# EPJ B Special Issue: Continuous Time Random Walk fifty years on

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- Published on 29 September 2016

This EPJ B Topical Issue aims to highlight the versatility and wide range of achievements of the continuous-time random walk (CTRW) formalism on the occasion of its fiftieth anniversary.

The concept of CTRW was introduced in 1965 by physicists Eliott W. Montroll and George H. Weiss as a way to give continuous and fluctuating interevent times. The CTRW is fundamentally different from the regular random flight or walk models as probability density of the flight or walk in the long-time (asymptotic) limit scales in a non-Gaussian way, being a serious and inspirational extension of the Gaussian one. Thus, the CTRW became a foundation of anomalous (dispersive, non-Gaussian) transport and diffusion opening a new trend in statistical physics, as well as condensed and soft matter physics, stimulating their very rapid expansion, even outside the traditional statistical physics. Since the canonical CTRW was first successfully applied by Scher and Lax in 1973 and independently by Moore one year later to describe anomalous transient photocurrent in an amorphous glassy material manifesting the power-law relaxation, this formalism has achieved much more than its original goal and now it has in many fields ranging from biology, telecommunication, finance, with extensions in econometrics and economics, and all the way to speech recognition.

Authors are invited to submit their work, in the usual form of an original contribution, directly to the Editorial Office through the Manuscript Management System, or by mail to This email address is being protected from spambots. You need JavaScript enabled to view it. indicating their wish to be considered for this Topical Issue. All submissions will be peer-reviewed according to the normal standards and criteria for quality of the journal.

**Deadline for submission: 6 July 2017 **

**Guest Editors:**

**Ryszard Kutner**, Faculty of Physics, University of Warsaw, Pasteur 5, PL-02093 Warsaw, Poland, This email address is being protected from spambots. You need JavaScript enabled to view it.**Jaume Masoliver**, Departament de Física Matèria Condensada and Institute of Complex Systems (UBICS), Universitat de Barcelona, Martí i Franquès 1, E-08028 Barcelona, Spain,This email address is being protected from spambots. You need JavaScript enabled to view it.

**List of confirmed authors**

Eli Barkai (Bar-Ilan University)

A fractional reaction-diffusion description of supply and demand

We suggest that the broad distribution of time scales in financial markets could be a crucial ingredient to reproduce realistic price dynamics in stylised Agent-Based Models. We propose a fractional reaction-diffusion model for the dynamics of liquidity in financial markets, where agents are very heteroge- neous in terms of their characteristic frequencies. Several features of our model are amenable to an exact analytical treatment. We find in particular that the impact is a concave function of the transacted volume (aka the \square-root impact law"), as in the normal diffusion limit. However, the impact kernel decays as t^-β with β = 1/2 in the diffusive case, which is inconsistent with market efficiency. In the sub-diffusive case the decay exponent β takes any value in [0; 1/2], and can be tuned to match the empirical value β 1=4. Numerical simulations confirm our theoretical results. Several extensions of the model are suggested.

Stas Burov (Bar-Ilan University)
We suggest that the broad distribution of time scales in financial markets could be a crucial ingredient to reproduce realistic price dynamics in stylised Agent-Based Models. We propose a fractional reaction-diffusion model for the dynamics of liquidity in financial markets, where agents are very heteroge- neous in terms of their characteristic frequencies. Several features of our model are amenable to an exact analytical treatment. We find in particular that the impact is a concave function of the transacted volume (aka the \square-root impact law"), as in the normal diffusion limit. However, the impact kernel decays as t^-β with β = 1/2 in the diffusive case, which is inconsistent with market efficiency. In the sub-diffusive case the decay exponent β takes any value in [0; 1/2], and can be tuned to match the empirical value β 1=4. Numerical simulations confirm our theoretical results. Several extensions of the model are suggested.

On the quantum CTRW approach

The concept of continuous-time random walk is generalized into the quantum approach using a completely positive map. This approach introduces in a phenomenological way the concept of disorder in the transport problem of a quantum open system. If the waiting-time of the continuous-time Renewal approach is exponential we recover a semigroup for a dissipative quantum walk. Two models of non-Markovian evolution have been solved considering different types of disorder.

D. del-Castillo-Negrete (Oak Ridge National Laboratory)
The concept of continuous-time random walk is generalized into the quantum approach using a completely positive map. This approach introduces in a phenomenological way the concept of disorder in the transport problem of a quantum open system. If the waiting-time of the continuous-time Renewal approach is exponential we recover a semigroup for a dissipative quantum walk. Two models of non-Markovian evolution have been solved considering different types of disorder.

Aleksei Chechkin (National Academy of Sciences of Ukraine, Kiev)

Sergey Denisov (Universität Augsburg)

Marco Dentz (Center for Strategic and International Studies, Washington)

Bartlomiej Dybiec (Jagiellonian University)

Sergei Fedotov (The University of Manchester)

Katarzyna Górska (Institute of Nuclear Physics, Polish Academy of Sciences, Kraków)

Fractional linear Boltzmann equation

The linear Boltzmann equation approach is generalized to describe fractional superdiffusive transport of the Levy walk type in arbitrary force fields. The time distribution between scattering events is assumed to have a mean value and infinite variance. It is completely characterized by two intermittent scattering rates, one normal and one fractional. Because the mean time between scattering events can be made arbitrary small, the retardation effects can be neglected, like in the standard linear Boltzmann equation. We formulate a general fractional linear Boltzmann equation approach, and exemplify it with a particularly simple case having the Bohm and Gross scattering integral. Here, at each scattering event the particle velocity is completely randomized and takes a value from equilibrium Maxwell distribution at a given finite temperature. We argue that this novel fractional kinetic equation provides a viable alternative to the fractional Kramers equation by Silbey and Barkai based on the picture of divergent mean time between scattering events. The range of applications is discussed.

