spectral formulation and wkb approximation for rare event statistics in reaction systems we develop a spectral formulation and a stationary wkb approximation for calculating the probabilities of rare events large deviations from the mean in systems of reacting particles with infinite range interaction describable by a master equation we compare the stationary wkb approximation to a recent time dependent semiclassical approximation developed for the same class of problems by elgart and kamenev phys rev e 70 41106 2004 as a benchmark we use an exactly solvable problem of the binary annihilation reaction 2a 0

a heterogeneous space time full approximation storage multilevel method for molecular dynamics simulations a heterogeneous space time full approximation storage hfas multilevel formulation for molecular dynamics simulations is developed the method consists of a waveform newton smoothing that produces initial space time iterates and a coarse model correction the formulation is coined as heterogeneous since it permits different interatomic potentials to be applied at different physical scales this results in a flexible framework for physics coupling time integration is performed in windows using the implicit newmark predictor corrector method that permits larger time integration steps than the explicit method the size of the time steps is governed by accuracy rather than by stability considerations of the algorithm we study three different variants of the method the picard iteration constrained dynamics and force splitting numerical examples show that fas based on force splitting provides significant time savings compared to standard explicit methods and alternative implicit space time schemes parallel studies of the picard iteration on harmonic problems illustrate the time parallelization effect that leads to a superior parallel performance compared to explicit methods

extinction rates for fluctuation induced metastabilities a real space wkb approach the extinction of a single species due to demographic stochasticity is analyzed the discrete nature of the individual agents and the poissonian noise related to the birth death processes result in local extinction of a metastable population as the system hits the absorbing state the fokker planck formulation of that problem fails to capture the statistics of large deviations from the metastable state while approximations appropriate close to the absorbing state become in general invalid as the population becomes large to connect these two regimes a real space wkb method based on the master equation is presented and is shown to yield an excellent approximation for the decay rate and the extreme events statistics all the way down to the absorbing state the details of the underlying microscopic process smeared out in a mean field treatment are shown to be crucial for an exact determination of the extinction exponent this general scheme is shown to reproduce the known results in the field to yield new corollaries and to fit quite precisely the numerical solutions moreover it allows for systematic improvement via a series expansion where the small parameter is the inverse of the number of individuals in the metastable state

maximization of the second non trivial neumann eigenvalue in this paper we prove that the second non trivial neumann eigenvalue of the laplace operator on smooth domains of r n with prescribed measure m attains its maximum on the union of two disjoint balls of measure m 2 as a consequence the polya conjecture for the neumann eigenvalues holds for the second eigenvalue and for arbitrary domains we moreover prove that a relaxed form of the same inequality holds in the context of non smooth domains and densities

epidemic models and percolation the authors argue that local epidemic models with immunisation are in the same universality class as percolation cluster growth models and show that the static exponents are equal to all orders in the epsilon expansion they calculate the dynamic exponent nu t 1 1 28 epsilon o epsilon 2 and relate this to other exponents involving the chemical distance

dimensional reduction of the master equation for stochastic chemical networks the reduced multiplane method chemical reaction networks which exhibit strong fluctuations are common in microscopic systems in which reactants appear in low copy numbers the analysis of these networks requires stochastic methods which come in two forms direct integration of the master equation and monte carlo simulations the master equation becomes infeasible for large networks because the number of equations increases exponentially with the number of reactive species monte carlo methods which are more efficient in integrating over the exponentially large phase space also become impractical due to the large amounts of noisy data that need to be stored and analyzed the recently introduced multiplane method a lipshtat and o biham phys rev lett 93 170601 2004 is an efficient framework for the stochastic analysis of large reaction networks it is a dimensional reduction method based on the master equation which provides a dramatic reduction in the number of equations without compromising the accuracy of the results the reduction is achieved by breaking the network into a set of maximal fully connected subnetworks maximal cliques a separate master equation is written for the reduced probability distribution associated with each clique with suitable coupling terms between them this method is highly efficient in the case of sparse networks in which the maximal cliques tend to be small however in dense networks some of the cliques may be rather large and the dimensional reduction is not as effective furthermore the derivation of the multiplane equations from the master equation is tedious and difficult here we present the reduced multiplane method in which the maximal cliques are broken down to the fundamental two vertex cliques the number of equations is further reduced making the method highly efficient even for dense networks moreover the equations take a simpler form which can be easily constructed using a diagrammatic procedure for any desired network architecture it is shown that the method provides accurate results for the population sizes of the reactive species and their reaction rates

