time dependent fiber bundles with local load sharing fiber bundle models where fibers have random lifetimes depending on their load histories are useful tools in explaining time dependent failure in heterogeneous materials such models shed light on diverse phenomena such as fatigue in structural materials and earthquakes in geophysical settings various asymptotic and approximate theories have been developed for bundles with various geometries and fiber load sharing mechanisms but numerical verification has been hampered by severe computational demands in larger bundles to gain insight at large size scales interest has returned to idealized fiber bundle models in 1d such simplified models typically assume either equal load sharing els among survivors or local load sharing lls where a failed fiber redistributes its load onto its two nearest flanking survivors such models can often be solved exactly or asymptotically in increasing bundle size n yet still capture the essence of failure in real materials the present work focuses on 1d bundles under lls as in previous works a fiber has failure rate following a power law in its load level with breakdown exponent ensuremath rho surviving fibers under fixed loads have remaining lifetimes that are independent and exponentially distributed we develop both new asymptotic theories and new computational algorithms that greatly increase the bundle sizes that can be treated in large replications e g one million fibers in thousands of realizations in particular we develop an algorithm that adapts several concepts and methods that are well known among computer scientists but relatively unknown among physicists to dramatically increase the computational speed with no attendant loss of accuracy we consider various regimes of ensuremath rho that yield drastically different behavior as n increases for 1 2l ensuremath rho l 1 els and lls have remarkably similar behavior they have identical lifetime distributions at ensuremath rho 1 with approximate gaussian bundle lifetime statistics and a finite limiting mean for ensuremath rho g1 this gaussian behavior also applies to els whereas lls behavior diverges sharply showing brittle weakest volume behavior in terms of characteristic elements derived from critical cluster formation for 0l ensuremath rho l1 2 els and lls again behave similarly but the bundle lifetimes are dominated by a few long lived fibers and show characteristics of strongest link extreme value distributions which apply exactly to ensuremath rho 0

numerical method for solving stochastic differential equations with poissonian white shot noise we propose a numerical integration scheme to solve stochastic differential equations driven by poissonian white shot noise our formula which is based on an integral equation which is equivalent to the stochastic differential equation utilizes a discrete time approximation with fixed integration time step we show that our integration formula approaches the euler formula if the poissonian noise approaches the gaussian white noise the accuracy and efficiency of the proposed algorithm are examined by studying the dynamics of an overdamped particle driven by poissonian white shot noise in a spatially periodic potential we find that the accuracy of the proposed algorithm only weakly depends on the parameters characterizing the poissonian white shot noise this holds true even if the limit of gaussian white noise is approached

numerical solution of differential equations with colored noise using the general theory of numerical integration of stochastic differential equations a constructive approach to numerical methods for a system with colored noise is proposed efficient methods up to the 5 2 strong order and up to the third weak order including runge kutta and implicit schemes are presented the algorithms are tested on the kubo oscillator

numerical simulations of stochastic differential equations a simple very accurate algorithm for numerical simulation of stochastic differential equations is described its relationship to colored noise is elucidated and exhibited by explicit results the especially delicate problem of mean first passage times is highlighted and highly accurate agreement between the numerical simulations and analytic results are shown

twenty five years after kls a celebration of non equilibrium statistical mechanics when lenz proposed a simple model for phase transitions in magnetism he couldn t have imagined that the ising model was to become a jewel in field of equilibrium statistical mechanics its role spans the spectrum from a good pedagogical example to a universality class in critical phenomena a quarter century ago katz lebowitz and spohn found a similar treasure by introducing a seemingly trivial modification to the ising lattice gas they took it into the vast realms of non equilibrium statistical mechanics an abundant variety of unexpected behavior emerged and caught many of us by surprise we present a brief review of some of the new insights garnered and some of the outstanding puzzles as well as speculate on the model s role in the future of non equilibrium statistical physics