The linear Boltzmann equation approach is generalized to describe fractional superdiffusive transport of the Levy walk type in arbitrary force fields. The time distribution between scattering events is assumed to have a mean value and infinite variance. It is completely characterized by two intermittent scattering rates, one normal and one fractional. Because the mean time between scattering events can be made arbitrary small, the retardation effects can be neglected, like in the standard linear Boltzmann equation. We formulate a general fractional linear Boltzmann equation approach, and exemplify it with a particularly simple case having the Bohm and Gross scattering integral. Here, at each scattering event the particle velocity is completely randomized and takes a value from equilibrium Maxwell distribution at a given finite temperature. We argue that this novel fractional kinetic equation provides a viable alternative to the fractional Kramers equation by Silbey and Barkai based on the picture of divergent mean time between scattering events. The range of applications is discussed.

Continuous-Time Random Walk with multi-step memory: An application to
market dynamics

A novel version of the Continuous-Time Random Walk (CTRW) model with memory is developed. This memory means the dependence between arbitrary number of successive jumps of the process, while waiting times between jumps are considered as i.i.d. random variables. The dependence was found by analysis of empirical histograms for the stochastic process of a single share price on a market within the high frequency time scale, and justified theoretically by considering bid-ask bounce mechanism containing some delay characteristic for any double-auction market. Our model turns out to be exactly analytically solvable, which enables a direct comparison of its predictions with their empirical counterparts, for instance, with empirical velocity autocorrelation function. Thus this paper significantly extends the capabilities of the CTRW formalism. (https://arxiv.org/abs/1305.6797)

Rudolf Hilfer (Universität Stuttgart)
A novel version of the Continuous-Time Random Walk (CTRW) model with memory is developed. This memory means the dependence between arbitrary number of successive jumps of the process, while waiting times between jumps are considered as i.i.d. random variables. The dependence was found by analysis of empirical histograms for the stochastic process of a single share price on a market within the high frequency time scale, and justified theoretically by considering bid-ask bounce mechanism containing some delay characteristic for any double-auction market. Our model turns out to be exactly analytically solvable, which enables a direct comparison of its predictions with their empirical counterparts, for instance, with empirical velocity autocorrelation function. Thus this paper significantly extends the capabilities of the CTRW formalism. (https://arxiv.org/abs/1305.6797)

Reiner Klages (Queen Mary, University of London)

Diego Krapf (Colorado State University)

Katja Lindeberg (University of California, San Diego)

Marcin Magdziarz (Wrocław University of Technology)

Francesco Mainardi (Università di Bologna)

Ralf Metzler (Universität Potsdam)

Miquel Montereo (Universitat de Barcelona)

Takashi Odagaki (CEO, Research Institute for Science Education, Inc., Japan)

Enrico Scalas (University of Sussex)

Harvey Scher (Weizmann Institute of Science)

Origins and applications of the Montroll-Weiss continuous time random walk

The Continuous Time Random Walk (CTRW) was introduced by Montroll and Weiss in 1965 in a purely mathematical paper. Its antecedents and later applications beginning in 1973 are discussed, especially for the case of fractal time where the mean waiting time between jumps is infinite.

The Continuous Time Random Walk (CTRW) was introduced by Montroll and Weiss in 1965 in a purely mathematical paper. Its antecedents and later applications beginning in 1973 are discussed, especially for the case of fractal time where the mean waiting time between jumps is infinite.

Direct and inverse problems in dispersive time-of-flight photocurrent revisited

Using the fact that the continuous time random walk (CTRW) scheme is a random process subordinated to a simple random walk under the operational time given by the number of steps taken by the walker up to a given time, we revisit the problem of strongly dispersive transport in disordered media, which first lead Scher and Montroll to introducing the power law waiting time distributions. Using subordination approach allows for disentangling the complexity of the problem, separating the solution of the boundary value problem (which is solved on the level of normal diffusive transport) from the influence of the waiting times, which allows for the solution of the direct problem in the whole time domain (including short times, out of reach of the initial approach), and simplifying strongly the analysis of the inverse problem. The analysis of the last shows that the current traced do not contain information sufficient for unique restoration of the waiting time probability densities, but define a single-parametric family of functions, all leading to the same photocurrent forms. The members of the family have the power-law tails which differ only by a prefactor, but may look astonishingly different at their body. The same applies to the multiple trapping model, mathematically equivalent to a special the limiting case of CTRW.

Using the fact that the continuous time random walk (CTRW) scheme is a random process subordinated to a simple random walk under the operational time given by the number of steps taken by the walker up to a given time, we revisit the problem of strongly dispersive transport in disordered media, which first lead Scher and Montroll to introducing the power law waiting time distributions. Using subordination approach allows for disentangling the complexity of the problem, separating the solution of the boundary value problem (which is solved on the level of normal diffusive transport) from the influence of the waiting times, which allows for the solution of the direct problem in the whole time domain (including short times, out of reach of the initial approach), and simplifying strongly the analysis of the inverse problem. The analysis of the last shows that the current traced do not contain information sufficient for unique restoration of the waiting time probability densities, but define a single-parametric family of functions, all leading to the same photocurrent forms. The members of the family have the power-law tails which differ only by a prefactor, but may look astonishingly different at their body. The same applies to the multiple trapping model, mathematically equivalent to a special the limiting case of CTRW.

The continuing relevance of Mandelbrot's non ergodic fractional renewal models of 1963-67

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. 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 http://arxiv.org/abs/1603.00738

Bruce J. West (United States Army Washington)
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. 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 http://arxiv.org/abs/1603.00738

Vasily Zaburdaev (Max-Planck-Institute for the Physics of Complex Systems)