disease extinction in the presence of random vaccination we investigate disease extinction in an epidemic model described by a birth death process we show that in the absence of vaccination the effective entropic barrier for extinction displays scaling with the distance to the bifurcation point with an unusual critical exponent even a comparatively weak poisson distributed random vaccination leads to an exponential increase in the extinction rate with the exponent that strongly depends on the vaccination parameters

extremal g invariant eigenvalues of the laplacian of g invariant metrics the study of extremal properties of the spectrum often involves restricting the metrics under consideration motivated by the work of abreu and freitas in the case of the sphere s 2 endowed with s 1 invariant metrics we consider the subsequence lambda k g of the spectrum of a riemannian manifold m which corresponds to metrics and functions invariant under the action of a compact lie group g if g has dimension at least 1 we show that the functional k g admits no extremal metric under volume preserving g invariant deformations if moreover m has dimension at least three then the functional lambda k g is unbounded when restricted to any conformal class of g invariant metrics of fixed volume as a special case of this we can consider the standard o n action on s n however if we also require the metric to be induced by an embedding of s n in mathbb r n 1 we get an optimal upper bound on lambda k g

rare event statistics in reaction diffusion systems we present an efficient method to calculate probabilities of large deviations from the typical behavior rare events in reaction diffusion systems this method is based on a semiclassical treatment of an underlying quantum hamiltonian encoding the system s evolution to this end we formulate the corresponding canonical dynamical system and investigate its phase portrait this method is presented for a number of pedagogical examples

population extinction risk in the aftermath of a catastrophic event we investigate how a catastrophic event modeled as a temporary fall of the reproduction rate increases the extinction probability of an isolated self regulated stochastic population using a variant of the verhulst logistic model as an example we combine the probability generating function technique with an eikonal approximation to evaluate the exponentially large increase in the extinction probability caused by the catastrophe this quantity is given by the eikonal action computed over the optimal path instanton of an effective classical hamiltonian system with a time dependent hamiltonian for a general catastrophe the eikonal equations can be solved numerically for simple models of catastrophic events analytic solutions can be obtained one such solution becomes quite simple close to the bifurcation point of the verhulst model the eikonal results for the increase in the extinction probability caused by a catastrophe agree well with numerical solutions of the master equation

extinction of metastable stochastic populations we investigate the phenomenon of extinction of a long lived self regulating stochastic population caused by intrinsic demographic noise extinction typically occurs via one of two scenarios depending on whether the absorbing state n 0 is a repelling scenario a or attracting scenario b point of the deterministic rate equation in scenario a the metastable stochastic population resides in the vicinity of an attracting fixed point next to the repelling point n 0 in scenario b there is an intermediate repelling point n n 1 between the attracting point n 0 and another attracting point n n 2 in the vicinity of which the metastable population resides the crux of the theory is a dissipative variant of wkb wentzel kramers brillouin approximation which assumes that the typical population size in the metastable state is large starting from the master equation we calculate the quasistationary probability distribution of the population sizes and the exponentially long mean time to extinction for each of the two scenarios when necessary the wkb approximation is complemented i by a recursive solution of the quasistationary master equation at small n and ii by the van kampen system size expansion valid near the fixed points of the deterministic rate equation the theory yields both entropic barriers to extinction and pre exponential factors and holds for a general set of multistep processes when detailed balance is broken the results simplify considerably for single step processes and near the characteristic bifurcations of scenarios a and b

persistence of instanton connections in chemical reactions with time dependent rates the evolution of a system of chemical reactions can be studied in the eikonal approximation by means of a hamiltonian dynamical system the fixed points of this dynamical system represent the different states in which the chemical system can be found and the connections among them represent instantons or optimal paths linking these states we study the relation between the phase portrait of the hamiltonian system representing a set of chemical reactions with constant rates and the corresponding system when these rates vary in time we show that the topology of the phase space is robust for small time dependent perturbations in concrete examples and state general results when possible this robustness allows us to apply some of the conclusions on the qualitative behavior of the autonomous system to the time dependent situation

extremal eigenvalues of the laplacian on euclidean domains and closed surfaces we investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the laplacian on euclidean domains of unit volume under dirichlet and neumann boundary conditions respectively in a second part we study sequences of extremal eigenvalues of the laplace beltrami operator on closed surfaces of unit area

