Special issues

Theoretical investigation of the fluctuations and anomalous dynamics in complex systems has been of interest for years. Modeling of random walks and stochastic processes in complex systems, including complex networks and graphs, requires an interdisciplinary approach due to the different applications in various fields, such as physics, biology, chemistry, engineering, computer science, economy, etc. This Focus Issue focuses on recent advances in the theory of anomalous diffusion and fluctuations in various complex systems.

Diffusion addresses the broadening of distribution functions in the course of time. When the variance grows linearly, we speak of diffusion; otherwise, the transport is anomalous. Anomalous transport is observed in a vast variety of systems like transport in porous media and other complex geometries, in solid-state disordered systems, crowded environments of cell interiors, the collective behavior of living organisms, and heat transport in low-dimensional systems. There is a wealth of theoretical research devoted to mechanisms that can induce anomalous transport. However, the past decade has vividly changed the field. Video Microscopy and particle tracking are providing a rapidly increasing wealth of highly-resolved experimental observations. Moreover, numerical simulations of ensembles of trajectories are now feasible also for disordered systems where one must average over many realizations of the geometry. The resulting data sets should best be addressed from a big data perspective to extract characteristic transport properties. Firstly, this poses challenges for the automatic data processing and parameter inference. Secondly, it calls for new mathematical perspectives that underpin the data analysis from a unified point of view. In particular, data sets are big enough now to address the anomalous decay of correlations in the dynamics and to search for universality in the transport. The aim of this Research Topic is to unify different visions, approaches and methodologies around the field of anomalous diffusion. We welcome contributions that review the current state of affairs in different fields of applications and mathematical models, opinion papers pointing towards urging open challenges, and original research articles. Altogether, they will provide a thorough overview of the status of the field, and insight into the main directions of current research.

The moment when a diffusing biomolecule first hits its binding-site, a groundwater tracer particle first exits into a catchment, or a financial-market stock crosses a given threshold value for the first time is called first-passage time (FPT). The statistics of FPT plays a crucial role in various fields of sciences and every-day life applications and there is currently a clear surge of applications of FPT concepts and tools to various disciplines, particularly in the life sciences and the upcoming field of sociophysics. With this issue we seek to showcase various aspects of first passage phenomena that are usually dispatched between applied mathematicians, theoretical physicists and applied sciences.

In this special issue we want to collect a timely range of applications of the basic idea presented in Smoluchowski's 1916 paper, as well as the mathematical physical theories that were inspired by that paper. The topics include: reaction kinetics; first passage theory; functionals (Feynman-Kac etc); facilitated diffusion and gene regulation; polymeric processes: looping, translocation; modifications in complex systems; anomalous diffusion, Levy walks. The discussed systems range from molecular processes over the dynamics in living biological cells to macroscopic systems such as random search processes in the motion patterns of animals and humans.

Biological physics is in its heyday: we are witnessing the great success of single molecule techniques, new possibilities for probing and manipulating biological systems, and new theoretical concepts. For instance, optical tweezers allow one to manipulate individual biopolymers and obtain previously inaccessible results. These include measurements of the interaction between single stranded DNA binding proteins and the DNA molecule itself, and confirmation of 1D motion along the DNA chain. New optical methods allow the motion of individual particles within a cell to be tracked in space and time. Such experimental advances also inspire new directions for theory. For instance, the exploration of fluctuation theorems has added a completely new perspective to statistical mechanics. This focus issue brings together a number of studies that exemplify this development. The papers range from conceptual statistical mechanical approaches to biopolymer physics to cell mechanics. From a physics perspective, cells are complex systems combining numerous processes that can either be driven by thermal fluctuations (such as the search processes of DNA binding proteins for their binding sites on the DNA) or by active motion fuelled by biochemical energy (such as molecular motors). A complete description of all these processes remains elusive. Moreover they cannot be simulated. The individual puzzles are beginning to be unraveled, but many questions remain. For example, how are intracellular transport and regulation affected by the high degree of molecular crowding? How is the arrangement of individual genes on the DNA connected with the specific spatial configuration of the DNA in the cell? It will be interesting to see what a similar survey produces in ten years' time. We would like to thank the Editorial Board of Physical Biology for the opportunity to organize this focus issue. Thanks also to IOP Publishing, and to Andrew Malloy in particular, for the professional handling of the issue. And of course we would like to thank all those who accepted our invitation and contributed interesting reports on the state-of-the-art in their respective fields.

Semi-classical dynamics of quantum wave packets spreading is studied for a kicked rotor. Quantum flights are established for a specific, “magic” value of a chaos control parameter when the classical stickiness of trajectories is most effective. By studying of a survival probability and distribution of the accelerations we identify the presence of quantum Lévy-type flights.