Any real physical process that produces entropy, dissipates energy as heat, or generates mechanic... more Any real physical process that produces entropy, dissipates energy as heat, or generates mechanical work must do so on a finite timescale. Recently derived thermodynamic speed limits place bounds on these observables using intrinsic timescales of the process. Here, we derive relationships for the thermodynamic speeds for any composite stochastic observable in terms of the timescales of its individual components. From these speed limits, we find bounds on thermal efficiency of stochastic processes exchanging energy as heat and work and bound the rate of entropy change in a system with entropy production and flow. Using the time set by an external clock, we find bounds on the first time to reach any value for the entropy production. As an illustration, we compute these bounds for Brownian particles diffusing in space subject to a constant-temperature heat bath and a time-dependent external force.
Journal of Physics A: Mathematical and Theoretical
Thermodynamic speed limits are a set of classical uncertainty relations that, so far, place globa... more Thermodynamic speed limits are a set of classical uncertainty relations that, so far, place global bounds on the stochastic dissipation of energy as heat and the production of entropy. Here, instead of constraints on these thermodynamic costs, we derive integral speed limits that are upper and lower bounds on a thermodynamic benefit—the minimum time for an amount of mechanical work to be done on or by a system. In the short time limit, we show how this extrinsic timescale relates to an intrinsic timescale for work, recovering the intrinsic timescales in differential speed limits from these integral speed limits and turning the first law of stochastic thermodynamics into a first law of speeds. As physical examples, we consider the work done by a flashing Brownian ratchet and the work done on a particle in a potential well subject to external driving.
Journal of Physics A: Mathematical and Theoretical
The field of movement ecology has seen a rapid increase in high-resolution data in recent years, ... more The field of movement ecology has seen a rapid increase in high-resolution data in recent years, leading to the development of numerous statistical and numerical methods to analyse relocation trajectories. Data are often collected at the level of the individual and for long periods that may encompass a range of behaviours. Here, we use the power spectral density (PSD) to characterise the random movement patterns of a black-winged kite (Elanus caeruleus) and a white stork (Ciconia ciconia). The tracks are first segmented and clustered into different behaviours (movement modes), and for each mode we measure the PSD and the ageing properties of the process. For the foraging kite we find 1/f noise, previously reported in ecological systems mainly in the context of population dynamics, but not for movement data. We further suggest plausible models for each of the behavioural modes by comparing both the measured PSD exponents and the distribution of the single-trajectory PSD to known theo...
We perform numerical studies of a thermally driven, overdamped particle in a random quenched forc... more We perform numerical studies of a thermally driven, overdamped particle in a random quenched force field, known as the Sinai model. We compare the unbounded motion on an infinite 1-dimensional domain to the motion in bounded domains with reflecting boundaries and show that the unbounded motion is at every time close to the equilibrium state of a finite system of growing size. This is due to time scale separation: inside wells of the random potential, there is relatively fast equilibration, while the motion across major potential barriers is ultraslow. Quantities studied by us are the time dependent mean squared displacement, the time dependent mean energy of an ensemble of particles, and the time dependent entropy of the probability distribution. Using a very fast numerical algorithm, we can explore times up top 1017 steps and thereby also study finite-time crossover phenomena.
Journal of Physics A: Mathematical and Theoretical
Recently, a large number of research teams from around the world collaborated in the so-called ‘a... more Recently, a large number of research teams from around the world collaborated in the so-called ‘anomalous diffusion challenge’. Its aim: to develop and compare new techniques for inferring stochastic models from given unknown time series, and estimate the anomalous diffusion exponent in data. We use various numerical methods to directly obtain this exponent using the path increments, and develop a questionnaire for model selection based on feature analysis of a set of known stochastic processes given as candidates. Here, we present the theoretical background of the automated algorithm which we put for these tasks in the diffusion challenge, as a counter to other pure data-driven approaches.
We provide a reply to a comment by I. Goychuk arXiv:1708.04155, version v2 from 27 Aug 2017 (not ... more We provide a reply to a comment by I. Goychuk arXiv:1708.04155, version v2 from 27 Aug 2017 (not under active consideration with Phys. Rev. Lett.), on our Letter E. Aghion, D. A. Kessler and E. Barkai, Phys. Rev. Lett., 118, 260601 (2017).
