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Alex Adamou (London Mathematical Laboratory)

Reallocating geometric Brownian motion

Geometric Brownian Motion is a simple random multiplicative growth process, used widely in finance. It is non-ergodic, in that the time-average growth rate of a single trajectory differs from the growth rate of the ensemble average. A finite sample of such motions is a toy model of a human economy, in which individual wealths evolve without interacting. Only recently have the statistical properties of the sample mean been understood, in the context of the Random Energy Model (Derrida 1980). In particular, a short-time regime exists where the sample is self-averaging. We add reallocation to this model. Individuals contribute proportionally to their wealth and receive an equal share of the amount collected. Sufficiently fast reallocation extends the self-averaging regime indefinitely. This maximises the time-average growth rate of the sample mean and yields a stationary wealth distribution. Slower reallocation results in a sample dominated by large deviations. Negative reallocation is qualitatively different: the sample splits into positive and negative wealths, whose distribution never equilibrates. Finally, we fit the model parameters to historical wealth inequality in the United States. We find that the effective reallocation rate is currently negative, i.e. from poorer to richer. Mainstream economic analyses miss such findings by assuming equilibrium.

Takuma Akimoto (Keio University)

Infinite invariant density in laser cooling

Cooling is not only a technogy in industry but also a basic science. How we can cool atoms ultimately is an important issue in science. One of typical methods to cool atoms is Doppler cooling, where Doppler effect generates a friction force to cool atoms. In other words, momentum of an atom performs random walk with bias. On the other hand, another cooling mechanism is proposed without using bias of random walk in momentum space. Instaed inhomogeneous diffusive environment in momentum space is utilized in this laser cooling. In this presentation, I will talk about infinite invariant density in inhomogeneous random walk, which is a model of laser cooling, and show a distributional limit theorem for time-averaged observables. Moreover, I will discuss another cooling model which is relevant to the inhomogeneous random walk of laser cooling and present the modified model to show that this cooling mechanism is robust in the sense that momentum goes to zero under weak noise (heating).

Eduardo G. Altmann (MPIPKS, Dresden)

Levy noise in rich-get-richer processes

Economical systems show a large inequality among agents. A remarkable manifestation are the fat tails found in the distributions not only of income and wealth, but also of the popularity of different products. The problem this talk eill address is to understand and model the dynamical process underlying such extreme inequality. Our work relies on the analysis of 10 Million time series of the daily number of views of YouTube videos. We model the growth in popularity using stochastic differential equations and find that the well-studied rich-get-richer mechanisms co-exist with Levy-type fluctuations, a combination that helps to understand the extremely low predictability of the most successful items.

Eli Barkai (Bar Elan Univ)

Some insights on 1/f noise: weak ergodicity breaking perspective

Starting with the work of Bernamont (1937) on resistance fluctuations, noisy signals of a vast number of natural processes exhibit 1/f power-spectrum. Such spectra are found in weather data, brain activity, currents of ion-channels and certain chaotic systems to name a few. The wide applicability of this spectrum resulted in conflicting theories distributed among many disciplines. A unifying feature is that 1/f power spectrum is non-integrable at low frequencies implying that the total energy in the system is infinite. As pointed out by Mandelbrot (1950's) this infrared catastrophe suggests that one should abandon the stationary mindset and hence go beyond the widely applicable Wiener-Khinchin formula for the power spectrum. Recent theoretical and experimental advances renewed the discussion on this old paradox, for example in the context of blinking quantum dots [1,2]. Importantly the removal of ensemble averaging in nano-scale measurement revealed time dependent spectrum, at least for nano-crystals. In this talk ageing, intermittency, weak ergodicity breaking, and critical exponents of the sample power spectrum are discussed within a theoretical framework which hopefully provides new insights on the 1/f enigma [1,3]. A general theoretical framework based on non-stationary but scale invariant correlation functions leads to an ageing Wiener-Khinchin theorem which replaces the standard spectral theory [3]. The non-integrable spectral density is reminiscent of the infinite invariant measure of the Pomeau Manneville intermittent map.