space time domain decomposition for parabolic problems we analyze a space time domain decomposition iteration for a model advection diffusion equation in one and two dimensions the discretization of this iteration is the block red black variant of the waveform relaxation method and our analysis provides new convergence results for this scheme the asymptotic convergence rate is super linear and it is governed by the diffusion of the error across the overlap between subdomains hence it depends on both the size of this overlap and the diffusion coefficient in the equation however it is independent of the number of subdomains provided the size of the overlap remains fixed the convergence rate for the heat equation in a large time window is initially linear and it deteriorates as the number of subdomains increases the duration of the transient linear regime is proportional to the length of the time window for advection dominated problems the convergence rate is initially linear and it improves as the the ratio of advection to diffusion increases moreover it is independent of the size of the time window and of the number of subdomains numerical calculations illustrate our analysis

a space time multilevel method for molecular dynamics simulations we present a novel multiscale approach for molecular dynamics simulations it is aimed at bridging discrete scales with either coarse grained discrete or continuum scales the method consists of the waveform relaxation scheme aimed at capturing the high frequency response of atomistic vibrations and a coarse scale solution in space and time aimed at resolving smooth features in both space and time domains of the discrete medium the method is implicit in space and time possesses superior stability properties and consequently enables larger time steps governed by accuracy considerations of coarse scale quantities of interest performance studies on polymer melts have shown significant speed up compared to the classical explicit methods in particular on parallel machines

bounds for eigenfunctions of the laplacian on noncompact riemannian manifolds we deal with eigenvalue problems for the laplacian on noncompact riemannian manifolds m of finite volume sharp conditions ensuring l q m and l infinity m bounds for eigenfunctions are exhibited

wilson renormalization of a reaction diffusion process abstract healthy and sick individuals a and b particles diffuse independently with diffusion constants da and db sick individuals upon encounter infect healthy ones at rate k but may also spontaneously recover at rate 1 the propagation of the epidemic therefore couples to the fluctuations in the total population density global extinction occurs below a critical value c of the spatially averaged total density the epidemic evolves as the diffusion reaction decay process a b 2b b a for which we write down the field theory the stationary state properties of this theory when da db were obtained by kree et al the critical behavior for da db remains unsolved

population extinction in a time modulated environment the extinction time of an isolated population can be exponentially reduced by a periodic modulation of its environment we investigate this effect using as an example a stochastic branching annihilation process with a time dependent branching rate the population extinction is treated in eikonal approximation where it is described as an instanton trajectory of a proper reaction hamiltonian the modulation of the environment perturbs this trajectory and synchronizes it with the modulation phase we calculate the corresponding change in the action along the instanton using perturbation techniques supported by numerical calculations the techniques include a first order theory with respect to the modulation amplitude a second order theory in the spirit of the kapitsa pendulum effect and adiabatic theory valid for low modulation frequencies

fermionic field theory for directed percolation in 1 1 dimensions we formulate directed percolation in 1 1 dimensions in the language of a reaction diffusion process with exclusion taking place in one space dimension we map the master equation that describes the dynamics of the system onto a quantum spin chain problem from there we build an interacting fermionic field theory of a new type we study the resulting theory using renormalization group techniques this yields numerical estimates for the critical exponents and provides a new alternative analytic systematic procedure to study low dimensional directed percolation

fluctuation statistics in networks a stochastic path integral approach we investigate the statistics of fluctuations in a classical stochastic network of nodes joined by connectors the nodes carry generalized charge that may be randomly transferred from one node to another our goal is to find the time evolution of the probability distribution of charges in the network the building blocks of our theoretical approach are 1 known probability distributions for the connector currents 2 physical constraints such as local charge conservation and 3 a time scale separation between the slow charge dynamics of the nodes and the fast current fluctuations of the connectors we integrate out fast current fluctuations and derive a stochastic path integral representation of the evolution operator for the slow charges the statistics of charge fluctuations may be found from the saddle point approximation of the action once the probability distributions on the discrete network have been studied the continuum limit is taken to obtain a statistical field theory we find a correspondence between the diffusive field theory and a langevin equation with gaussian noise sources leading nevertheless to nontrivial fluctuation statistics to complete our theory we demonstrate that the cascade diagrammatics recently introduced by nagaev naturally follows from the stochastic path integral by generalizing the principle of minimal correlations we extend the diagrammatics to calculate current correlation functions for an arbitrary network one primary application of this formalism is that of full counting statistics fcs the motivation for why it was developed in the first place we stress however that the formalism is suitable for general classical stochastic problems as an alternative approach to the traditional master equation or doi peliti technique the formalism is illustrated with several examples both instantaneous and time averaged charge fluctuation statistics in a mesoscopic chaotic cavity as well as the fcs and new results for a generalized diffusive wire