Various mathematical Black-Scholes-Merton-like models of option pricing employ the paradigmatic s... more Various mathematical Black-Scholes-Merton-like models of option pricing employ the paradigmatic stochastic process of geometric Brownian motion (GBM). The innate property of such models and of real stock-market prices is the roughly exponential growth of prices with time [on average, in crisis-free times]. We here explore the ensemble-and time averages of a multiplicative-noise stochastic process with power-law-like time-dependent volatility, σ (t) ∼ t α , named scaled GBM (SGBM). For SGBM, the mean-squared displacement (MSD) computed for an ensemble of statistically equivalent trajectories can grow faster than exponentially in time, while the time-averaged MSD (TAMSD)-based on a sliding-window averaging along a single trajectory-is always linear at short lag times. The proportionality factor between these the two averages of the time series is /T at short lag times, where T is the trajectory length, similarly to GBM. This discrepancy of the scaling relations and pronounced nonequivalence of the MSD and TAMSD at /T 1 is a manifestation of weak ergodicity breaking for standard GBM and for SGBM with σ (t)-modulation, the main focus of our analysis. The analytical predictions for the MSD and mean TAMSD for SGBM are in quantitative agreement with the results of stochastic computer simulations.
We study a method for detecting the origins of anomalous diffusion, when it is observed in an ens... more We study a method for detecting the origins of anomalous diffusion, when it is observed in an ensemble of times-series, generated experimentally or numerically, without having knowledge about the exact underlying dynamics. The reasons for anomalous diffusive scaling of the mean-squared displacement are decomposed into three root causes: increment correlations are expressed by the ‘Joseph effect’ (Mandelbrot and Wallis 1968 Water Resour. Res. 4 909), fat-tails of the increment probability density lead to a ‘Noah effect’ (Mandelbrot and Wallis 1968 Water Resour. Res. 4 909), and non-stationarity, to the ‘Moses effect’ (Chen et al 2017 Phys. Rev. E 95 042141). After appropriate rescaling, based on the quantification of these effects, the increment distribution converges at increasing times to a time-invariant asymptotic shape. For different processes, this asymptotic limit can be an equilibrium state, an infinite-invariant, or an infinite-covariant density. We use numerical methods of ...
We analyze the citation time-series of manuscripts in three different fields of science; physics,... more We analyze the citation time-series of manuscripts in three different fields of science; physics, social science and technology. The evolution of the time-series of the yearly number of citations, namely the citation trajectories, diffuse anomalously, their variance scales with time ∝t 2H , where H ≠ 1/2. We provide detailed analysis of the various factors that lead to the anomalous behavior: non-stationarity, long-ranged correlations and a fat-tailed increment distribution. The papers exhibit a high degree of heterogeneity across the various fields, as the statistics of the highest cited papers is fundamentally different from that of the lower ones. The citation data is shown to be highly correlated and non-stationary; as all the papers except the small percentage of them with high number of citations, die out in time.
We examine the bi-scaling behavior of Lévy walks with nonlinear coupling, where , the particle di... more We examine the bi-scaling behavior of Lévy walks with nonlinear coupling, where , the particle displacement during each step, is coupled to the duration of the step, ⌧ , by ⇠ ⌧. An example of such a process is regular Lévy walks, where = 1. In recent years such processes were shown to be highly useful for analysis of a class of Langevin dynamics, in particular a system of Sisyphus laser-cooled atoms in an optical lattice, where = 3/2. We discuss the well-known decoupling approximation used to describe the central part of the particles' position distribution, and use the recently introduced infinite-covariant density approach to study the large fluctuations. Since the density of the step displacements is fat-tailed, the last travel event must be treated with care for the latter. This e↵ect requires a modification of the Montroll-Weiss equation, an equation which has proved important for the analysis of many microscopic models.
We study a particle immersed in a heat bath, in the presence of an external force which decays at... more We study a particle immersed in a heat bath, in the presence of an external force which decays at least as rapidly as 1=x, e.g., a particle interacting with a surface through a Lennard-Jones or a logarithmic potential. As time increases, our system approaches a non-normalizable Boltzmann state. We study observables, such as the energy, which are integrable with respect to this asymptotic thermal state, calculating both time and ensemble averages. We derive a useful canonical-like ensemble which is defined out of equilibrium, using a maximum entropy principle, where the constraints are normalization, finite averaged energy, and a meansquared displacement which increases linearly with time. Our work merges infinite-ergodic theory with Boltzmann-Gibbs statistics, thus extending the scope of the latter while shedding new light on the concept of ergodicity.
We introduce three strategies for the analysis of financial time series based on time averaged ob... more 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.