[1] M. Niemann, H. Kantz, E. Barkai Fluctuations of 1/f noise and the low frequency cutoff paradox Phys. Rev. Lett. 110, 140603 (2013).

[2] S. Sadegh, E. Barkai, and D. Krapf 1/f noise for intermittent quantum dots exhibits non-stationarity and critical exponents New. J. of Physics 16113054 (2014).

[3] N. Leibovich and E. Barkai, Aging Wiener-Khinchin Theorem Phys. Rev. Lett.115, 080602 (2015).

Anna Bodrova (Moscow State University)

Brownian motion with time-dependent diffusion coefficient

Brownian motion with time-dependent diffusion coefficient is ubiquitous in nature. It has been observed for the mobility of proteins in cell membranes, motion of molecules in porous environment, water diffusion in brain measured in terms of magnetic resonance imaging and also in media with time-dependent temperature such as free cooling granular materials or melting snow. We investigate a new type of anomalous diffusion processes governed by an underdamped Langevin equation with time-dependent diffusion and friction coefficients and discuss possible applications to real physical systems such as free cooling granular materials. We show that for certain range of parameter values the overdamped limit for the Langevin equation does not exist.

Nikolai Brilliantov (University of Leicester)

Intermediate regimes in granular Brownian motion: superdiffusion and subdiffusion

Brownian motion in a granular gas in a homogeneous cooling state is studied theoretically and by means of molecular dynamics. We use the simplest first-principles model for the impact-velocity dependent restitution coefficient, as it follows for the model of viscoelastic spheres. We reveal that for a wide range of initial conditions the ratio of granular temperatures of Brownian and bath particles demonstrates complicated nonmonotonic behavior, which results in a transition between different regimes of Brownian dynamics: It starts from the ballistic motion, switches later to a superballistic one, and turns at still later times into subdiffusion; eventually normal diffusion is achieved. Our theory agrees very well with the molecular dynamics results, although extreme computational costs prevented us from detecting the final diffusion regime. Qualitatively, the reported intermediate diffusion regimes are generic for granular gases with any realistic dependence of the restitution coefficient on the impact velocity.

Sandra Chapman (Warwick)

Avalanching systems under intermediate driving rate

The paradigm of self-organized criticality (SOC) has found application in understanding scaling and bursty transport in driven, dissipative systems including in astrophysical systems as opposed to controlled laboratory experiments. SOC is, however, a limiting process that occurs as the ratio of driving rate to dissipation rate is taken to zero. We consider the more realistic scenario of finite sized systems with finite driving rate. We show that similarity analysis identifies a control parameter RA which is analogous to the Reynolds number RE of turbulence in that it relates to the number of excited degrees of freedom, that is, the range of spatio-temporal scales over which one finds scaling behaviour. However for avalanching systems the number of excited degrees of freedom is maximal at the zero driving rate, SOC limit, in the opposite sense to fluid turbulence. Practically, at finite RE or RA one observes scaling over a finite range which for turbulence, increases with RE and for SOC, decreases with increasing RA suggesting an observable trend to distinguish them. We use the BTW sandpile model to explore this idea and find that whilst avalanche distributions can, depending on the details of the driving, reflect this behaviour, finite range power spectra do not and thus are not clear discriminators of an SOC state.

