transition phenomena in cellular automata rule space abstract we define several qualitative classes of cellular automata ca behavior based on various statistical measures and describe how the space of all cellular automata is organized as a cellular automaton is changed by varying entries in its rule table abrupt changes in qualitative behavior may occur these abrupt changes have the character of bifurcations in smooth dynamical systems or of phase transitions in statistical mechanical systems the most complex ca rules exhibit many of the characteristics of second order transitions suggesting an association between computation complexity and critical phenomena

kramers like escape driven by fractional gaussian noise we investigate the escape from a potential well of a test particle driven by fractional gaussian noise with hurst exponent 0 h 1 from a numerical analysis we demonstrate the exponential distribution of escape times from the well and analyze in detail the dependence of the mean escape time on the hurst exponent h and the particle diffusivity d we observe different behavior for the subdiffusive antipersistent and superdiffusive persistent domains in particular we find that the escape becomes increasingly faster for decreasing values of h consistent with previous findings on the first passage behavior approximate analytical calculations are shown to support the numerically observed dependencies

the attractor basin portrait of a cellular automaton local space time structures such as domains and the intervening dislocations dominate a wide class of cellular automaton ca behavior for such spatially extended dynamics regular domains vicinities and attractors are introduced as organizing principles to identify the discretized analogs of attractors basins and separatrices structures used in classifying dissipative continuous state dynamical systems we describe the attractor basin portrait of nonlinear elementary ca rule 18 whose global dynamics is largely determined by a single regular attracting domain the latter s basin is analyzed in terms of subbasin and portal structures associated with particle annihilation the conclusion is that the computational complexity of such ca is more apparent than real transducer machines are constructed that automatically identify domain and dislocation structures in space time count the number of dislocations in a spatial pattern and implement an isomorphism between rule 18 and rule 90 we use a transducer to trace dislocation trajectories and confirm that in rule 18 isolated dislocation trajectories as well as a dislocation gas agree extremely well with the classical model of annihilating diffusive particles the ca efficiently transforms randomness of an initial pattern ensemble into a random walk of dislocations in space time

annealed and quenched inhomogeneous cellular automata inca a probabilistic one dimensional cellular automaton model by domany and kinzel is mapped into an inhomogeneous cellular automaton with the boolean functions xor and and as transition rules wolfram s classification is recovered by varying the frequency of these two simple rules and by quenching or annealing the inhomogeneity in particular class 4 is related to critical behavior in directed percolation also the critical slowing down of second order phase transitions is related to a stochastic version of the classical halting problem of computation theory

evolving cellular automata to perform computations mechanisms and impediments abstract we present results from experiments in which a genetic algorithm ga was used to evolve cellular automata cas to perform a particular computational task one dimensional density classification we look in detail at the evolutionary mechanisms producing the ga s behavior on this task and the impediments faced by the ga in particular we identify four epochs of innovation in which new ca strategies for solving the problem are discovered by the ga describe how these strategies are implemented in ca rule tables and identify the ga mechanisms underlying their discovery the epochs are characterized by a breaking of the task s symmetries on the part of the ga the symmetry breaking results in a short term fitness gain but ultimately prevents the discovery of the most highly fit strategies we discuss the extent to which symmetry breaking and other impediments are general phenomena in any ga search

analysis of short subdiffusive time series scatter of the time averaged mean squared displacement we analyse the statistical behaviour of short time series in systems performing subdiffusion comparing the non ergodic continuous time random walk model to the ergodic fractional brownian motion we demonstrate that the scatter between individual trajectories is not purely dominated by finite sample size effects but preserves some of the characteristics of the individual processes in particular we show that the distribution of the time averaged mean squared displacements allows one to clearly distinguish between the two stochastic mechanisms even for a very short time series

statistical properties of functionals of the paths of a particle diffusing in a one dimensional random potential we present a formalism for obtaining the statistical properties of functionals and inverse functionals of the paths of a particle diffusing in a one dimensional quenched random potential we demonstrate the implementation of the formalism in two specific examples 1 where the functional corresponds to the local time spent by the particle around the origin and 2 where the functional corresponds to the occupation time spent by the particle on the positive side of the origin within an observation time window of size t we compute the disorder average distributions of the local time the inverse local time the occupation time and the inverse occupation time and show that in many cases disorder modifies the behavior drastically

viscoelastic subdiffusion from anomalous to normal we study viscoelastic subdiffusion in bistable and periodic potentials within the generalized langevin equation approach our results justify the ultra slow fluctuating rate view of the corresponding bistable non markovian dynamics which displays bursting and anticorrelation of the residence times in two potential wells the transition kinetics is asymptotically stretched exponential when the potential barrier v 0 several times exceeds thermal energy k b t v 0 ensuremath sim 2 10 k b t and it cannot be described by the non markovian rate theory nmrt the well known nmrt result approximates however ever better with the increasing barrier height the most probable logarithm of the residence times moreover the rate description is gradually restored when the barrier height exceeds a fuzzy borderline which depends on the power law exponent of free subdiffusion ensuremath alpha such a potential free subdiffusion is ergodic surprisingly in periodic potentials it is not sensitive to the barrier height in the long time asymptotic limit however the transient to this asymptotic regime is extremally slow and it does profoundly depend on the barrier height the time scale of such subdiffusion can exceed the mean residence time in a potential well or in a finite spatial domain by many orders of magnitude all these features are in sharp contrast with an alternative subdiffusion mechanism involving jumps among traps with the divergent mean residence time in these traps

quasi steady state analysis of two dimensional random intermittent search processes we use perturbation methods to analyze a two dimensional random intermittent search process in which a searcher alternates between a diffusive search phase and a ballistic movement phase whose velocity direction is random a hidden target is introduced within a rectangular domain with reflecting boundaries if the searcher moves within range of the target and is in the search phase it has a chance of detecting the target a quasi steady state analysis is applied to the corresponding chapman kolmogorov equation this generates a reduced fokker planck description of the search process involving a nonzero drift term and an anisotropic diffusion tensor in the case of a uniform direction distribution for which there is zero drift and isotropic diffusion we use the method of matched asymptotics to compute the mean first passage time mfpt to the target under the assumption that the detection range of the target is much smaller than the size of the domain we show that an optimal search strategy exists consistent with previous studies of intermittent search in a radially symmetric domain that were based on a decoupling or moment closure approximation we also show how the decoupling approximation can break down in the case of biased search processes finally we analyze the mfpt in the case of anisotropic diffusion and find that anisotropy can be useful when the searcher starts from a fixed location