Large-deviations theory deals with tails of probability distributions and the rare events of rand... more Large-deviations theory deals with tails of probability distributions and the rare events of random processes, for example spreading packets of particles. Mathematically, it concerns the exponential fall-of of the density of thin-tailed systems. Here we investigate the spatial density Pt(x) of laser cooled atoms, where at intermediate length scales the shape is fat-tailed. We focus on the rare events beyond this range, which dominate important statistical properties of the system. Through a novel friction mechanism induced by the laser fields, the density is explored with the recently proposed nonnormalized infinite-covariant density approach. The small and large fluctuations give rise to a bi-fractal nature of the spreading packet.
We present an exact solution for the distribution of sample averaged monomer to monomer distance ... more We present an exact solution for the distribution of sample averaged monomer to monomer distance of ring polymers. For non-interacting and local-interaction models these distributions correspond to the distribution of the area under the reflected Bessel bridge and the Bessel excursion respectively, and are shown to be identical in dimension d ≥ 2, albeit with pronounced finite size effects at the critical dimension, d = 2. A symmetry of the problem reveals that dimension d and 4 - d are equivalent, thus the celebrated Airy distribution describing the areal distribution of the d = 1 Brownian excursion describes also a polymer in three dimensions. For a self-avoiding polymer in dimension d we find numerically that the fluctuations of the scaled averaged distance are nearly identical in dimension d = 2, 3 and are well described to a first approximation by the non-interacting excursion model in dimension 5.
We investigate the overdamped stochastic dynamics of a particle in an asymptotically flat externa... more We investigate the overdamped stochastic dynamics of a particle in an asymptotically flat external potential field, in contact with a thermal bath. For an infinite system size, the particles may escape the force field and diffuse freely at large length scales. The partition function diverges and hence the standard canonical ensemble fails. This is replaced with tools stemming from infinite ergodic theory. Boltzmann-Gibbs statistics, even though not normalized, still describes integrable observables, like energy and occupation times. The Boltzmann infinite density is derived heuristically using an entropy maximization principle, as well as via a first-principles calculation using an eigenfunction expansion in the continuum of low-energy states. A generalized virial theorem is derived, showing how the virial coefficient describes the delay in the diffusive spreading of the particles, found at large distances. When the process is non-recurrent, e.g. diffusion in three dimensions with a Coulomb-like potential, we use weighted time averages to restore basic canonical relations between time and ensemble averages.
Any real physical process that produces entropy, dissipates energy as heat, or generates mechanic... more Any real physical process that produces entropy, dissipates energy as heat, or generates mechanical work must do so on a finite timescale. Recently derived thermodynamic speed limits place bounds on these observables using intrinsic timescales of the process. Here, we derive relationships for the thermodynamic speeds for any composite stochastic observable in terms of the timescales of its individual components. From these speed limits, we find bounds on thermal efficiency of stochastic processes exchanging energy as heat and work and bound the rate of entropy change in a system with entropy production and flow. Using the time set by an external clock, we find bounds on the first time to reach any value for the entropy production. As an illustration, we compute these bounds for Brownian particles diffusing in space subject to a constant-temperature heat bath and a time-dependent external force.
Journal of Physics A: Mathematical and Theoretical
Thermodynamic speed limits are a set of classical uncertainty relations that, so far, place globa... more Thermodynamic speed limits are a set of classical uncertainty relations that, so far, place global bounds on the stochastic dissipation of energy as heat and the production of entropy. Here, instead of constraints on these thermodynamic costs, we derive integral speed limits that are upper and lower bounds on a thermodynamic benefit—the minimum time for an amount of mechanical work to be done on or by a system. In the short time limit, we show how this extrinsic timescale relates to an intrinsic timescale for work, recovering the intrinsic timescales in differential speed limits from these integral speed limits and turning the first law of stochastic thermodynamics into a first law of speeds. As physical examples, we consider the work done by a flashing Brownian ratchet and the work done on a particle in a potential well subject to external driving.
Journal of Physics A: Mathematical and Theoretical
The field of movement ecology has seen a rapid increase in high-resolution data in recent years, ... more The field of movement ecology has seen a rapid increase in high-resolution data in recent years, leading to the development of numerous statistical and numerical methods to analyse relocation trajectories. Data are often collected at the level of the individual and for long periods that may encompass a range of behaviours. Here, we use the power spectral density (PSD) to characterise the random movement patterns of a black-winged kite (Elanus caeruleus) and a white stork (Ciconia ciconia). The tracks are first segmented and clustered into different behaviours (movement modes), and for each mode we measure the PSD and the ageing properties of the process. For the foraging kite we find 1/f noise, previously reported in ecological systems mainly in the context of population dynamics, but not for movement data. We further suggest plausible models for each of the behavioural modes by comparing both the measured PSD exponents and the distribution of the single-trajectory PSD to known theo...