Marco Dentz (Barcelona)

Self-Averaging and Ergodicity of Subdiffusion in d-Dimensional Quenched Random Media

We study self-averaging and ergodicity of diffusion in d–dimensional quenched random media characterized by spatially variable mobility coefficient, which accounts for particle retention due to physical and chemical interactions with the medium. These type of models find applications for the quantification of diffusion in a variety of systems, including radionuclide migration in low permeability media, gas production in shales, the motion of charge carriers in amorphous semi-conductors, and the motion of proteins in living cells. It can be shown that this type of heterogeneous diffusion problems are equivalent to the quenched random trap model. We focus on the mean square displacement m(t) = x(t)2 of diffusing particles, where the angular brackets denote the noise averge in a single disorder realization. For this class of models, it has been well-known that the ensemble average mean-square displacement m(t) of a diffusing particle evolves subdiffusively as m(t) ∝ tγ with 0 < γ < 1 in the presence of quenched random traps with an associated trapping time distribution that evolves as a power-law ψ(t) ∝ t−1−β for 0 < β < 1. In d > 2 spatial dimensions, the average number of new sites sampled by the diffusing particles increases in average with the number of random walk steps. As a consequence, subsequent trapping times may be considered independent and diffusion describes in average a continuous time random walk (CTRW). This is different for d ≤ 2 for which the disorder sampling is less efficient and the average number of new sites visited increases sublinearly with the number of steps of the random walker. Here we focus on the self-averaging properties and ergodicity of the mean-square displacement. This means, on one hand, we evaluate how well the noise averaged m(t) in a single disorder realization represents the temporal average (ergodicity), and how well the
disorder averaged m(t) describes m(t) in single disorder realizations (self- averaging). Self-averaging is achieved if particles sample in a single disorder realization a representative part of the heterogeneity spectrum. Ergodicity is related to the efficient disorder sampling along a single particle trajectory. We find that m(t) is not self-averaging for d < 2 due to the inefficient disorder sampling by random motion in single realizations. All the particles experience the same sequence of random traps in a single realization. Disorder sampling is minimum. For d ≥ 2 in contrast, the efficient sampling of heterogeneity by the space random walk renders m(t) self-averaging. We find that diffusion exhibits weak ergodicty breaking in any dimension.

Itzhak Fouxon (Yonsei University)

CTRW for tracer motion through porous media

In this talk we describe continuous time random walk (CTRW) model for the motion of tracer through porous medium. This motion manifests anomalous diffusion whose laws are not completely understood. The model is based on experimental data. We derive the laws of diffusion and demonstrate that depending on the model parameter wide range of laws are possible - from superdiffusion to subdiffusion, including possible logarithmic corrections. In certain regime the laws coincide with those found previously in a different model raising the question of universality. Finally we stress the difference between different types of separable CTRW in modeling.

Holger Kantz (Dresden)

WEB and ageing in dynamical systems

We study Pomenau-Manneville type maps which are known to exhibit very strong intermittency and an invariant distribution which cannot be normalized. When integrating the map's output in such a situation, one finds anomalous diffusion and ageing well known from CTRWs. We discuss the consequences of the lack of a normalisable invariant measure of the maps, and we introduce a class of time-continuous physical systems, namely damped and driven oscillators with a specific potential, which exhibit the same behaviour.

This work was done together with my master student Philipp Meyer

Bronius Kaulakys (Vilnius)

Diffusion-like processes yielding 1/f noise with different distributions of observables

Internal mechanism leading to the emergence of the widely occurring 1/f noise still remains an open issue. We show that a couple of seemingly different models exhibiting a 1/f spectrum are due to similar scaling properties of the signals [1]. We use the self-similarity properties of the model based on the stochastic differential equations (SDEs) with respect to the nonlinear transformations of the variables of these equations and show that 1/f noise of the observable may yield from the power-law transformations of well-known standard processes, like the Brownian motion, Bessel and similar stochastic processes [2]. For the modeling of the so-called ‘inverse cubic law’ of the cumulative distributions of the long-memory fluctuations of market indicators, we propose the SDEs giving these features [3]. This is achieved using the suggestion that when the market evolves from calm to violent behavior there is a decrease of the delay time of multiplicative feedback of the system in comparison to the driving noise correlation time. This results in a transition from the Itô to the Stratonovich sense of the SDE and yields a long-range memory process.