escape rates in periodically driven markov processes we present an approximate analytical expression for the escape rate of time dependent driven stochastic processes with an absorbing boundary such as the driven leaky integrateand fire model for neural spiking the novel approximation i s based on a discrete state markovian modeling of the full long time dynamics with time dependent rates it is valid in a wide parameter regime beyond the restraining limits of weak driving linear response and or weak noise the scheme is carefully tested and yields excellent agreement with three different numerical methods based on the langevin equation the fokker planck equation and an integral equation

persistent diffusion on a line we consider solutions to the telegraph equation describing persistent diffusion on a line under various initial conditions the first passage time distribution is evaluated in closed form biased persistent diffusion is also considered a direct derivation of the telegraph equation from the stochastic equation for the displacement is presented in an appendix

stochastic giant resonance the model of an electric circuit with dichotomous resistance is investigated it is shown that the dichotomous resistance can induce a phenomenon of stochastic giant resonance for the signal to noise ratio snr as a function of input signal frequency this phenomenon is a direct consequence of the existence of a zero noise power spectrum with a nonzero signal power spectrum for the same parameters in addition two kinds of the usual stochastic resonance phenomena have been obtained for the snr one is as a function of the correlation time of the asymmetric dichotomous resistance the other is as a function of the input signal frequency

dynamic critical behavior of failure and plastic deformation in the random fiber bundle model the random fiber bundle rfb model with the strength of the fibers distributed uniformly within a finite interval is studied under the assumption of global load sharing among all unbroken fibers of the bundle at any fixed value of the applied stress sigma load per fiber initially present in the bundle the fraction u t sigma of fibers that remain unbroken at successive time steps t is shown to follow simple recurrence relations the model is found to have stable fixed point u filled sigma for applied stress in the range 0 infinity the avalanche size distribution for this mean field dynamics of failure at sigma sigma c has been studied the elastic response of the rfb model has also been studied analytically for a specific probability distribution of fiber strengths where the bundle shows plastic behavior before complete failure following an initial linear response

on the relation between white shot noise gaussian white noise and the dichotomic markov process it is shown that the dichotomic markov process converges to a white shot noise interpreted according to the stratonovich integration rule in the joint limit in which the average duration of one of the states goes to zero and the value at this state goes to infinity a further limit procedure allows us to obtain gaussian white noise from white shot noise these results are applied to the problem of noise induced transitions it is shown that white shot noise can give rise to transitions which do not occur for gaussian white noise the above results are finally generalized in introducing compound dichotomic markov processes

correlation between eigenvalue spectra and dynamics of neural networks this letter presents a study of the correlation between the eigenvalue spectra of synaptic matrices and the dynamical properties of asymmetric neural networks with associative memories for this type of neural network it was found that there are essentially two different dynamical phases the chaos phase with almost all trajectories converging to a single chaotic attractor and the memory phase with almost all trajectories being attracted toward fixed point attractors acting as memories we found that if a neural network is designed in the chaos phase the eigenvalue spectrum of its synaptic matrix behaves like that of a random matrix i e all eigenvalues lie uniformly distributed within a circle in the complex plan and if it is designed in the memory phase the eigenvalue spectrum will split into two parts one part corresponds to a random background the other part equal in number to the memory attractors the mechanism for these phenomena is discussed in this letter

elasticity and failure of a set of elements loaded in parallel the author studies the elasticity and failure characteristics of a system built by association in parallel of links with identical spring constant but random failure thresholds the occurrence of the first link rupture which is related to the theory of extreme order statistics is contrasted with the global failure for which a central limit theorem holds he also discusses the similarities and differences between the democratic and hierarchical models which differ in the mechanism by which the stress supported by links which have failed is transferred to the other stronger links this problem gives the simplest model of elastic non linear behaviour before global rupture occurs

