Next step in random walks: Understanding mechanisms behind complex spreading phenomena
- Michael Urbakh (School of Chemistry, Tel Aviv University, Tel Aviv, Israel)
- Eli Barkai (Dept. of Physics, Bar-Ilan University, Ramat-Gan, 52900 Israel, Israel)
- Sergey Denisov (Dept. of Theor. Physics I, Universität Augsburg, D-86135 Augsburg, Germany)
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Motivation and the idea of the workshop
Spreading is an omnipresent phenomenon which plays either negative or positive role, depending on what is spreading, an invasive pathogen or holes in a semiconductor. There are many facets of spreading that have been studied in different fields.
There is a huge variety of real-life examples of spreading; bio-dispersals are the most illustrative. On the micro time-space scales compared with the lifetime and areal of a single mover, an atom migrating over a substrate  or a foraging animal , spreading splits into a set of point-like random processes, so that individual trajectories look like trajectories of random walkers. It was therefore very natural that the paradigm of random walks heavily influenced the development of the fields where spreadings play the key role – solid state electronics, turbulence, molecular bio-physics, ecology, and others. At the beginning, Gaussian random walks, as a well-established con-cept, were extensively used. Then in many labs, it was observed that the obtained data do not fit this model, so new tools and models were demanded. The complexity of the observed phenomena can be captured in more detail with such updates as continuous-time random  and Lévy  walks.
Let's briefly discuss the Lévy walks (LWs). They are random processes in which a walker makes long excursions interrupted by re-orientation events . Unlike ordinary random walks, step sizes in LWs are broadly distributed resulting in a scale free evolution, deeply related to the idea of fractals. Since the concept was born in 1985 , LWs have found a striking number of applica-tions in diverse fields, including optics, dynamical chaos, turbulence, many-body physics (both quantum and classical ones), biophysics, behavioral science, and even robotics. LWs remain a hot research field but most of the application-oriented studies are plagued by a solution focused atti-tude: LWs are implemented now as a built-in core of search algorithms for autonomous robots  – simply because albatrosses produce Lévy walk-like motional patterns when preying on the fish  and the albatrosses are very good in it. At the same time no one would believe that an albatross utilizes LW concept, by independently drawing a length of the next flight from a power-law tailed probability distribution. Albatrosses evidently “do not care about math” [J. Travis, 2007]. Motion of any animal is a product of a complex multi-layered activity of informational circuits which are constantly processing external signals and generating internal signals to control organism's mo-tion. An anomalous dispersal pattern appears as the result of this activity and not as a result of knowledge of some mathematical models.
This leads us to the idea of our workshop: We want to change the “cargo-cult” paradigm prevail-ing now on the field of anomalous diffusion and random walks when it comes to their practical applications. Namely, it is not that experimental data should be analyzed in the view of the exist-ing random walk and diffusion models (and, if they fit badly, then either experimental data have to be thrown away or theory needs to be slightly adjusted) but models themselves have to be con-structed in a way as to capture essential physics behind the emerging spreading. That simply means that physical mechanisms running the spreading have to be understood first by those theo-reticians who want to describe them with their mathematical constructions. The focus of the pro-posed workshop is to leave the phenomenological stage of the theory and bring together experts who work on the basics mechanism still covering a large body of models and systems.
Existing models such as LWs  and fractional Fokker-Planck equations  have a strong appeal – they are very well developed, they are famous and have very good reputations and agenda. It is very tempting therefore to use them immediately when an experimentalist or a field ecologist comes with the statical data and ask “Could you please explain it with your theories?”. But even is the matching is perfect, it does not serve an explanation. The explanation is encoded in the data and in order to extract it, the theoretician has, first of all, to understand the process which pro-duced this data. Bacteria do not care about us and our Lévy walks even though they also produce LW-like patterns when migrating ; instead we should care about the bacteria and understand what drives them in a such a way that their trajectory resembles LWs to great details.
Spreading of cold atoms in dissipative optical potentials is an example where this path is already taken. At first, a specific classical diffusion equation was derived by Marksteiner et al.  to cap-ture the specific cooling mechanism (essentially quantum by its nature) governing the dynamics of atoms; then, by unrevealing this equation into a Langevin dynamics it was possible to demon-strate that on the microscopic level trajectories of individual atoms appear as LWs . In such a way, a LW-like process has been derived from physics. It is a noteworthy that the LW-like spread-ing of cold atoms was observed ‘in vivo’ and Lévy scaling of the atomic cloud was confirmed by measurements . Yet these experiments have also revealed that a simple LW description does not capture all features of the observed phase space dynamics.
In a very different direction another microscopic origin of anomalous diffusion of bacteria was recently developed. Ariel et al. suggested a model based on chaotic motion and collective motion of the bacteria .
These two examples are only part of a trend of a maturing field switching from phenomenological methods to deeper modeling, and our primary goal is to help diffuse these new ideas among the relevant practitioners.
The purpose of the proposed three-day workshop is to bring together researchers using random walk models as every-day tools in their studies and discuss whether is possible to go beyond phe-nomenological approaches and thus depart from the existing paradigm. The key point is to ex-change new ideas, concepts, and technical (computational) means which can be used for this pur-pose. In addition, this exchange could create a new interface for researchers working with differ-ent type of diffusion and random walk processes, nonlinear dynamics, ecology, solid state physics etc. Several researchers dealing with the field collected data will be included as speakers.
More information is on CECAM TAU Node http://www3.tau.ac.il/cecam/
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