After demonstrating the appearance of 1/f noise in the earlier proposed point process model, we generalize it starting from a SDE which describes a Brownian-like motion in the internal (operational) time. We consider this equation together with an additional equation relating the internal time to the external (physical) time [4]. We generalize this model and propose a system of two coupled nonlinear stochastic differential equations [5]. The equations are derived from the scaling properties necessary for the achievement of 1/f noise. The proposed coupled stochastic differential equations allow us to obtain a 1/f spectrum in a wide range of frequencies together with the almost arbitrary steady-state density of the signal.

[1] J. Ruseckas and B. Kaulakys, J. Stat. Mech. P06005 (2014).

[2] B. Kaulakys, M. Alaburda, and J. Ruseckas, Mod. Phys. Lett. B 29, 1550223 (2015).

[3] B. Kaulakys, M. Alaburda, and J. Ruseckas, J. Stat. Mech. P054035 (2016).

[4] J. Ruseckas, R. Kazakevicius, and B. Kaulakys, J. Stat. Mech. P054022 (2016).

[5] J. Ruseckas, R. Kazakevicius, and B. Kaulakys, J. Stat. Mech. P043209 (2016).

Ralf Metzler (Potsdam)

Anomalous diffusion in membranes and cells

I will present results from Molecular Dynamics simulations of pure and crowded lipid bilayer systems, giving evidence of anomalous diffusion. While in the pure lipid bilayer this anomaly is very short ranged, the addition of colesterols or the passage to the gel phase leads to extended anomalous diffusion. The character of the dynamics corresponds to that of the fractional Brownian motion in the dilute bilayer, changing to non-Gaussian motion when the bilayer is crowded with proteins. Real biological membranes show macroscopic anomalous diffusion, which is evidenced from superresolution microscopy experiments. In particular the motion becomes non-ergodic and ageing. In the second part of the talk I will also cover non-ergodic and ageing motion in bulk liquids.


[1] R Metzler, J-H Jeon, and AG Cherstvy, Biochim Biophys Acta,
at press; DOI:10.1016/j.bbamem.2016.01.022.

[2] J-H Jeon, M Javanainen, H Martinez-Seara, R Metzler, and I
Vattulainen, Phys Rev X 6, 021006 (2016).

[3] AV Weigel, B Simon, MM Tamkun and D Krapf, Proc Natl Acad Sci USA
108, 6438 (2011).

[4] R Metzler, J-H Jeon, AG Cherstvy, and E Barkai, Phys Chem Chem
Phys 16, 24128 (2014)

Guenter Radons (Chemnitz)

The (5+1) infinities for space-time coupled Lévy walks

We recently detected and analyzed in detail the phenomenon of weak ergodicity breaking (WEB), i.e. the inequivalence of ensemble- and time-averaged squared displacements, for randomly accelerated particles (RAP) [1]. These results, which are relevant for anomalous chaotic diffusion in Hamiltonian systems, for passive tracer transport in turbulent flows, and many other systems showing momentum diffusion, motivated us to study some related models. One of them, namely the space-time correlated Lévy walk, introduced in the context of turbulence [2], is considered as physically most relevant due to the finite velocities appearing in its definition. Therefore, after showing the basic WEB results for RAP, the WEB related “phase diagram”, obtained by varying the two defining exponents of the Lévy walk, is discussed in detail. Apart from the well-known boundary between stationary and non-stationary increments indicated by a diverging mean residence time, we find even in the stationary regime five more transition lines, each associated with a diverging quantity. These lines intersect and divide the stationary regime into nine different “phases”, each characterized by certain combinations of diverging and non-diverging characteristic quantities.