turbulent pattern bases for cellular automata abstract unpredictable patterns generated by cellular automata ca can be decomposed with respect to a turbulent positive entropy rate pattern basis the resulting filtered patterns uncover significant structural organization in a ca s dynamics and information processing capabilities we illustrate the decompostion technique by analyzing a binary range 2 cellular automaton having two invariant chaotic domains of different complexities and entropies once identified the domains are seen to organize the ca s state space and to dominate its evolution starting from the domains structures we show how to construct a finite state transducer that performs nonlinear spatial filtering such that the resulting space time patterns reveal the domains and the intervening walls and dislocations to show the statistical consequences of domain detection we compare the entropy and complexity densities of each domain with the globally averaged quantities a more graphical comparison uses difference patterns and difference plumes which trace the space time influence of a single site perturbation we also investigate the diversity of walls and particles emanating from the interface between two adjacent domains

global mean first passage times of random walks on complex networks we present a general framework applicable to a broad class of random walks on complex networks which provides a rigorous lower bound for the mean first passage time of a random walker to a target site averaged over its starting position the so called global mean first passage time gmfpt this bound is simply expressed in terms of the equilibrium distribution at the target and implies a minimal scaling of the gmfpt with the network size we show that this minimal scaling which can be arbitrarily slow is realized under the simple condition that the random walk is transient at the target site and independently of the small world scale free or fractal properties of the network last we put forward that the gmfpt to a specific target is not a representative property of the network since the target averaged gmfpt satisfies much more restrictive bounds

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

fractional brownian motion and motion governed by the fractional langevin equation in confined geometries motivated by subdiffusive motion of biomolecules observed in living cells we study the stochastic properties of a non brownian particle whose motion is governed by either fractional brownian motion or the fractional langevin equation and restricted to a finite domain we investigate by analytic calculations and simulations how time averaged observables e g the time averaged mean squared displacement and displacement correlation are affected by spatial confinement and dimensionality in particular we study the degree of weak ergodicity breaking and scatter between different single trajectories for this confined motion in the subdiffusive domain the general trend is that deviations from ergodicity are decreased with decreasing size of the movement volume and with increasing dimensionality we define the displacement correlation function and find that this quantity shows distinct features for fractional brownian motion fractional langevin equation and continuous time subdiffusion such that it appears an efficient measure to distinguish these different processes based on single particle trajectory data

scale invariant motion in intermittent chaotic systems we investigate the dynamics generated from iterated maps and analyze the motion in terms of the probabilistic continuous time random walk ctrw approach two different ctrw models are considered i particles jump between sites turning points or ii particles move at a constant velocity between sites and choose a new direction at random for both models we study the mean squared displacement mathit r 2 t and the propagator p r t the probability to be at location r at time t having started at the origin at t 0 iterated maps are used to generate both dispersive and enhanced diffusion and the results are analyzed using the ctrw framework and scaling arguments for the case of dispersive motion we discuss the problem of the stationary state and point out its relevance

random walk in a random medium in one dimension applying scaling and universality arguments the long time behavior of the probability distribution for a random walk in a one dimensional random medium satisfying sinai s constraint is obtained analytically the convergence to this asymptotic limit and the fluctuations of this distribution are evaluated by solving numerically the stochastic equations for this walk

physics like models of computation abstract reversible cellular automata are computer models that embody discrete analogues of the classical physics notions of space time locality and microscopic reversibility they are offered as a step towards models of computation that are closer to fundamental physics

survival probability of diffusion with trapping in cellular neurobiology the problem of diffusion with absorption and trapping sites arises in the theory of molecular signaling inside and on the membranes of biological cells in particular this problem arises in the case of spine dendrite communication where the number of calcium ions modeled as random particles is regulated across the spine microstructure by pumps which play the role of killing sites while the end of the dendritic shaft is an absorbing boundary we develop a general mathematical framework for diffusion in the presence of absorption and killing sites and apply it to the computation of the time dependent survival probability of ions we also compute the ratio of the number of absorbed particles at a specific location to the number of killed particles we show that the ratio depends on the distribution of killing sites the biological consequence is that the position of the pumps regulates the fraction of calcium ions that reach the dendrite

on form factors in nested bethe ansatz systems we investigate form factors of local operators in a multi component quantum nonlinear schrodinger model a prototype theory solvable by the so called nested bethe ansatz we determine the analytic properties of the infinite volume form factors using the coordinate bethe ansatz solution and establish a connection with the finite volume matrix elements in the two component models we derive a set of recursion relations for the magnonic form factors which are the matrix elements on the nested bethe ansatz states in certain simple cases involving states with only one spin impurity we obtain explicit solutions for the recursion relations

lax pair and super yangian symmetry of the nonlinear super schrodinger equation we consider a version of the nonlinear schrodinger equation with m bosons and n fermions we first solve the classical and quantum versions of this equation using a super zamolodchikov faddeev zf algebra then we prove that the hierarchy associated to this model admits a super yangian y gl m n symmetry we exhibit the corresponding classical and quantum lax pairs finally we construct explicitly the super yangian generators in terms of the canonical fields on the one hand and in terms of the zf algebra generators on the other hand the latter construction uses the well bred operators introduced recently

