We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

where A represents the magnetic vector potential, is an integral of the hydromagnetic equations. This -integral made it possible to formulate a variational principle for the force-free magnetic fields. The integral expresses the fact that motions cannot transform a given field in an entirely arbitrary different field, if the conductivity of the medium isconsidered infinite. In this paper we shall show that the full set of hydromagnetic equations admit five more integrals, besides the energy integral, if dissipative processes are absent. These integrals, as we shall presently verify, are I2 =fbHvdV, (2)

Proceedings of the National Academy of Sciences

Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion.

/pdf/anomalous-diffusion-models-and-their-properties-non-50bjeilyc7.pdf

Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking.

The physics of one-dimensional interacting bosonic systems is reviewed. Beginning with results from exactly solvable models and computational approaches, the concept of bosonic Tomonaga-Luttinger liquids relevant for one-dimensional Bose fluids is introduced, and compared with Bose-Einstein condensates existing in dimensions higher than one. The effects of various perturbations on the Tomonaga-Luttinger liquid state are discussed as well as extensions to multicomponent and out of equilibrium situations. Finally, the experimental systems that can be described in terms of models of interacting bosons in one dimension are discussed.

/pdf/one-dimensional-bosons-from-condensed-matter-systems-to-4vp39jxn82.pdf

One dimensional bosons: From condensed matter systems to ultracold gases

Journal of Physics A - Mathematical and General, Special Issue. SI Aug 11 2006 ?? Preface

Nonergodicity observed in single-particle tracking experiments is usually modeled by transient trapping rather than spatial disorder. We introduce models of a particle diffusing in a medium consisting of regions with random sizes and random diffusivities. The particle is never trapped but rather performs continuous Brownian motion with the local diffusion constant. Under simple assumptions on the distribution of the sizes and diffusivities, we find that the mean squared displacement displays subdiffusion due to nonergodicity for both annealed and quenched disorder. The model is formulated as a walk continuous in both time and space, similar to the Levy walk.

Nonergodic subdiffusion from Brownian motion in an inhomogeneous medium.

Molecular transport in living systems regulates numerous processes underlying biological function. Although many cellular components exhibit anomalous diffusion, only recently has the subdiffusive motion been associated with nonergodic behavior. These findings have stimulated new questions for their implications in statistical mechanics and cell biology. Is nonergodicity a common strategy shared by living systems? Which physical mechanisms generate it? What are its implications for biological function? Here, we use single-particle tracking to demonstrate that the motion of dendritic cell-specific intercellular adhesion molecule 3-grabbing nonintegrin (DC-SIGN), a receptor with unique pathogen-recognition capabilities, reveals nonergodic subdiffusion on living-cell membranes In contrast to previous studies, this behavior is incompatible with transient immobilization, and, therefore, it cannot be interpreted according to continuous-time random-walk theory. We show that the receptor undergoes changes of diffusivity, consistent with the current view of the cell membrane as a highly dynamic and diverse environment. Simulations based on a model of an ordinary random walk in complex media quantitatively reproduce all our observations, pointing toward diffusion heterogeneity as the cause of DC-SIGN behavior. By studying different receptor mutants, we further correlate receptor motion to its molecular structure, thus establishing a strong link between nonergodicity and biological function. These results underscore the role of disorder in cell membranes and its connection with function regulation. Because of its generality, our approach offers a framework to interpret anomalous transport in other complex media where dynamic heterogeneity might play a major role, such as those found, e.g., in soft condensed matter, geology, and ecology.

/pdf/weak-ergodicity-breaking-of-receptor-motion-in-living-cells-sj2hb7dw9y.pdf

Weak Ergodicity Breaking of Receptor Motion in Living Cells Stemming from Random Diffusivity

Molecular transport in living systems regulates numerous processes underlying biological function. Although many cellular components exhibit anomalous diffusion, only recently has the subdiffusive motion been associated with nonergodic behavior. These findings have stimulated new questions for their implications in statistical mechanics and cell biology. Is nonergodicity a common strategy shared by living systems? Which physical mechanisms generate it? What are its implications for biological function? Here, we use single particle tracking to demonstrate that the motion of DC-SIGN, a receptor with unique pathogen recognition capabilities, reveals nonergodic subdiffusion on living cell membranes. In contrast to previous studies, this behavior is incompatible with transient immobilization and therefore it can not be interpreted according to continuous time random walk theory. We show that the receptor undergoes changes of diffusivity, consistent with the current view of the cell membrane as a highly dynamic and diverse environment. Simulations based on a model of ordinary random walk in complex media quantitatively reproduce all our observations, pointing toward diffusion heterogeneity as the cause of DC-SIGN behavior. By studying different receptor mutants, we further correlate receptor motion to its molecular structure, thus establishing a strong link between nonergodicity and biological function. These results underscore the role of disorder in cell membranes and its connection with function regulation. Due to its generality, our approach offers a framework to interpret anomalous transport in other complex media where dynamic heterogeneity might play a major role, such as those found, e.g., in soft condensed matter, geology and ecology.

Weak ergodicity breaking of receptor motion in living cells stemming from random diffusivity

The concentration and distribution of quantum entanglement is an essential ingredient in emerging quantum information technologies. Much theoretical and experimental effort has been expended in understanding how to distribute entanglement in one-dimensional networks. However, as experimental techniques in quantum communication develop, protocols for multi-dimensional systems become essential. Here, we focus on recent theoretical developments in protocols for distributing entanglement in regular and complex networks, with particular attention to percolation theory and network-based error correction.

https://iopscience.iop.org/article/10.1088/0034-4885/76/9/096001/pdf

Gerald J. Lapeyre

Papers

Nonergodic subdiffusion from Brownian motion in an inhomogeneous medium.

Weak Ergodicity Breaking of Receptor Motion in Living Cells Stemming from Random Diffusivity

Weak ergodicity breaking of receptor motion in living cells stemming from random diffusivity

Distribution of entanglement in large-scale quantum networks.

Electronic Structure and Kinetics of the Oxidation of Ba and Sr