[1] T. Albers and G. Radons, Physical Review Letters 113, 184101 (2014)
[2] M. Shlesinger, B. West, and J. Klafter, Physical Review Letters 58, 1100 (1987)

Yuzuru Sato (Hokkaido University and LML)

Anomalous diffusion in random dynamical systems

Diffusion based on stochastic chaos in random dynamical systems is studied. There is noise-induced anomalous diffusion and ageing in a spatially extended dynamical systems, where the unperturbed version is a well-established model for deterministic diffusion. The power law exponents for the mean square displacement and the waiting time distribution match to predictions by continuous time random walk theory. When we fix a particular noise realisation, we find
noise-induced synchronization of jump times and the breaking of self-averaging, which is similar to weak ergodicity breaking in weakly chaotic dynamics. Robustness of this phenomenon are also briefly discussed by revealing noise-induced super/sub-diffusion in a class of nonlinear maps.

This is a joint work with Prof. Rainer Klages at Queen Mary University of London.

Enrico Scalas (Sussex)

Solvable non-Markovian dynamic network

Non-Markovian processes are widespread in natural and human-made systems. Here, a non-Markovian dynamic
network is considered with random link activation and deletion (RLAD) and heavy tailed Mittag-Leffler distribution for the inter-event times. We derive an analytically and computationally tractable system of Kolmogorov-like forward equations utilising the Caputo derivative for the probability of having a given number of active links in the network and solve them. Simulations for the RLAD are also studied for power-law inter-event times and we show excellent agreement with the Mittag-Leffler model. This agreement holds even when the RLAD network dynamics is coupled with the susceptible-infected-susceptible (SIS) spreading dynamics. Thus, the analytically solvable Mittag-Leffler model provides an excellent approximation to the case when the network dynamics is characterised by power-law distributed inter-event times. The ergodic properties of the model are discussed.

Jakub Ślęzak (Wrocław)

Ergodicity in Fourier space for Gaussian models

I will show that the classical Maruyama mixing theorem allows to reformulate the ergodic condition using the Fourier transform in an unexpected, but simple manner. For the Gaussian processes the ergodicity is equivalent to a particular form of its harmonizable representation and the corresponding spectral measure. I will explain this idea considering elemental examples, like harmonic oscillator with random initial conditions, and more advanced ones, like generalised Langevin equation. This change of perspective provides mathematical and statistical tools to study ergodicity for both the theoretical (model oriented) and practical (data oriented) approach. Moreover, in line with this approach, I propose some qualitative measures of non-ergodicity. The discussed methodology leads to strict results only in the Gaussian case, however I will comment on possibility to consider also non-Gaussian models in a similar manner, treating the results as a study of “linear-dependence” type of ergodicity, important from the practical point of view.


[1] G. Maruyama, “Infinitely divisible processes”, Theory Probab. Appl. 15:1, 3–23 (1970)
[2] M. Magdziarz, “A note on Maruyama's mixing theorem”, Stochastic Process. Appl. 119, 3416–3434 (2009)
[3] H. Dym, H. P. McKean, “Gaussian Processes, Function Theory, and the Inverse Spectral Problem”, Academic Press (1976)

Igor Sokolov (Humboldt University, Berlin)

Nonspectral relaxation patterns in anomalous diffusion

The relaxation of a dissipative system to its equilibrium state often shows a multiexponential pattern with relaxation rates, which are typically considered to be independent of the initial conditions. The rates follow from the spectrum of a Hermitian operator obtained by a similarity transformation of the initial Fokker-Planck operator. However, some initial conditions are mapped by this similarity transformation to functions which grow at infinity. These cannot be expanded in terms of the eigenfunctions of a Hermitian operator, and show different relaxation patterns. Considering the exactly solvable examples of Gaussian and generalized Levy Ornstein-Uhlenbeck processes (OUPs) we show that the relaxation rates belong to the Hermitian spectrum only if the initial condition belongs to the domain of attraction of the stable distribution defining the noise. While for an ordinary OUP initial conditions leading to nonspectral relaxation can be considered exotic, for generalized OUPs driven by Levy noise, such initial conditions are the rule.

