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

anomalous diffusion in correlated continuous time random walks we demonstrate that continuous time random walks in which successive waiting times are correlated by gaussian statistics lead to anomalous diffusion with the mean squared displacement r2 t t2 3 long ranged correlations of the waiting times with a power law exponent 0 2 give rise to subdiffusion of the form r2 t t 1 in contrast correlations in the jump lengths are shown to produce superdiffusion we show that in both cases weak ergodicity breaking occurs our results are in excellent agreement with simulations

optimal estimates of the diffusion coefficient of a single brownian trajectory modern developments in microscopy and image processing are revolutionizing areas of physics chemistry and biology as nanoscale objects can be tracked with unprecedented accuracy the goal of single particle tracking is to determine the interaction between the particle and its environment the price paid for having a direct visualization of a single particle is a consequent lack of statistics here we address the optimal way of extracting diffusion constants from single trajectories for pure brownian motion it is shown that the maximum likelihood estimator is much more efficient than the commonly used least squares estimate furthermore we investigate the effect of disorder on the distribution of estimated diffusion constants and show that it increases the probability of observing estimates much smaller than the true average value

one dimensional hard rod caricature of hydrodynamics navier stokes correction for local equilibrium initial states a one dimensional system consisting of identical hard rod particles of length a is studied in the hydrodynamical limit a navier stokes correction to the euler equation is found for an initial local equilibrium family of states of constant density the correction is given at by a non linear second order differential operator acting on f q v the hydrodynamical density at a point of the species of fluid with velocity

intrinsic randomness of transport coefficient in subdiffusion with static disorder fluctuations in the time averaged mean square displacement for random walks on hypercubic lattices with static disorder are investigated it is analytically shown that the diffusion coefficient becomes a random variable as a manifestation of weak ergodicity breaking for two and higher dimensional systems the distribution function of the diffusion coefficient is found to be the mittag leffler distribution which is the same as for the continuous time random walk whereas for one dimensional systems a different distribution a modified mittag leffler distribution arises we also present a comparison of these two distributions in terms of an ergodicity breaking parameter and show that the modified mittag leffler distribution has a larger deviation from ergodicity some remarks on similarities between these results and observations in biological experiments are presented

a new class of long tailed pausing time densities for the ctrw we present some asymptotic results for the family of pausing time densities having the asymptotic t property t t ln1 t t 1 in particular we show that for this class of pausing time densities the mean squared displacement r2 t is asymptotically proportional to ln t t and the asymptotic distribution of the displacement has a negative exponential form

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

continuous time random walk with a superheavy tailed distribution of waiting times we study the long time behavior of the probability density associated with the decoupled continuous time random walk which is characterized by a superheavy tailed distribution of waiting times it is shown that if the random walk is unbiased biased and the jump distribution has a finite second moment then the properly scaled probability density converges in the long time limit to a symmetric two sided an asymmetric one sided exponential density the convergence occurs in such a way that all the moments of the probability density grow slower than any power of time as a consequence the reference random walk can be viewed as a generic model of superslow diffusion a few examples of superheavy tailed distributions of waiting times that give rise to qualitatively different laws of superslow diffusion are considered

asymptotic solutions of decoupled continuous time random walks with superheavy tailed waiting time and heavy tailed jump length distributions we study the long time behavior of decoupled continuous time random walks characterized by superheavy tailed distributions of waiting times and symmetric heavy tailed distributions of jump lengths our main quantity of interest is the limiting probability density of the position of the walker multiplied by a scaling function of time we show that the probability density of the scaled walker position converges in the long time limit to a nondegenerate one only if the scaling function behaves in a certain way this function as well as the limiting probability density are determined in explicit form also we express the limiting probability density which has heavy tails in terms of the fox h function and find its behavior for small and large distances

localization of random walks in one dimensional random environments we consider a random walk on the one dimensional semi lattice 0 1 2 we prove that the moving particle walks mainly in a finite neighbourhood of a point depending only on time and a realization of the random environment the size of this neighbourhood is estimated the limit parameters of the walks are also determined

weakly non ergodic statistical physics for weakly non ergodic systems the probability density function of a time average observable overline mathcal o is f alpha overline mathcal o 1 over pi lim epsilon to 0 mbox im sum j 1 l p mathrm eq j overline mathcal o mathcal o j i epsilon alpha 1 over sum j 1 l p mathrm eq j overline mathcal o mathcal o j i epsilon alpha where mathcal o j is the value of the observable when the system is in state j 1 l pjeq is the probability that a member of an ensemble of systems occupies state j in equilibrium for a particle undergoing a fractional diffusion process in a binding force field with thermal detailed balance conditions pjeq is boltzmann s canonical probability within the unbiased sub diffusive continuous time random walk model the exponent 0 1 is the anomalous diffusion exponent x2 t found for free boundary conditions when 1 ergodic statistical mechanics is recovered lim alpha to 1 f alpha overline mathcal o delta overline mathcal o langle mathcal o rangle we briefly discuss possible physical applications in single particle experiments

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 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

anomalous diffusion testing ergodicity breaking in experimental data recent advances in single molecule experiments show that various complex systems display nonergodic behavior in this paper we show how to test ergodicity and ergodicity breaking in experimental data exploiting the so called dynamical functional we introduce a simple test which allows us to verify ergodic properties of a real life process the test can be applied to a large family of stationary infinitely divisible processes we check the performance of the test for various simulated processes and apply it to experimental data describing the motion of mrna molecules inside live escherichia coli cells we show that the data satisfy necessary conditions for mixing and ergodicity the detailed analysis is presented in the supplementary material

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

compensating asynchrony effects in the calculation of financial correlations abstract we present a method to compensate statistical errors in the calculation of correlations on asynchronous time series the method is based on the assumption of an underlying time series we set up a model and apply it to financial data to examine the decrease of calculated correlations towards smaller return intervals epps effect we show that the discovered statistical effect is a major cause of the epps effect hence we are able to quantify and to compensate it using only trading prices and trading times

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

exact results on sinai s diffusion we study the continuum version of sinai s problem of a random walker in a random force field in one dimension a method of stochastic representation is used to represent various probability distributions in this problem mean probability density function and first passage time distributions this method reproduces already known rigorous results and also confirms directly some recent results derived using approximation schemes we demonstrate clearly in the sinai scaling regime that the disorder dominates the problem and that the thermal distributions tend to zero one laws

the limit distribution of sinai s random walk in random environment abstract recently sinai proved that if x n is a one dimensional random walk in random environment which is recurrent then log n 2 x n converges in distribution here we calculate the limit distribution explicitly

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

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