We perform numerical studies of a thermally driven, overdamped particle in a random quenched forc... more We perform numerical studies of a thermally driven, overdamped particle in a random quenched force field, known as the Sinai model. We compare the unbounded motion on an infinite 1-dimensional domain to the motion in bounded domains with reflecting boundaries and show that the unbounded motion is at every time close to the equilibrium state of a finite system of growing size. This is due to time scale separation: inside wells of the random potential, there is relatively fast equilibration, while the motion across major potential barriers is ultraslow. Quantities studied by us are the time dependent mean squared displacement, the time dependent mean energy of an ensemble of particles, and the time dependent entropy of the probability distribution. Using a very fast numerical algorithm, we can explore times up top 1017 steps and thereby also study finite-time crossover phenomena.
Journal of Physics A: Mathematical and Theoretical
Recently, a large number of research teams from around the world collaborated in the so-called ‘a... more Recently, a large number of research teams from around the world collaborated in the so-called ‘anomalous diffusion challenge’. Its aim: to develop and compare new techniques for inferring stochastic models from given unknown time series, and estimate the anomalous diffusion exponent in data. We use various numerical methods to directly obtain this exponent using the path increments, and develop a questionnaire for model selection based on feature analysis of a set of known stochastic processes given as candidates. Here, we present the theoretical background of the automated algorithm which we put for these tasks in the diffusion challenge, as a counter to other pure data-driven approaches.
We provide a reply to a comment by I. Goychuk arXiv:1708.04155, version v2 from 27 Aug 2017 (not ... more We provide a reply to a comment by I. Goychuk arXiv:1708.04155, version v2 from 27 Aug 2017 (not under active consideration with Phys. Rev. Lett.), on our Letter E. Aghion, D. A. Kessler and E. Barkai, Phys. Rev. Lett., 118, 260601 (2017).
Various mathematical Black-Scholes-Merton-like models of option pricing employ the paradigmatic s... more Various mathematical Black-Scholes-Merton-like models of option pricing employ the paradigmatic stochastic process of geometric Brownian motion (GBM). The innate property of such models and of real stock-market prices is the roughly exponential growth of prices with time [on average, in crisis-free times]. We here explore the ensemble-and time averages of a multiplicative-noise stochastic process with power-law-like time-dependent volatility, σ (t) ∼ t α , named scaled GBM (SGBM). For SGBM, the mean-squared displacement (MSD) computed for an ensemble of statistically equivalent trajectories can grow faster than exponentially in time, while the time-averaged MSD (TAMSD)-based on a sliding-window averaging along a single trajectory-is always linear at short lag times. The proportionality factor between these the two averages of the time series is /T at short lag times, where T is the trajectory length, similarly to GBM. This discrepancy of the scaling relations and pronounced nonequivalence of the MSD and TAMSD at /T 1 is a manifestation of weak ergodicity breaking for standard GBM and for SGBM with σ (t)-modulation, the main focus of our analysis. The analytical predictions for the MSD and mean TAMSD for SGBM are in quantitative agreement with the results of stochastic computer simulations.
We study a method for detecting the origins of anomalous diffusion, when it is observed in an ens... more We study a method for detecting the origins of anomalous diffusion, when it is observed in an ensemble of times-series, generated experimentally or numerically, without having knowledge about the exact underlying dynamics. The reasons for anomalous diffusive scaling of the mean-squared displacement are decomposed into three root causes: increment correlations are expressed by the ‘Joseph effect’ (Mandelbrot and Wallis 1968 Water Resour. Res. 4 909), fat-tails of the increment probability density lead to a ‘Noah effect’ (Mandelbrot and Wallis 1968 Water Resour. Res. 4 909), and non-stationarity, to the ‘Moses effect’ (Chen et al 2017 Phys. Rev. E 95 042141). After appropriate rescaling, based on the quantification of these effects, the increment distribution converges at increasing times to a time-invariant asymptotic shape. For different processes, this asymptotic limit can be an equilibrium state, an infinite-invariant, or an infinite-covariant density. We use numerical methods of ...