statistical dynamics of the royal road genetic algorithm abstract metastability is a common phenomenon many evolutionary processes both natural and artificial alternate between periods of stasis and brief periods of rapid change in their behavior in this paper an analytical model for the dynamics of a mutation only genetic algorithm ga is introduced that identifies a new and general mechanism causing metastability in evolutionary dynamics the ga s population dynamics is described in terms of flows in the space of fitness distributions the trajectories through fitness distribution space are derived in closed form in the limit of infinite populations we then show how finite populations induce metastability even in regions where fitness does not exhibit a local optimum in particular the model predicts the occurrence of fitness epochs periods of stasis in population fitness distributions at finite population size and identifies the locations of these fitness epochs with the flow s hyperbolic fixed points this enables exact predictions of the metastable fitness distributions during the fitness epochs as well as giving insight into the nature of the periods of stasis and the innovations between them all these results are obtained as closed form expressions in terms of the ga s parameters an analysis of the jacobian matrices in the neighborhood of an epoch s metastable fitness distribution allows for the calculation of its stable and unstable manifold dimensions and so reveals the state space s topological structure more general quantitative features of the dynamics fitness fluctuation amplitudes epoch stability and speed of the innovations are also determined from the jacobian eigenvalues the analysis shows how quantitative predictions for a range of dynamical behaviors that are specific to the finite population dynamics can be derived from the solution of the infinite population dynamics the theoretical predictions are shown to agree very well with statistics from ga simulations we also discuss the connections of our results with those from population genetics and molecular evolution theory

a hierarchical classification of cellular automata abstract a central issue in the theory of cellular automata is classification a classification is a structure imposed on the space of automata which groups together cellular automata with related properties in this paper a classification based on the action of cellular automata on n step markov measures is presented this classification is hierarchical and parametric an algorithm is described which efficiently constructs all rules in a given class at a given level of the hierarchy the utility of the classification is explored in the study of the invariant statistical properties of next nearest neighbor cellular automata on two states per cell

nonergodicity fluctuations and criticality in heterogeneous diffusion processes we study the stochastic behavior of heterogeneous diffusion processes with the power law dependence d x ensuremath sim x ensuremath alpha of the generalized diffusion coefficient encompassing sub and superdiffusive anomalous diffusion based on statistical measures such as the amplitude scatter of the time averaged mean squared displacement of individual realizations the ergodicity breaking and non gaussianity parameters as well as the probability density function p x t we analyze the weakly nonergodic character of the heterogeneous diffusion process and particularly the degree of irreproducibility of individual realizations as we show the fluctuations between individual realizations increase with growing modulus ensuremath alpha of the scaling exponent the fluctuations appear to diverge when the critical value ensuremath alpha 2 is approached while for even larger ensuremath alpha the fluctuations decrease again at criticality the power law behavior of the mean squared displacement changes to an exponentially fast growth and the fluctuations of the time averaged mean squared displacement do not converge for increasing number of realizations from a systematic comparison we observe some striking similarities of the heterogeneous diffusion process with the familiar subdiffusive continuous time random walk process with power law waiting time distribution and diverging characteristic waiting time

local information transfer as a spatiotemporal filter for complex systems we present a measure of local information transfer derived from an existing averaged information theoretical measure namely transfer entropy local transfer entropy is used to produce profiles of the information transfer into each spatiotemporal point in a complex system these spatiotemporal profiles are useful not only as an analytical tool but also allow explicit investigation of different parameter settings and forms of the transfer entropy metric itself as an example local transfer entropy is applied to cellular automata where it is demonstrated to be a useful method of filtering for coherent structure more importantly local transfer entropy provides the first quantitative evidence for the long held conjecture that the emergent traveling coherent structures known as particles both gliders and domain walls which have analogs in many physical processes are the dominant information transfer agents in cellular automata

co evolving non uniform cellular automata to perform computations a major impediment of cellular automata ca stems from the difficulty of utilizing their complex behavior to perform useful computations recent studies by packard and mitchell et al have shown that cas can be evolved to perform a computational task in this paper non uniform cas are studied where each cell may contain a different rule in contrast to the original uniform model we describe experiments in which non uniform cas are evolved to perform the computational task using a local co evolutionary algorithm for radius r 3 we attain peak performance values of 0 92 comparable to those obtained for uniform cas 0 93 0 95 this is notable considering the huge search spaces involved much larger than the uniform case smaller radius cas previously unstudied in this context attain performance values of 0 93 0 94 for r 1 this is considerably higher than the maximal possible uniform ca performance of 0 83 suggesting that non uniformity reduces connectivity requirements we thus demonstrate that 1 non uniform cas can attain high computational performance and 2 such systems can be evolved rather than designed

is there a sharp phase transition for deterministic cellular automata abstract previous work has suggested that there is a kind of phase transition between deterministic automata exhibiting periodic behavior and those exhibiting chaotic behavior however unlike the usual phase transitions of physics this transition takes place over a range of values of the parameter rather than at a specific value the present paper asks whether the transition can be made sharp either by taking the limit of an infinitely large rule table or by changing the parameter in terms of which the space of automata is explored we find strong evidence that for the class of automata we consider the transition does become sharp in the limit of an infinite number of symbols the size of the neighborhood being held fixed our work also suggests an alternative parameter in terms of which it is likely that the transition will become fairly sharp even if one does not increase the number of symbols in the course of our analysis we find that mean field theory which is our main tool gives surprisingly good predictions of the statistical properties of the class of automata we consider

localisation and universal fluctuations in ultraslow diffusion processes we study ultraslow diffusion processes with logarithmic mean squared displacement msd comparison of annealed renewal continuous time random walks ctrws with logarithmic waiting time distribution and sinai diffusion in quenched random landscapes reveals striking similarities despite the great differences in their physical nature in particular they exhibit a weakly non ergodic disparity of the time averaged and ensemble averaged msds remarkably for the ctrw we observe that the fluctuations of time averages become universal with an exponential suppression of mobile trajectories we discuss the fundamental connection between the golosov localization effect and non ergodicity in the sense of the disparity between ensemble averaged msd and time averaged msd