crossover behavior in a mixed mode fiber bundle model we introduce a mixed mode load sharing scheme in a fiber bundle model this model reduces exactly to equal load sharing els and local load sharing lls models at the two extreme limits of a single load sharing parameter we identify two distinct regimes a the mean field regime where the els mode dominates and b the short range regime dominated by the lls mode the crossover behavior is explored through a numerical study of the strength variation the avalanche statistics susceptibility and relaxation time variations the correlations among the broken fibers and their cluster analysis analyzing the moments of the cluster size distributions we locate the crossover point of these regimes we thus conclude that even in one dimension the fiber bundle model shows crossover behavior from mean field to short range interactions

a phase transition in the first passage of a brownian process through a fluctuating boundary with implications for neural coding finding the first time a fluctuating quantity reaches a given boundary is a deceptively simple looking problem of vast practical importance in physics biology chemistry neuroscience economics and industrial engineering problems in which the bound to be traversed is itself a fluctuating function of time include widely studied problems in neural coding such as neuronal integrators with irregular inputs and internal noise we show that the probability p t that a gauss markov process will first exceed the boundary at time t suffers a phase transition as a function of the roughness of the boundary as measured by its holder exponent h the critical value occurs when the roughness of the boundary equals the roughness of the process so for diffusive processes the critical value is hc 1 2 for smoother boundaries h 1 2 the probability density is a continuous function of time for rougher boundaries h 1 2 the probability is concentrated on a cantor like set of zero measure the probability density becomes divergent almost everywhere either zero or infinity the critical point hc 1 2 corresponds to a widely studied case in the theory of neural coding in which the external input integrated by a model neuron is a white noise process as in the case of uncorrelated but precisely balanced excitatory and inhibitory inputs we argue that this transition corresponds to a sharp boundary between rate codes in which the neural firing probability varies smoothly and temporal codes in which the neuron fires at sharply defined times regardless of the intensity of internal noise

response to a periodic stimulus in a perfect integrate and fire neuron model driven by colored noise the output interspike interval statistics of a stochastic perfect integrate and fire neuron model driven by an additive exogenous periodic stimulus is considered the effect of temporally correlated random activity of synaptic inputs is modeled by an additive symmetric dichotomous noise using a first passage time formulation exact expressions for the output interspike interval density and for the serial correlation coefficient are derived in the nonsteady regime and their dependence on input parameters e g the noise correlation time and amplitude as well as the frequency of an input current is analyzed it is shown that an interplay of a periodic forcing and colored noise can cause a variety of nonequilibrium cooperation effects such as sign reversals of the interspike interval correlations versus noise switching rate as well as versus the frequency of periodic forcing a power law like decay of oscillations of the serial correlation coefficients in the long lag limit amplification of the output signal modulation in the instantaneous firing rate of the neural response etc the features of spike statistics in the limits of slow and fast noises are also discussed

computing a non maxwellian velocity distribution from first principles we investigate a family of single particle anomalous velocity distribution by solving a particular class of stochastic liouville equations the stationary state is obtained analytically and the maxwell boltzmann distribution is reobtained in a particular limit we discuss the comparison with other different methods to obtain the stationary state extensions when the models cannot be solved in an exact way are also pointed out in connection with the one ficton approximation

stochastic calculus in physics the relationship of the ito stratonovich stochastic calculus to studies of weakly colored noise is explained a functional calculus approach is used to obtain an effective fokker planck equation for the weakly colored noise regime in a smooth limit this representation produces the stratonovich version of the ito stratonovich calculus for white noise it also provides an approach to steady state behavior for strongly colored noise numerical simulation algorithms are explored and a novel suggestion is made for efficient and accurate simulation of white noise equations

numerical integration of multiplicative noise stochastic differential equations a two stage runge kutta algorithm for vector ito and by transform also stratonovich stochastic differential equations with multiplicative noise has been developed the method is second order accurate but for vanishing drift the algorithm yields a martingale independent of step size several examples are included to illustrate our method a discussion of errors shows that sample size can be as important as truncation

