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Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories
(2021)
Extensive time-series encoding the position of particles such as viruses, vesicles, or individualproteins are routinely garnered insingle-particle tracking experiments or supercomputing studies.They contain vital clues on how viruses spread or drugs may be delivered in biological cells.Similar time-series are being recorded of stock values in financial markets and of climate data.Such time-series are most typically evaluated in terms of time-averaged mean-squareddisplacements (TAMSDs), which remain random variables for finite measurement times. Theirstatistical properties are different for differentphysical stochastic processes, thus allowing us toextract valuable information on the stochastic process itself. To exploit the full potential of thestatistical information encoded in measured time-series we here propose an easy-to-implementand computationally inexpensive new methodology, based on deviations of the TAMSD from itsensemble average counterpart. Specifically, we use the upper bound of these deviations forBrownian motion (BM) to check the applicability of this approach to simulated and real data sets.By comparing the probability of deviations fordifferent data sets, we demonstrate how thetheoretical bound for BM reveals additional information about observed stochastic processes. Weapply the large-deviation method to data sets of tracer beads tracked in aqueous solution, tracerbeads measured in mucin hydrogels, and of geographic surface temperature anomalies. Ouranalysis shows how the large-deviation properties can be efficiently used as a simple yet effectiveroutine test to reject the BM hypothesis and unveil relevant information on statistical propertiessuch as ergodicity breaking and short-time correlations.
A rapidly increasing number of systems is identified in which the stochastic motion of tracer particles follows the Brownian law < r(2)(t)> similar or equal to Dt yet the distribution of particle displacements is strongly non-Gaussian. A central approach to describe this effect is the diffusing diffusivity (DD) model in which the diffusion coefficient itself is a stochastic quantity, mimicking heterogeneities of the environment encountered by the tracer particle on its path. We here quantify in terms of analytical and numerical approaches the first passage behaviour of the DD model. We observe significant modifications compared to Brownian-Gaussian diffusion, in particular that the DD model may have a faster first passage dynamics. Moreover we find a universal crossover point of the survival probability independent of the initial condition.
Leveraging large-deviation statistics to decipher the stochastic properties of measured trajectories
(2021)
Extensive time-series encoding the position of particles such as viruses, vesicles, or individualproteins are routinely garnered insingle-particle tracking experiments or supercomputing studies.They contain vital clues on how viruses spread or drugs may be delivered in biological cells.Similar time-series are being recorded of stock values in financial markets and of climate data.Such time-series are most typically evaluated in terms of time-averaged mean-squareddisplacements (TAMSDs), which remain random variables for finite measurement times. Theirstatistical properties are different for differentphysical stochastic processes, thus allowing us toextract valuable information on the stochastic process itself. To exploit the full potential of thestatistical information encoded in measured time-series we here propose an easy-to-implementand computationally inexpensive new methodology, based on deviations of the TAMSD from itsensemble average counterpart. Specifically, we use the upper bound of these deviations forBrownian motion (BM) to check the applicability of this approach to simulated and real data sets.By comparing the probability of deviations fordifferent data sets, we demonstrate how thetheoretical bound for BM reveals additional information about observed stochastic processes. Weapply the large-deviation method to data sets of tracer beads tracked in aqueous solution, tracerbeads measured in mucin hydrogels, and of geographic surface temperature anomalies. Ouranalysis shows how the large-deviation properties can be efficiently used as a simple yet effectiveroutine test to reject the BM hypothesis and unveil relevant information on statistical propertiessuch as ergodicity breaking and short-time correlations.
Based on the space-fractional Fokker-Planck equation with a delta-sink term, we study the efficiency of random search processes based on Levy flights with power-law distributed jump lengths in the presence of an external drift, for instance, an underwater current, an airflow, or simply the preference of the searcher based on prior experience. While Levy flights turn out to be efficient search processes when the target is upstream relative to the starting point, in the downstream scenario, regular Brownian motion turns out to be advantageous. This is caused by the occurrence of leapovers of Levy flights, due to which Levy flights typically overshoot a point or small interval. Studying the solution of the fractional Fokker-Planck equation, we establish criteria when the combination of the external stream and the initial distance between the starting point and the target favours Levy flights over the regular Brownian search. Contrary to the common belief that Levy flights with a Levy index alpha = 1 (i.e. Cauchy flights) are optimal for sparse targets, we find that the optimal value for alpha may range in the entire interval (1, 2) and explicitly include Brownian motion as the most efficient search strategy overall.
