the whole population? Here we answer this question for a connected network of model habitat patches with different carrying capacities.

q-bio.PE↗

arXiv2012

Multiple extinction routes in stochastic population models

Isolated populations ultimately go extinct because of the intrinsic noise of elementary processes. In multi-population systems extinction of a population may occur via more than one route. We investigate this generic situation in a simple predator-prey (or infected-susceptible) model. The predator and prey populations may coexist for a long time but ultimately both go extinct. In the first extinction route the predators go extinct first, whereas the prey thrive for a long time and then also go extinct. In the second route the prey go extinct first causing a rapid extinction of the predators. Assuming large sub-population sizes in the coexistence state, we compare the probabilities of each of the two extinction routes and predict the most likely path of the sub-populations to extinction. We also suggest an effective three-state master equation for the probabilities to observe the coexistence state, the predator-free state and the empty state.

q-bio.PE↗