burst avalanches in solvable models of fibrous materials we review limiting models for fracture in bundles of fibers with statistically distributed thresholds for breakdown of individual fibers during the breakdown process avalanches consisting of simultaneous rupture of several fibers occur and the distribution d delta of the magnitude delta of such avalanches is the central characteristics in our analysis for a bundle of parallel fibers two limiting models of load sharing are studied and contrasted the global model in which the load carried by a bursting fiber is equally distributed among the surviving members and the local model in which the nearest surviving neighbors take up the load for the global model we investigate in particular the conditions on the threshold distribution which would lead to anomalous behavior i e deviations from the asymptotics d delta sim delta 5 2 known to be the generic behavior for the local model no universal power law asymptotics exists but we show for a particular threshold distribution how the avalanche distribution can nevertheless be explicitly calculated in the large bundle limit

phase transition in fiber bundle models with recursive dynamics we study the phase transition in a class of fiber bundle models in which the fiber strengths are distributed randomly within a finite interval and global load sharing is assumed the dynamics is expressed as recursion relations for the redistribution of the applied stress and the evolution of the surviving fraction of fibers we show that an irreversible phase transition of second order occurs from a phase of partial failure to a phase of total failure when the initial applied stress just exceeds a critical value the phase transition is characterized by static and dynamic critical properties we calculate exactly the critical value of the initial stress for three models of this kind each with a different distribution of fiber strengths we derive exact expressions for the order parameter the susceptibility to changes in the initial applied stress and the critical relaxation of the surviving fraction of fibers for all the three models the static and dynamic critical exponents obtained from these expressions are found to be universal

self similarity of waiting times in fracture systems experimental and numerical results are presented for a fracture experiment carried out on a fiber reinforced element under flexural loading and a statistical analysis is performed for acoustic emission waiting time distributions by an optimization procedure a recently proposed scaling law describing these distributions for different event magnitude scales is confirmed by both experimental and numerical data thus reinforcing the idea that fracture of heterogeneous materials has scaling properties similar to those found for earthquakes analysis of the different scaling parameters obtained for experimental and numerical data leads us to formulate the hypothesis that the type of scaling function obtained depends on the level of correlation among fracture events in the system

an integral algorithm for numerical integration of one dimensional additive colored noise problems we present an integral closed algorithm for solving a langevin equation driven by an additive colored noise both the mean first passage time in a bistable system and the diffusion current in a titled periodic potential are calculated and the comparison with existing algorithms is carried out the dependence of the numerical results on the time steps is studied our algorithm is shown to have high accuracy and stability

ito versus stratonovich a survey is given of the facts and fancies concerning the nonlinear langevin or ito equation actually it is merely a pre equation which becomes an equation when an interpretation rule is added the rules of ito and stratonovich differ but both are mathematically consistent and therefore equally admissible conventions the reason why they seem to lead to physical differences is that the langevin approach used to arrive at the equation involves a tacit assumption for systems with external noise this assumption can be justified and it is then clear that the stratonovich rule applies systems with internal noise however can only be properly described by a master equation and the ito stratonovich controversy never enters afterward one is free to model the resulting fluctuations either with an ito or a stratonovich scheme but that does not lead to any new information

a haar like construction for the ornstein uhlenbeck process the classical haar construction of brownian motion uses a binary tree of triangular wedge shaped functions this basis has compactness properties which make it especially suited for certain classes of numerical algorithms we present a similar basis for the ornstein uhlenbeck process in which the basis elements approach asymptotically the haar functions as the index increases and preserve the following properties of the haar basis all basis elements have compact support on an open interval with dyadic rational endpoints these intervals are nested and become smaller for larger indices of the basis element and for any dyadic rational only a finite number of basis elements is nonzero at that number thus the expansion in our basis when evaluated at a dyadic rational terminates in a finite number of steps we prove the covariance formulae for our expansion and discuss its statistical interpretation