spectral theory of metastability and extinction in a branching annihilation reaction we apply the spectral method recently developed by the authors to calculate the statistics of a reaction limited multistep birth death process or chemical reaction that includes as elementary steps branching a 2a and annihilation 2a 0 the spectral method employs the generating function technique in conjunction with the sturm liouville theory of linear differential operators we focus on the limit when the branching rate is much higher than the annihilation rate and obtain accurate analytical results for the complete probability distribution including large deviations of the metastable long lived state and for the extinction time statistics the analytical results are in very good agreement with numerical calculations furthermore we use this example to settle the issue of the lacking boundary condition in the spectral formulation

long time step methods for oscillatory differential equations considered are numerical integration schemes for nondissipative dynamical systems in which multiple time scales are present it is assumed that one can do an explicit separation of the rhs forces into fast forces and slow forces such that i the fast forces contain the high frequency part of the solution ii the fast forces are conservative and iii the reduced problem consisting only of the fast forces can be integrated much more cheaply than the full problem the fast forces are allowed to have low frequency components particular applications for which the schemes are intended include n body problems for which most of the forces are slow and nonlinear wave phenomena for which the fast forces can be propagated by spectral methods the assumption of cheap integration of fast forces implies that the overall cost of integration is primarily determined by the step size used to sample the slow forces a long time step method is one in which this step size exceeds half the period of the fastest normal mode present in the full system an existing method that comes close to qualifying is the impulse method also known as verlet i and r respa it is shown that it might fail though for a couple of reasons first it suffers a serious loss of accuracy if the step size is near a multiple of the period of a normal mode and second it is unstable if the step size is near a multiple of half the period of a normal mode proposed in this paper is a mollified impulse method having an error bound that is independent of the frequency of the fast forces it is also shown to possess superior stability properties theoretical results are supplemented by numerical experiments the method is efficient and reasonably easy to implement

maximization of the second positive neumann eigenvalue for planar domains we prove that the second positive neumann eigenvalue of a bounded simply connected planar domain of a given area does not exceed the first positive neumann eigenvalue on a disk of a twice smaller area this estimate is sharp and attained by a sequence of domains degenerating to a union of two identical disks in particular this result implies the polya conjecture for the second neumann eigenvalue the proof is based on a combination of analytic and topological arguments as a by product of our method we obtain an upper bound on the second eigenvalue for conformally round metrics on odd dimensional spheres

rattle a velocity version of the shake algorithm for molecular dynamics calculations an algorithm called rattle for integrating the equations of motion in molecular dynamics calculations for molecular models with internal constraints is presented the algorithm is similar to shake which is one of the standard methods for performing such calculations rattle calculates the positions and velocities at the next time from the positions and velocities at the present time step without requiring information about the earlier history like shake it is based on the verlet algorithm and retains the simplicity of using cartesian coordinates for each of the atoms to describe the configuration of a molecule with internal constraints rattle guarantees that the coordinates and velocities of the atoms in a molecule satisfy the internal constraints at each time step rattle has two advantages over shake on computers of fixed precision it is of higher precision than shake since it deals directly with the velocities it is easier to modify rattle for use with the recently developed constant temperature and constant pressure molecular dynamics methods and with the nonequilibrium molecular dynamics methods that make use of rescaling of the atomic velocities

reducing the extinction risk of stochastic populations via nondemographic noise we consider nondemographic noise in the form of uncertainty in the reaction step size and reveal a dramatic effect this noise may have on the stability of self regulating populations employing the reaction scheme ma ka but allowing e g the product number k to be a priori unknown and sampled from a given distribution we show that such nondemographic noise can greatly reduce the population s extinction risk compared to the fixed k case our analysis is tested against numerical simulations and by using empirical data of different species we argue that certain distributions may be more evolutionary beneficial than others