We present a detailed analysis of the eigenfunctions of the Fokker-Planck operator for the Levy-OUP, discuss their asymptotic behavior and recurrence relations, and give explicit expressions in coordinate space for the special cases of the OUP with Cauchy white noise, together with the transformation kernel, which maps the fractional Fokker-Planck operator of the Cauchy-OUP to the non-fractional Fokker-Planck operator of the usual Gaussian OUP. We also describe how non-spectral relaxation can be observed in bounded random variables of the Levy-Ornstein-Uhlenbeck process and their correlation functions. The first nonspectral mode is shown to govern the relaxation process and allows for estimation of the initial distribution's Levy index. We also discuss how both both (spectral and nonspectral) relaxation rates can be extracted from a stochastic data sample.

R. Toenjes, I.M. Sokolov and E.B. Postnikov, Phys. Rev. Lett. 110, 150602 (2013)
R. Toenjes, I.M. Sokolov and E.B. Postnikov, Eur. Phys. J. B 87 (12), 287 (2014)
F. Thiel, I.M. Sokolov, and E.B. Postnikov, Phys. Rev. E 93, 052104 (2016)

Nick Watkins (LSE)

Mandelbrot's fractional renewal models for 1/f noise: what can they tell us today?

The problem of 1/f noise has been with us for about a century. Because it is so often framed in Fourier spectral language, the most famous solutions have tended to be the stationary long range dependent (LRD) models such as Mandelbrot's fractional Gaussian noise. In view of the increasing importance to physics of non-ergodic fractional renewal models, I present preliminary results of my research into the history of Mandelbrot's very little known work in that area from 1963-67 [1-4]. I speculate about how the lack of awareness of this work in the physics and statistics communities may have affected the development of complexity science, and I discuss the differences between the Hurst effect, 1/f noise and LRD, concepts which are often treated as equivalent. See


[1] J. M. Berger and B. B. Mandelbrot, "A New Model for Error Clustering in Telephone Circuits", IBM Technical Journal (July 1963).

[2] B. B. Mandelbrot, “Self-similar error clusters in communications systems, and the concept of conditional stationarity”, IEEE Trans. on Communications Technology, COM-13, 71-90 (1965).

[3] B. B. Mandelbrot, “Time varying Channels, 1/f noises, and the Infrared Catastrophe: Or why does the low frequency energy sometimes seem infinite ?”, IEEE Communication Convention, Boulder, Colorado (1965).

[4] Mandelbrot, "Some Noises With 1/f Spectrum, a Bridge Between Direct Current and White Noise", IEEE Trans. Inf. Theory, 13(2), 289 (1967).

Alex Weron (Hugo Steinhaus Center, Wroclaw)

Ergodicity testing of ARFIMA processes with applications to single particle tracking data

In this talk we will present asymptotic behaviour of a dynamical functional (DF) for a stable autoregressive fractionally integrated moving average (ARFIMA) process. We derive an analytical formula for this important statistics and analyse it as a diagnostic tool for ergodic properties. Moreover two estimators for the DF are presented in details. The obtained results point to the very fast convergence of the DF and show that even for short trajectories one may obtain reliable conclusions on the ergodicity breaking of the ARFIMA process. We use the obtained theoretical results to illustrate how the DF statistics can be used in the verification of the proper model for an analysis of some biophysical SPT experimental data.

[1] M. Magdziarz and A. Weron, Anomalous diffusion: Testing ergodicity breaking in experimental data, Phys. Rev. E 84,
051138 (2011).
[2] J. Janczura and A. Weron, Ergodicity testing for anomalous diffusion: Small sample statistics, J. Chem. Phys. 142, 144103
[3] H.Loch, J.Janczura and A.Weron, Ergodicity testing using an analytical formula for a dynamical functional of alpha stable ARFIMA processes, Phys. Rev. E 93, 043317 (2016).
[4] Y. Lanoiselee and D. S. Grebenkov, Revealing nonergodic dynamics in living cells from a single particle trajectory, Phys. Rev. E 93, 052146 (2016).