pattern recognition using asymmetric attractor neural networks the asymmetric attractor neural networks designed by the monte carlo mc adaptation rule are shown to be promising candidates for pattern recognition in such a neural network with relatively low symmetry when the members of a set of template patterns are stored as fixed point attractors their attraction basins are shown to be isolated islands embedded in a chaotic sea the sizes of these islands can be controlled by a single parameter we show that these properties can be used for effective pattern recognition and rejection in our method the pattern to be identified is attracted to a template pattern or a chaotic attractor if the difference between the pattern to be identified and the template pattern is smaller than a predescribed threshold the pattern is attracted to the template pattern automatically and thus is identified as belonging to this template pattern otherwise it wanders in a chaotic attractor for ever and thus is rejected as an unknown pattern the maximum sizes of these islands allowed by this kind of neural networks are determined by a modified mc adaptation rule which are shown to be able to dramatically enlarge the sizes of the islands we illustrate the use of our method for pattern recognition and rejection with an example of recognizing a set of chinese characters

effect of discontinuity in the threshold distribution on the critical behavior of a random fiber bundle the critical behavior of a random fiber bundle model with mixed uniform distribution of threshold strengths and global load sharing rule is studied with a special emphasis on the nature of distribution of avalanches for different parameters of the distribution the discontinuity in the threshold strength distribution of fibers nontrivially modifies the critical stress as well as puts a restriction on the allowed values of parameters for which the recursive dynamics approach holds good the discontinuity leads to a nonuniversal behavior in the avalanche size distribution for smaller values of avalanche size we observe that apart from the mean field behavior for larger avalanches a new behavior for smaller avalanche size is observed as a critical threshold distribution is approached the phenomenological understanding of the above result is provided using the exact analytical result for the avalanche size distribution most interestingly the prominence of nonuniversal behavior in avalanche size distribution depends on the system parameters

accurate monte carlo tests of the stochastic ginzburg landau model with multiplicative colored noise an accurate and fast monte carlo algorithm is proposed for solving the ginzburg landau equation with multiplicative colored noise the stable cases of solution for choosing time steps and trajectory numbers are discussed

effects of transient subordinators on the firing statistics of a neuron model driven by dichotomous noise the behavior of a stochastic perfect integrate and fire pif model of neurons is considered the effect of temporally correlated random activity of synaptic inputs is modeled as a combination of an asymmetric dichotomous noise and a random operation time in the form of an inverse strictly increasing levy type subordinator using a first passage time formulation we find exact expressions for the output interspike interval isi statistics particularly it is shown that at some parameter regimes the isi density exhibits a multimodal structure moreover it is demonstrated that the coefficient of variation the serial correlation coefficient and the fano factor display a nonmonotonic dependence on the mean input current i e the isi s regularity is maximized at an intermediate value of the features of spike statistics analytically revealed in our study are compared with previously obtained results for a perfect integrate and fire neuron model driven by dichotomous noise without subordination

stochastic resonance reverse resonance and resonant activation induced by a multi state noise abstract in this paper we investigate the periodic response for a linear system driven by a multiplicative multi state noise which is composed of the multiplication of two dichotomous noises to an input temporal oscillatory signal and the escape of brownian particles over the fluctuating potential barrier for a system with a piece wise linear potential and driven by an additive multi state noise which is also composed of the multiplication of two dichotomous noises for the first system we get the stochastic resonance phenomenon for the amplitude of the periodic response vs the two dichotomous noise strengths and the phenomenon of reverse resonance for the amplitude of the periodic response vs k which represents the asymmetry degree of the dichotomous noises for the second system we obtain the resonant activation phenomenon for which the mean first passage time of the brownian particles over the fluctuating potential barrier shows a minimum as the function of the transition rates of the multi state noise