We study transient work fluctuation relations (FRs) for Gaussian stochastic systems generating anomalous diffusion. For this purpose we use a Langevin approach by employing two different types of additive noise: (i) internal noise where the fluctuation dissipation relation of the second kind (FDR II) holds, and (ii) external noise without FDR II. For internal noise we demonstrate that the existence of FDR II implies the existence of the fluctuation dissipation relation of the first kind (FDR I), which in turn leads to conventional (normal) forms of transient work FRs. For systems driven by external noise we obtain violations of normal FRs, which we call anomalous FRs. We derive them in the long-time limit and demonstrate the existence of logarithmic factors in FRs for intermediate times. We also outline possible experimental verifications.
We introduce three strategies for the analysis of financial time series based on time averaged observables. These comprise the time averaged mean squared displacement (MSD) as well as the ageing and delay time methods for varying fractions of the financial time series. We explore these concepts via statistical analysis of historic time series for several Dow Jones Industrial indices for the period from the 1960s to 2015. Remarkably, we discover a simple universal law for the delay time averaged MSD. The observed features of the financial time series dynamics agree well with our analytical results for the time averaged measurables for geometric Brownian motion, underlying the famed Black–Scholes–Merton model. The concepts we promote here are shown to be useful for financial data analysis and enable one to unveil new universal features of stock market dynamics.
We introduce three strategies for the analysis of financial time series based on time averaged observables. These comprise the time averaged mean squared displacement (MSD) as well as the ageing and delay time methods for varying fractions of the financial time series. We explore these concepts via statistical analysis of historic time series for several Dow Jones Industrial indices for the period from the 1960s to 2015. Remarkably, we discover a simple universal law for the delay time averaged MSD. The observed features of the financial time series dynamics agree well with our analytical results for the time averaged measurables for geometric Brownian motion, underlying the famed Black–Scholes–Merton model. The concepts we promote here are shown to be useful for financial data analysis and enable one to unveil new universal features of stock market dynamics.
We introduce three strategies for the analysis of financial time series based on time averaged observables. These comprise the time averaged mean squared displacement (MSD) as well as the ageing and delay time methods for varying fractions of the financial time series. We explore these concepts via statistical analysis of historic time series for several Dow Jones Industrial indices for the period from the 1960s to 2015. Remarkably, we discover a simple universal law for the delay time averaged MSD. The observed features of the financial time series dynamics agree well with our analytical results for the time averaged measurables for geometric Brownian motion, underlying the famed Black-Scholes-Merton model. The concepts we promote here are shown to be useful for financial data analysis and enable one to unveil new universal features of stock market dynamics.
The passive and active motion of micron-sized tracer particles in crowded liquids and inside living biological cells is ubiquitously characterised by 'viscoelastic' anomalous diffusion, in which the increments of the motion feature long-ranged negative and positive correlations. While viscoelastic anomalous diffusion is typically modelled by a Gaussian process with correlated increments, so-called fractional Gaussian noise, an increasing number of systems are reported, in which viscoelastic anomalous diffusion is paired with non-Gaussian displacement distributions. Following recent advances in Brownian yet non-Gaussian diffusion we here introduce and discuss several possible versions of random-diffusivity models with long-ranged correlations. While all these models show a crossover from non-Gaussian to Gaussian distributions beyond some correlation time, their mean squared displacements exhibit strikingly different behaviours: depending on the model crossovers from anomalous to normal diffusion are observed, as well as a priori unexpected dependencies of the effective diffusion coefficient on the correlation exponent. Our observations of the non-universality of random-diffusivity viscoelastic anomalous diffusion are important for the analysis of experiments and a better understanding of the physical origins of 'viscoelastic yet non-Gaussian' diffusion.
The passive and active motion of micron-sized tracer particles in crowded liquids and inside living biological cells is ubiquitously characterised by 'viscoelastic' anomalous diffusion, in which the increments of the motion feature long-ranged negative and positive correlations. While viscoelastic anomalous diffusion is typically modelled by a Gaussian process with correlated increments, so-called fractional Gaussian noise, an increasing number of systems are reported, in which viscoelastic anomalous diffusion is paired with non-Gaussian displacement distributions. Following recent advances in Brownian yet non-Gaussian diffusion we here introduce and discuss several possible versions of random-diffusivity models with long-ranged correlations. While all these models show a crossover from non-Gaussian to Gaussian distributions beyond some correlation time, their mean squared displacements exhibit strikingly different behaviours: depending on the model crossovers from anomalous to normal diffusion are observed, as well as a priori unexpected dependencies of the effective diffusion coefficient on the correlation exponent. Our observations of the non-universality of random-diffusivity viscoelastic anomalous diffusion are important for the analysis of experiments and a better understanding of the physical origins of 'viscoelastic yet non-Gaussian' diffusion.