improved estimation of anomalous diffusion exponents in single particle tracking experiments the mean square displacement is a central tool in the analysis of single particle tracking experiments shedding light on various biophysical phenomena frequently parameters are extracted by performing time averages on single particle trajectories followed by ensemble averaging this procedure however suffers from two systematic errors when applied to particles that perform anomalous diffusion the first is significant at short time lags and is induced by measurement errors the second arises from the natural heterogeneity in biophysical systems we show how to estimate and correct these two errors and improve the estimation of the anomalous parameters for the whole particle distribution as a consequence we manage to characterize ensembles of heterogeneous particles even for rather short and noisy measurements where regular time averaged mean square displacement analysis fails we apply this method to both simulations and in vivo measurements of telomere diffusion in 3t3 mouse embryonic fibroblast cells the motion of telomeres is found to be subdiffusive with an average exponent constant in time individual telomere exponents are normally distributed around the average exponent the proposed methodology has the potential to improve experimental accuracy while maintaining lower experimental costs and complexity

sensitivity to noise and ergodicity of an assembly line of cellular automata that classifies density we investigate the sensitivity of the composite cellular automaton of h fuk ifmmode acute s else s fi phys rev e 55 r2081 1997 to noise and assess the density classification performance of the resulting probabilistic cellular automaton pca numerically we conclude that the composite pca performs the density classification task reliably only up to very small levels of noise in particular it cannot outperform the noisy gacs kurdyumov levin automaton an imperfect classifier for any level of noise while the original composite ca is nonergodic analyses of relaxation times indicate that its noisy version is an ergodic automaton with the relaxation times decaying algebraically over an extended range of parameters with an exponent very close possibly equal to the mean field value

inverse geometric approach for the simulation of close to circular growth the case of multicellular tumor spheroids abstract we demonstrate the power of genetic algorithms to construct a cellular automata model simulating the growth of 2d close to circular clusters revealing the desired properties such as the growth rate and at the same time the fractal behavior of their contours the possible application of the approach in the field of tumor modeling is outlined

random walk to a nonergodic equilibrium concept random walk models such as the trap model continuous time random walks and comb models exhibit weak ergodicity breaking when the average waiting time is infinite the open question is what statistical mechanical theory replaces the canonical boltzmann gibbs theory for such systems in this paper a nonergodic equilibrium concept is investigated for a continuous time random walk model in a potential field in particular we show that in the nonergodic phase the distribution of the occupation time of the particle in a finite region of space approaches u or w shaped distributions related to the arcsine law we show that when conditions of detailed balance are applied these distributions depend on the partition function of the problem thus establishing a relation between the nonergodic dynamics and canonical statistical mechanics in the ergodic phase the distribution function of the occupation times approaches a ensuremath delta function centered on the value predicted based on standard boltzmann gibbs statistics the relation of our work to single molecule experiments is briefly discussed

ageing and confinement in non ergodic heterogeneous diffusion processes we study the effects of ageing the time delay between initiation of the physical process at t 0 and start of observation at some time t 0 a and spatial confinement on the properties of heterogeneous diffusion processes hdps with deterministic power law space dependent diffusivities d xd x 0 from analysis of the ensemble and time averaged mean squared displacements and the ergodicity breaking parameter quantifying the inherent degree of irreproducibility of individual realizations of the hdp we obtain striking similarities to ageing subdiffusive continuous time random walks with scale free waiting time distributions we also explore how both processes can be distinguished for confined hdps we study the long time saturation of the ensemble and time averaged particle displacements as well as the magnitude of the inherent scatter of time averaged displacements and contrast the outcomes to the results known for other anomalous diffusion processes under confinement

fractional langevin equation we investigate fractional brownian motion with a microscopic random matrix model and introduce a fractional langevin equation we use the latter to study both subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath we further compare fractional brownian motion with the fractal time process the respective mean square displacements of these two forms of anomalous diffusion exhibit the same power law behavior here we show that their lowest moments are actually all identical except the second moment of the velocity this provides a simple criterion that enable us to distinguish these two non markovian processes

markov additive processes i eigenvalue properties and limit theorems on etudie d un point de vue general les proprietes d existence et de regularite des valeurs propres et des fonctions propres de l operateur de feynman kac

subdiffusion in time averaged confined random walks certain techniques characterizing diffusive processes such as single particle tracking or molecular dynamics simulation provide time averages rather than ensemble averages whereas the ensemble averaged mean squared displacement msd of an unbounded continuous time random walk ctrw with a broad distribution of waiting times exhibits subdiffusion the time averaged msd delta2 does not we demonstrate that in contrast to the unbounded ctrw in which delta2 is linear in the lag time delta the time averaged msd of the ctrw of a walker confined to a finite volume is sublinear in delta i e for long lag times delta2 approximately delta1 alpha the present results permit the application of ctrw to interpret time averaged experimental quantities

first passages in bounded domains when is the mean first passage time meaningful we study the first passage statistics to adsorbing boundaries of a brownian motion in bounded two dimensional domains of different shapes and configurations of the adsorbing and reflecting boundaries from extensive numerical analysis we obtain the probability p ensuremath omega distribution of the random variable ensuremath omega ensuremath tau 1 ensuremath tau 1 ensuremath tau 2 which is a measure for how similar the first passage times ensuremath tau 1 and ensuremath tau 2 are of two independent realizations of a brownian walk starting at the same location we construct a chart for each domain determining whether p ensuremath omega represents a unimodal bell shaped form or a bimodal m shaped behavior while in the former case the mean first passage time mfpt is a valid characteristic of the first passage behavior in the latter case it is an insufficient measure for the process strikingly we find a distinct turnover between the two modes of p ensuremath omega characteristic for the domain shape and the respective location of absorbing and reflective boundaries our results demonstrate that large fluctuations of the first passage times may occur frequently in two dimensional domains rendering quite vague the general use of the mfpt as a robust measure of the actual behavior even in bounded domains in which all moments of the first passage distribution exist

zero constant formula for first passage observables in bounded domains in this letter we develop an analytical approach which provides an explicit determination of mean first passage times mfpts for random walks in bounded domains for a wide class of transport processes in particular we derive for the first time explicit expressions of mfpts for emblematic models of transport in complex media such as diffusion on deterministic and random fractals this approach relies on a scale invariance hypothesis and a large volume expansion of the mfpt which actually proves to be very accurate even for small system sizes as shown by numerical simulations this explicit determination of mfpts can be straightforwardly generalized to further useful first passage observables such as occupation times and splitting probabilities