failure properties of loaded fiber bundles having a lower cutoff in fiber threshold distribution the presence of lower cutoff in fiber threshold distribution may affect the failure properties of a bundle of fibers subjected to external load we investigate this possibility both in an equal load sharing els model and in a local load sharing lls one using analytic as well as numerical methods in the els model the critical strength gets modified and beyond a certain lower cutoff level the whole bundle fails instantly brittle failure after the first fiber ruptures although the dynamic exponents for the order parameter susceptibility and relaxation time remain unchanged the avalanche size distribution shows a gradual deviation from the mean field power law a similar instant failure situation occurs in the lls model at a lower cutoff level which reduces to that of the equivalent els model at higher high enough dimensions also the system size variation of the bundle s strength and the avalanche statistics show strong dependence on the lower cutoff level

numerical method for solving stochastic differential equations with dichotomous noise we propose a numerical method for solving stochastic differential equations with dichotomous markov noise the numerical scheme is formulated such that i the stochastic formula used follows the stratonovichtaylor form over the entire range of noise correlation times including the gaussian white noise limit and ii the method is readily applicable to dynamical systems driven by arbitrary types of noise provided there exists a way to describe the random increment of the stochastic process expressed in the stratonovich taylor form we further propose a simplified taylor scheme that significantly reduces the computation time while still satisfying the moment properties up to the required order the accuracies and efficiencies of the proposed algorithms are validated by applying the schemes to two prototypical model systems that possess analytical solutions

second order algorithm for simulating stochastic differential equations with white noises abstract the second order algorithm for simulating stochastic differential equations with gaussian white noises is presented these stochastic differential equations are universal type among these gaussian white noises come from different sources specifically the proposed algorithm extends previous first order algorithm for stochastic differential equations with different white noises and second order algorithm for stochastic differential equations with same white noise in practice it is proved that this algorithm is scientific

the availability of logical operation induced by dichotomous noise for a nonlinear bistable system instead of a continuous system driven by gaussian white noise logical stochastic resonance will be investigated in a nonlinear bistable system with two thresholds driven by dichotomous noise which shows a phenomenon different from gaussian white noise we can realize two parallel logical operations by simply adjusting the values of these two thresholds besides to quantify the reliability of obtaining the correct logic output we numerically calculate the success probability and effects of dichotomous noise on the success probability are observed these observations show that the reliability of realizing logical operation in the bistable system can be improved through optimizing parameters of dichotomous noise

stochastic processes driven by dichotomous markov noise some exact dynamical results stochastic processes defined by a general langevin equation of motion where the noise is the non gaussian dichotomous markov noise are studied a non fokker planck master differential equation is deduced for the probability density of these processes two different models are exactly solved in the second one a nonequilibrium bimodal distribution induced by the noise is observed for a critical value of its correlation time critical slowing down does not appear in this point but in another one

computation of lyapunov spectra effect of interactive noise and application to a chemical oscillator abstract a refined version of the sano sawada algorithm to compute lyapunov spectra has been implemented and tested with several models which include periodic chaotic and hyperchaotic attractors under conditions designed to mimic experimental situations it was found that reliable lyapunov exponents can be obtained for time series of moderate length and point density also in the presence of interactive noise as long as its amplitude does not exceed 1 of the oscillation amplitude the algorithm was then applied to experimental data from the co oxidation on pt 110 clearly showing the occurrence of chaos as well as hyperchaos

numerical simulations of generalized langevin equations with deeply asymptotic parameters a unified algorithm for solving langevin equations with deeply asymptotic parameters is proposed and tested the method consists of identifying solvable linear friction and implementing the force evaluations by use of the runge kutta method we apply the present scheme to the periodic motion of an overdamped particle subjected to a multiplicative white noise the accurate calculations for the temporal velocity of the particle and its correlation function can be realized by introducing an inertial term it is shown that the fluctuation around the steady quantity increases with decreasing time step in the overdamped white noise algorithm however a massive white noise technique greatly reduces this spurious drift and the result can converge to the correct value if the added inertia approaches zero the other application is the simulation of generalized langevin equation with an exponential memory friction this allows us to treat a weak non markovian process