We analyze the citation time-series of manuscripts in three different fields of science; physics,... more We analyze the citation time-series of manuscripts in three different fields of science; physics, social science and technology. The evolution of the time-series of the yearly number of citations, namely the citation trajectories, diffuse anomalously, their variance scales with time ∝t 2H , where H ≠ 1/2. We provide detailed analysis of the various factors that lead to the anomalous behavior: non-stationarity, long-ranged correlations and a fat-tailed increment distribution. The papers exhibit a high degree of heterogeneity across the various fields, as the statistics of the highest cited papers is fundamentally different from that of the lower ones. The citation data is shown to be highly correlated and non-stationary; as all the papers except the small percentage of them with high number of citations, die out in time.
We examine the bi-scaling behavior of Lévy walks with nonlinear coupling, where , the particle di... more We examine the bi-scaling behavior of Lévy walks with nonlinear coupling, where , the particle displacement during each step, is coupled to the duration of the step, ⌧ , by ⇠ ⌧. An example of such a process is regular Lévy walks, where = 1. In recent years such processes were shown to be highly useful for analysis of a class of Langevin dynamics, in particular a system of Sisyphus laser-cooled atoms in an optical lattice, where = 3/2. We discuss the well-known decoupling approximation used to describe the central part of the particles' position distribution, and use the recently introduced infinite-covariant density approach to study the large fluctuations. Since the density of the step displacements is fat-tailed, the last travel event must be treated with care for the latter. This e↵ect requires a modification of the Montroll-Weiss equation, an equation which has proved important for the analysis of many microscopic models.
We study a particle immersed in a heat bath, in the presence of an external force which decays at... more We study a particle immersed in a heat bath, in the presence of an external force which decays at least as rapidly as 1=x, e.g., a particle interacting with a surface through a Lennard-Jones or a logarithmic potential. As time increases, our system approaches a non-normalizable Boltzmann state. We study observables, such as the energy, which are integrable with respect to this asymptotic thermal state, calculating both time and ensemble averages. We derive a useful canonical-like ensemble which is defined out of equilibrium, using a maximum entropy principle, where the constraints are normalization, finite averaged energy, and a meansquared displacement which increases linearly with time. Our work merges infinite-ergodic theory with Boltzmann-Gibbs statistics, thus extending the scope of the latter while shedding new light on the concept of ergodicity.
We introduce three strategies for the analysis of financial time series based on time averaged ob... more 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.
Large-deviations theory deals with tails of probability distributions and the rare events of rand... more Large-deviations theory deals with tails of probability distributions and the rare events of random processes, for example spreading packets of particles. Mathematically, it concerns the exponential fall-of of the density of thin-tailed systems. Here we investigate the spatial density Pt(x) of laser cooled atoms, where at intermediate length scales the shape is fat-tailed. We focus on the rare events beyond this range, which dominate important statistical properties of the system. Through a novel friction mechanism induced by the laser fields, the density is explored with the recently proposed nonnormalized infinite-covariant density approach. The small and large fluctuations give rise to a bi-fractal nature of the spreading packet.
We present an exact solution for the distribution of sample averaged monomer to monomer distance ... more We present an exact solution for the distribution of sample averaged monomer to monomer distance of ring polymers. For non-interacting and local-interaction models these distributions correspond to the distribution of the area under the reflected Bessel bridge and the Bessel excursion respectively, and are shown to be identical in dimension d ≥ 2, albeit with pronounced finite size effects at the critical dimension, d = 2. A symmetry of the problem reveals that dimension d and 4 - d are equivalent, thus the celebrated Airy distribution describing the areal distribution of the d = 1 Brownian excursion describes also a polymer in three dimensions. For a self-avoiding polymer in dimension d we find numerically that the fluctuations of the scaled averaged distance are nearly identical in dimension d = 2, 3 and are well described to a first approximation by the non-interacting excursion model in dimension 5.
We investigate the overdamped stochastic dynamics of a particle in an asymptotically flat externa... more We investigate the overdamped stochastic dynamics of a particle in an asymptotically flat external potential field, in contact with a thermal bath. For an infinite system size, the particles may escape the force field and diffuse freely at large length scales. The partition function diverges and hence the standard canonical ensemble fails. This is replaced with tools stemming from infinite ergodic theory. Boltzmann-Gibbs statistics, even though not normalized, still describes integrable observables, like energy and occupation times. The Boltzmann infinite density is derived heuristically using an entropy maximization principle, as well as via a first-principles calculation using an eigenfunction expansion in the continuum of low-energy states. A generalized virial theorem is derived, showing how the virial coefficient describes the delay in the diffusive spreading of the particles, found at large distances. When the process is non-recurrent, e.g. diffusion in three dimensions with a Coulomb-like potential, we use weighted time averages to restore basic canonical relations between time and ensemble averages.
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Papers by Erez Aghion