ergodic properties of fractional brownian langevin motion we investigate the time average mean square displacement overline ensuremath delta 2 mathbf x t mathbf ensuremath int 0 t ensuremath ensuremath delta x t ensuremath ensuremath delta ensuremath x t ensuremath 2 d t ensuremath t ensuremath ensuremath delta for fractional brownian langevin motion where x t is the stochastic trajectory and ensuremath delta is the lag time unlike the previously investigated continuous time random walk model overline ensuremath delta 2 converges to the ensemble average x 2 ensuremath sim t 2h in the long measurement time limit the convergence to ergodic behavior is slow however and surprisingly the hurst exponent h frac 3 4 marks the critical point of the speed of convergence when hl frac 3 4 the ergodicity breaking parameter e b mathbf overline ensuremath delta 2 mathbf x t mathbf mathbf 2 ensuremath overline ensuremath delta 2 mathbf x t mathbf 2 mathbf overline ensuremath delta 2 mathbf x t mathbf 2 ensuremath sim k h ensuremath delta t ensuremath 1 when h frac 3 4 e b ensuremath sim frac 9 16 mathrm ln phantom rule 0 2em 0ex t ensuremath delta t ensuremath 1 and when frac 3 4 lhl1 e b ensuremath sim k h ensuremath delta 4 ensuremath 4h t 4h ensuremath 4 in the ballistic limit h ensuremath rightarrow 1 ergodicity is broken and e b ensuremath sim 2 the critical point h frac 3 4 is marked by the divergence of the coefficient k h fractional brownian motion as a model for recent experiments of subdiffusion of mrna in the cell is briefly discussed and a comparison with the continuous time random walk model is made

deducing local rules for solving global tasks with random boolean networks abstract it has been shown that uniform as well as non uniform cellular automata ca can be evolved to perform certain computational tasks random boolean networks are a generalization of two state cellular automata where the interconnection topology and the cell s rules are specified at random here we present a novel analytical approach to find the local rules of random boolean networks rbns to solve the global density classification and the synchronization task from any initial configuration we quantitatively and qualitatively compare our results with previously published work on cellular automata and show that randomly interconnected automata are computationally more efficient in solving these two global tasks our approach also provides convergence and quality estimates and allows the networks to be randomly rewired during operation without affecting the global performance finally we show that rbns outperform small world topologies on the density classification task and that they perform equally well on the synchronization task our novel approach and the results may have applications in designing robust complex networks and locally interacting distributed computing systems for solving global tasks

state dependent diffusion thermodynamic consistency and its path integral formulation the friction coefficient of a particle can depend on its position as it does when the particle is near a wall we formulate the dynamics of particles with such state dependent friction coefficients in terms of a general langevin equation with multiplicative noise whose evaluation requires the introduction of specific rules two common conventions the ito and the stratonovich provide alternative rules for evaluation of the noise but other conventions are possible we show that the requirement that a particle s distribution function approach the boltzmann distribution at long times dictates that a drift term must be added to the langevin equation this drift term is proportional to the derivative of the diffusion coefficient times a factor that depends on the convention used to define the multiplicative noise we explore the consequences of this result in a number of examples with spatially varying diffusion coefficients we also derive a path integral representation for arbitrary interpretation of the noise and use it in a perturbative study of correlations in a simple system

fractional brownian motion and the critical dynamics of zipping polymers we consider two complementary polymer strands of length l attached by a common end monomer the two strands bind through complementary monomers and at low temperatures form a double stranded conformation zipping while at high temperature they dissociate unzipping this is a simple model of dna or rna hairpin formation here we investigate the dynamics of the strands at the equilibrium critical temperature t t c using monte carlo rouse dynamics we find that the dynamics is anomalous with a characteristic time scaling as l 2 26 2 exceeding the rouse time l 2 18 we investigate the probability distribution function velocity autocorrelation function survival probability and boundary behavior of the underlying stochastic process these quantities scale as expected from a fractional brownian motion with a hurst exponent h 0 44 1 we discuss similarities to and differences from unbiased polymer translocation

diffusion in periodic correlated random forcing landscapes we study the dynamics of a brownian particle in a strongly correlated quenched random potential defined as a periodically extended with period l finite trajectory of a fractional brownian motion with arbitrary hurst exponent h in 0 1 while the periodicity ensures that the ultimate long time behavior is diffusive the generalised sinai potential considered here leads to a strong logarithmic confinement of particle trajectories at intermediate times these two competing trends lead to dynamical frustration and result in a rich statistical behavior of the diffusion coefficient d l although one has the typical value d rm typ l sim exp beta l h we show via an exact analytical approach that the positive moments k 0 scale like langle d k l rangle sim exp c k beta l h 1 1 h and the negative ones as langle d k l rangle sim exp a k beta l h 2 c and a being numerical constants and beta the inverse temperature these results demonstrate that d l is strongly non self averaging we further show that the probability distribution of d l has a log normal left tail and a highly singular one sided log stable right tail reminiscent of a lifshitz singularity

first passage times for random walks in bounded domains we present a novel computational method of first passage times between a starting site and a target site of regular bounded lattices we derive accurate expressions for all the moments of this first passage time validated by numerical simulations their range of validity is discussed we also consider the case of a starting site and two targets in addition we present the extension to continuous brownian motion these results are of great relevance to any system involving diffusion in confined media

spontaneous symmetry breaking in the non linear schrodinger hierarchy with defect we introduce and solve the one dimensional quantum non linear schrodinger nls equation for an n component field defined on the real line with a defect sitting at the origin the quantum solution is constructed using the quantum inverse scattering method based on the concept of reflection transmission rt algebras recently introduced the symmetry of the model is generated by the reflection and transmission defect generators defining a defect subalgebra we classify all the corresponding reflection and transmission matrices this provides the possible boundary conditions obeyed by the canonical field and we compute these boundary conditions explicitly finally we exhibit a phenomenon of spontaneous symmetry breaking induced by the defect and identify the unbroken generators as well as the exact remaining symmetry