solvability of the master equation for dichotomous flow we consider the one dimensional stochastic flow mathrm x ifmmode dot else fi f x g x ensuremath xi t where ensuremath xi t is a dichotomous markov noise and use a simple procedure to identify the conditions under which the integro differential equation satisfied by the total probability density p x t of the driven variable can be reduced to a differential equation of finite order this generalizes the enumeration of the solvable cases

stochastic algorithms for discontinuous multiplicative white noise stochastic differential equations with multiplicative noise need a mathematical prescription due to different interpretations of the stochastic integral this fact implies specific algorithms to perform numerical integrations or simulations of the stochastic trajectories moreover if the multiplicative noise function is not continuous then the standard algorithms cannot be used we present an explicit algorithm to avoid this problem and we apply it to a well controlled example finally we discuss on the existence of higher order algorithms for this specific situation

statistics of a leaky integrate and fire model of neurons driven by dichotomous noise the behavior of a stochastic leaky integrate and fire model of neurons is considered the effect of temporally correlated random neuronal input is modeled as a colored two level dichotomous markovian noise relying on the riemann method exact expressions for the output interspike interval density and for the serial correlation coefficient are derived and their dependence on noise parameters such as correlation time and amplitude is analyzed particularly noise induced sign reversal and a resonancelike amplification of the kurtosis of the interspike interval distribution are established the features of spike statistics analytically revealed in our study are compared with recently obtained results for a perfect integrate and fire neuron model

effect of correlated dichotomous noises on stochastic resonance in a linear system a linear system driven by correlated asymmetric dichotomous noises and periodic signal was investigated in the overdamped case the exact expressions of output signal amplitude and signal to noise ratio snr of the system were derived by means of numerical calculations we found that i at some fixed multiplicative noise intensities the output signal amplitude with frequency exhibits the structure of a weak peak even no peak as the dichotomous noise is asymmetric ii in the case of asymmetric dichotomous noise the signal frequency can cause non monotonous behavior of the output signal amplitude with respect to multiplicative noise intensity iii the curve of snr with frequency has a weak peak and a trough in the case of symmetric dichotomous noise but no peak with asymmetric iv whether the multiplicative noise is symmetric or asymmetric the noise can enhance response of the system v the snr increases with the correlation strength between the two noises decreasing in addition the plane of multiplicative noise intensity versus noise symmetric parameter was plotted

a fast algorithm for the first passage times of gauss markov processes with holder continuous boundaries even for simple diffusion processes treating first passage problems analytically proves intractable for generic barriers and existing numerical methods are inaccurate and computationally costly here we present a novel numerical method that is faster and has more tightly controlled accuracy our algorithm is a probabilistic variant of dichotomic search for the computation of first passage times through non negative homogeneously holder continuous boundaries by gauss markov processes these include the ornstein uhlenbeck process underlying the ubiquitous leaky integrate and fire model of neuronal excitation our method evaluates discrete points in a sample path exactly and refines this representation recursively only in regions where a passage is rigorously estimated to be probable e g when close to the boundary

random fiber bundle with many discontinuities in the threshold distribution we study the breakdown of a random fiber bundle model rfbm with n discontinuities in the threshold distribution using the global load sharing scheme in other words n 1 different classes of fibers identified on the basis of their threshold strengths are mixed such that the strengths of the fibers in the ith class are uniformly distributed between the values sigma2i 2 and sigma2i 1 where 1 or i or n 1 moreover there is a gap in the threshold distribution between ith and i 1 th class we show that although the critical stress depends on the parameter values of the system the critical exponents are identical to that obtained in the recursive dynamics of a rfbm with a uniform distribution and global load sharing the avalanche size distribution on the other hand shows a nonuniversal non power law behavior for smaller values of avalanche sizes which becomes prominent only when a critical distribution is approached we establish that the behavior of the avalanche size distribution for an arbitrary n is qualitatively similar to a rfbm with a single discontinuity in the threshold distribution n 1 especially when the density and the range of threshold values of fibers belonging to strongest n 1 th class is kept identical in all the cases