first passage time statistics for two channel diffusion we present rigorous results for the mean first passage time and first passage time statistics for two channel markov additive diffusion in a 3 dimensional spherical domain inspired by biophysical examples we assume that the particle can only recognise the target in one of the modes which is shown to effect a non trivial first passage behaviour we also address the scenario of intermittent immobilisation in both cases we prove that despite the perfectly non recurrent motion of two channel markov additive diffusion in 3 dimensions the first passage statistics at long times do not display poisson like behaviour if none of the phases has a vanishing diffusion coefficient this stands in stark contrast to the standard one channel markov diffusion counterpart we also discuss the relevance of our results in the context of cellular signalling

topological evolution for embodied cellular automata behavioral variation in ca are traditionally created through various update rules instead we change behavior through topological variation of the substrate the automata of actuated cells are taken to define the morphology of a soft robot this morphology is optimized with an evolutionary algorithm to maximize locomotion a diversity of effective behaviors and morphologies are demonstrated in this work we introduce a novel method for creating behaviors in cellular automata optimizing the topology of the cellular substrate while maintaining a single simple update rule we study the effect of altering the shape of a 3d cellular automaton and local signaling ability of each of its cells on the ability of that automaton as a whole to give rise to emergent locomotion behavior this system optimizes for the physically embodied interactions between a cellular automaton with an external physically simulated world rather than optimizing directly for a computational ability internal to the automaton itself we give each cell in the automaton the ability to have an internal excited state and also the ability to perform a physical action volumetric contraction and expansion as a result of that state we then employ an evolutionary algorithm to optimize for the locomotion ability of the robot resulting from the behavior of this embodied automaton we demonstrate a number of diverse topologies which lead to effective locomotion behaviors in this paradigm we believe that creating complex behavior from simple rules in a complex substrate not only opens up questions about cellular automata but also provides insights towards the study of morphological computation and embodied cognition

performance of the majority voting rule in solving the density classification problem in high dimensions the density classification problem dcp is one of the most widely studied problems in the theory of cellular automata after it was shown that the dcp cannot be solved perfectly the research in this area has been focused on finding better rules that could solve the dcp approximately in this paper we argue that the majority voting rule in high dimensions can achieve high performance in solving the dcp and that its performance increases with dimension we support this conjecture with arguments based on the mean field approximation and direct computer simulations

optimization and universality of brownian search in a basic model of quenched heterogeneous media the kinetics of a variety of transport controlled processes can be reduced to the problem of determining the mean time needed to arrive at a given location for the first time the so called mean first passage time mfpt problem the occurrence of occasional large jumps or intermittent patterns combining various types of motion are known to outperform the standard random walk with respect to the mfpt by reducing oversampling of space here we show that a regular but spatially heterogeneous random walk can significantly and universally enhance the search in any spatial dimension in a generic minimal model we consider a spherically symmetric system comprising two concentric regions with piecewise constant diffusivity the mfpt is analyzed under the constraint of conserved average dynamics that is the spatially averaged diffusivity is kept constant our analytical calculations and extensive numerical simulations demonstrate the existence of an optimal heterogeneity minimizing the mfpt to the target we prove that the mfpt for a random walk is completely dominated by what we term direct trajectories towards the target and reveal a remarkable universality of the spatially heterogeneous search with respect to target size and system dimensionality in contrast to intermittent strategies which are most profitable in low spatial dimensions the spatially inhomogeneous search performs best in higher dimensions discussing our results alongside recent experiments on single particle tracking in living cells we argue that the observed spatial heterogeneity may be beneficial for cellular signaling processes

inequivalence of time and ensemble averages in ergodic systems exponential versus power law relaxation in confinement single particle tracking has become a standard tool for the investigation of diffusive properties especially in small systems such as biological cells usually the resulting time series are analyzed in terms of time averages over individual trajectories here we study confined normal as well as anomalous diffusion modeled by fractional brownian motion and the fractional langevin equation and show that even for such ergodic systems time averaged quantities behave differently from their ensemble averaged counterparts irrespective of how long the measurement time becomes knowledge of the exact behavior of time averages is therefore fundamental for the proper physical interpretation of measured time series in particular for extraction of the relaxation time scale from data

survival of an evasive prey abstract we study the survival of a prey that is hunted by n predators the predators perform independent random walks on a square lattice with v sites and start a direct chase whenever the prey appears within their sighting range the prey is caught when a predator jumps to the site occupied by the prey we analyze the efficacy of a lazy minimal effort evasion strategy according to which the prey tries to avoid encounters with the predators by making a hop only when any of the predators appears within its sighting range otherwise the prey stays still we show that if the sighting range of such a lazy prey is equal to 1 lattice spacing at least 3 predators are needed in order to catch the prey on a square lattice in this situation we establish a simple asymptotic relation lnp ev t n v 2lnp imm t between the survival probabilities of an evasive and an immobile prey hence when the density n v of the predators is low 1 the lazy evasion strategy leads to the spectacular increase of the survival probability we also argue that a short sighting prey its sighting range is smaller than the sighting range of the predators undergoes an effective superdiffusive motion as a result of its encounters with the predators whereas a far sighting prey performs a diffusive type motion

solving the density classification problem with a large diffusion and small amplification cellular automaton abstract one of the most studied inverse problems in cellular automata cas is the density classification problem it consists in finding a ca such that given any initial configuration of 0s and 1s it converges to the all 1 fixed point configuration if the fraction of 1s is greater than the critical density 1 2 and it converges to the all 0 fixed point configuration otherwise in this paper we propose an original approach to solve this problem by designing a ca inspired by two mechanisms that are ubiquitous in nature diffusion and nonlinear sigmoidal response this ca which is different from the classical ones because it has many states has a success ratio of 100 and works for any system size any dimension and any critical density

filling of a poisson trap by a population of random intermittent searchers we extend the continuum theory of random intermittent search processes to the case of n independent searchers looking to deliver cargo to a single hidden target located somewhere on a semi infinite track each searcher randomly switches between a stationary state and either a leftward or rightward constant velocity state we assume that all of the particles start at one end of the track and realize sample trajectories independently generated from the same underlying stochastic process the hidden target is treated as a partially absorbing trap in which a particle can only detect the target and deliver its cargo if it is stationary and within range of the target the particle is removed from the system after delivering its cargo as a further generalization of previous models we assume that up to n successive particles can find the target and deliver its cargo assuming that the rate of target detection scales as 1 n we show that there exists a well defined mean field limit n rightarrow infty in which the stochastic model reduces to a deterministic system of linear reaction hyperbolic equations for the concentrations of particles in each of the internal states these equations decouple from the stochastic process associated with filling the target with cargo the latter can be modeled as a poisson process in which the time dependent rate of filling lambda t depends on the concentration of stationary particles within the target domain hence we refer to the target as a poisson trap we analyze the efficiency of filling the poisson trap with n particles in terms of the waiting time density f n t the latter is determined by the integrated poisson rate mu t int 0 t lambda s ds which in turn depends on the solution to the reaction hyperbolic equations we obtain an approximate solution for the particle concentrations by reducing the system of reaction hyperbolic equations to a scalar advection diffusion equation using a quasi steady state analysis we compare our analytical results for the mean field model with monte carlo simulations for finite n we thus determine how the mean first passage time mfpt for filling the target depends on n and n

on the field dependence of random walks in the presence of random fields numerical simulations and scaling arguments are used to study the field dependence of a random walk in a one dimensional system with a bias field on each site the bias is taken randomly with equal probability to be e or e the probability density p x t is found to scale asymptotically as left a e beta 2 ln 2 t right exp left left x a e beta 2 ln 2 t right alpha right witha e ln 1 e 1 e 4 25 and 1 25 the mean square displacement scales as langle x 2 rangle sim a e beta f ta beta e where f u ln4u asymptotically

a single predator charging a herd of prey effects of self volume and predator prey decision making we study the degree of success of a single predator hunting a herd of prey on a two dimensional square lattice landscape we explicitly consider the self volume of the prey restraining their dynamics on the lattice the movement of both predator and prey is chosen to include an intelligent decision making step based on their respective sighting ranges the radius in which they can detect the other species prey cannot recognise each other besides the self volume interaction after spotting each other the motion of prey and predator turns from a nearest neighbour random walk into directed escape or chase respectively we consider a large range of prey densities and sighting ranges and compute the mean first passage time for a predator to catch a prey as well as characterise the effective dynamics of the hunted prey we find that the prey s sighting range dominates their life expectancy and the predator profits more from a bad eyesight of the prey than from his own good eye sight we characterise the dynamics in terms of the mean distance between the predator and the nearest prey it turns out that effectively the dynamics of this distance coordinate can be captured in terms of a simple ornstein uhlenbeck picture reducing the many body problem to a simple two body problem by imagining predator and nearest prey to be connected by an effective hookean bond all features of the model such as prey density and sighting ranges merge into the effective binding constant

scaled brownian motion as a mean field model for continuous time random walks we consider scaled brownian motion sbm a random process described by a diffusion equation with explicitly time dependent diffusion coefficient d t d0t 1 batchelor s equation which for 1 is often used for fitting experimental data for subdiffusion of unclear genesis we show that this process is a close relative of subdiffusive continuous time random walks and describes the motion of the rescaled mean position of a cloud of independent walkers it shares with subdiffusive continuous time random walks its nonstationary and nonergodic properties the nonergodicity of sbm does not however go hand in hand with strong difference between its different realizations its heterogeneity ergodicity breaking parameter tends to zero for long trajectories

power spectra and random walks in intermittent chaotic systems abstract we investigate anomalous diffusion and the corresponding power spectra generated by iterated maps and analyze the motion in terms of the probabilistic continous time random walk approach both stationary and non stationary conditions are considered demonstrating the dependence of the mean squared displacement and of the power spectra on the initial conditions the theoretical results are corroborated by numerical calculations and excellent and agreement is obtained

infinite densities for levy walks motion of particles in many systems exhibits a mixture between periods of random diffusive like events and ballistic like motion in many cases such systems exhibit strong anomalous diffusion where low order moments ensuremath langle x t q ensuremath rangle with q below a critical value q c exhibit diffusive scaling while for qg q c a ballistic scaling emerges the mixed dynamics constitutes a theoretical challenge since it does not fall into a unique category of motion e g the known diffusion equations and central limit theorems fail to describe both aspects in this paper we resolve this problem by resorting to the concept of infinite density using the widely applicable l evy walk model we find a general expression for the corresponding non normalized density which is fully determined by the particles velocity distribution the anomalous diffusion exponent ensuremath alpha and the diffusion coefficient k ensuremath alpha we explain how infinite densities play a central role in the description of dynamics of a large class of physical processes and discuss how they can be evaluated from experimental or numerical data

self reproduction in cellular automata abstract self reproduction in cellular automata is discussed with reference to the models of von neumann and codd the conclusion is drawn that although the capacity for universal construction is a sufficient condition for self reproduction it is not a necessary condition slightly more liberal criteria for what constitutes genuine self reproduction are introduced and a simple self reproducing structure is exhibited which satisfies these new criteria this structure achieves its simplicity by storing its description in a dynamic loop rather than on a static tape

the gacs kurdyumov levin automaton revisited we study a one dimensional cellular automaton that was originally proposed as a candidate for exhibiting nonergodic behavior under noise we prove that the deterministic model has the eroder property for two and only two invariant states moreover we give the best possible estimates for the corresponding erosion times we then review the results we have obtained from extensive computer simulations for the stochastic model and for a mixed model and argue that they suggest numerical and heuristic evidence in favour of ergodic behavior for all nonzero values of the noise parameter

time averaged mean square displacement for switching diffusion we consider a classic two state switching diffusion model from a single particle tracking perspective the mean and the variance of the time averaged mean square displacement tamsd are computed exactly when the measurement time i e the trajectory duration is comparable to or smaller than the mean residence times in each state the ergodicity breaking parameter is shown to take arbitrarily large values suggesting an apparent weak ergodicity breaking for this ergodic model in this regime individual random trajectories are not representative while the related tamsd curves exhibit a broad spread in agreement with experimental observations in living cells and complex fluids switching diffusions can thus present in some cases an ergodic alternative to commonly used and inherently non ergodic continuous time random walks that capture similar features

ageing first passage time density in continuous time random walks and quenched energy landscapes we study the first passage dynamics of an ageing stochastic process in the continuous time random walk ctrw framework in such ctrw processes the test particle performs a random walk in which successive steps are separated by random waiting times distributed in terms of the waiting time probability density function an ageing stochastic process is defined by the explicit dependence of its dynamic quantities on the ageing time ta the time elapsed between its preparation and the start of the observation subdiffusive ageing ctrws with describe systems such as charge carriers in amorphous semiconducters tracer dispersion in geological and biological systems or the dynamics of blinking quantum dots we derive the exact forms of the first passage time density for an ageing subdiffusive ctrw in the semi infinite confined and biased case finding different scaling regimes for weakly intermediately and strongly aged systems these regimes with different scaling laws are also found when the scaling exponent is in the range for sufficiently long ta we compare our results with the ageing motion of a test particle in a quenched energy landscape we test our theoretical results in the quenched landscape against simulations only when the bias is strong enough the correlations from returning to previously visited sites become insignificant and the results approach the ageing ctrw results with small bias or without bias the ageing effects disappear and a change in the exponent compared to the case of a completely annealed landscape can be found reflecting the build up of correlations in the quenched landscape

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

leapover lengths and first passage time statistics for levy flights exact results for the first passage time and leapover statistics of symmetric and one sided levy flights lfs are derived lfs with a stable index alpha are shown to have leapover lengths that are asymptotically power law distributed with an index alpha for one sided lfs and surprisingly with an index alpha 2 for symmetric lfs the first passage time distribution scales like a power law with an index 1 2 as required by the sparre andersen theorem for symmetric lfs whereas one sided lfs have a narrow distribution of first passage times the exact analytic results are confirmed by extensive simulations

yangian symmetry in the nonlinear schrodinger hierarchy we study the yangian symmetry of the multicomponent quantum nonlinear schrodinger hierarchy in the framework of the quantum inverse scattering method we give an explicit realization of the yangian generators in terms of the deformed oscillators algebra which naturally occurs in this framework

parameter characterization of two dimensional cellular automata rule space cellular automata ca are discrete dynamical systems formed by a great number of identical and simple components with local connectivity it has been proved that the problem of forecasting the dynamical behavior of ca is undecidable based upon parameterizations of ca rule space several approximations have been investigated the majority of these studies are focused on the one dimensional ca the present work generalizes for the two dimensional space the definition of three parameters previously published in the one dimensional context sensitivity neighborhood dominance and activity propagation as an example of the application of such generalized parameters some simulations about the computational task known as the density classification task are reported

self similar gaussian processes for modeling anomalous diffusion we study some gaussian models for anomalous diffusion which include the time rescaled brownian motion two types of fractional brownian motion and models associated with fractional brownian motion based on the generalized langevin equation gaussian processes associated with these models satisfy the anomalous diffusion relation which requires the mean square displacement to vary with t ensuremath alpha 0l ensuremath alpha l2 however these processes have different properties thus indicating that the anomalous diffusion relation with a single parameter is insufficient to characterize the underlying mechanism although the two versions of fractional brownian motion and time rescaled brownian motion all have the same probability distribution function the slepian theorem can be used to compare their first passage time distributions which are different finally in order to model anomalous diffusion with a variable exponent ensuremath alpha t it is necessary to consider the multifractional extensions of these gaussian processes

random walkers in one dimensional random environments exact renormalization group analysis sinai s model of diffusion in one dimension with random local bias is studied by a real space renormalization group which yields exact results at long times the effects of an additional small uniform bias force are also studied we obtain analytically the scaling form of the distribution of the position x t of a particle the probability of it not returning to the origin and the distributions of first passage times in an infinite sample as well as in the presence of a boundary and in a finite but large sample we compute the distribution of the meeting time of two particles in the same environment we also obtain a detailed analytic description of the thermally averaged trajectories by computing quantities such as the joint distribution of the number of returns and of the number of jumps forward these quantities obey multifractal scaling characterized by generalized persistence exponents ensuremath theta g which we compute in the presence of a small bias the number of returns to the origin becomes finite characterized by a universal scaling function which we obtain the full statistics of the distribution of successive times of return of thermally averaged trajectories is obtained as well as detailed analytical information about correlations between directions and times of successive jumps the two time distribution of the positions of a particle x t and x t ensuremath with tg t ensuremath is also computed exactly it is found to exhibit aging with several time regimes characterized by different behaviors in the unbiased case for t ensuremath t ensuremath ensuremath sim t ensuremath ensuremath alpha with ensuremath alpha g1 it exhibits a mathrm ln t mathrm ln t ensuremath scaling with a singularity at coinciding rescaled positions x t x t ensuremath this singularity is a novel feature and corresponds to particles that remain in a renormalized valley for closer times ensuremath alpha l1 the two time diffusion front exhibits a quasiequilibrium regime with a mathrm ln t ensuremath t ensuremath mathrm ln t ensuremath behavior which we compute the crossover to a t t ensuremath aging form in the presence of a small bias is also obtained analytically rare events corresponding to intermittent splitting of the thermal packet between separated wells which dominate some averaged observables are also characterized in detail connections with the green function of a one dimensional schr odinger problem and quantum spin chains are discussed

quantum inverse scattering method for multicomponent non linear schrodinger model of bosons or fermions with repulsive coupling the generalisation of the quantum inverse scattering method to the study of direct and inverse problems for the multicomponent non linear schrodinger model of bosons or fermions with repulsive coupling is made two sets of yang baxter equations are solved to obtain commutation relations between the scattering state operators the eigenfunctions have been constructed for the infinite number of conserved quantities and the eigenvalues of the first three conserved quantities number of particles momentum and energy are obtained the global izergin korepin relations and the relations between the quantum jost functions are derived and from them the quantum gel fand levitan equations are established finally the series expansion for the field operators in terms of the scattering state operators is written out explicitly