Telegraphic transport processes and their fractional generalization: a review and some extensions

We address the problem of telegraphic transport in several dimensions. We review the derivation of two and three dimensional telegrapher's equations¿as well as their fractional generalizations¿from microscopic random walk models for transport (normal and anomalous). We also present new results on solutions of the higher dimensional fractional equations

From semi-Markov random evolutions to scattering transport and superdiffusion

We here study random evolutions on Banach spaces, driven by a class of semi-Markov processes. The expectation (in the sense of Bochner) of such evolutions is shown to solve some abstract Cauchy problems. Further, the abstract telegraph (damped wave) equation is generalized to the case of semi-Markov perturbations. A special attention is devoted to semi-Markov models of scattering transport processes which can be represented through these evolutions. In particular, we consider random flights with infinite mean flight times which turn out to be governed by a semi-Markov generalization of a linear Boltzmann equation; their scaling limit is proved to converge to superdiffusive transport processes

Relativistic Brownian Motion

Stimulated by experimental progress in high energy physics and astrophysics, the unification of relativistic and stochastic concepts has re-attracted considerable interest during the past decade. Focusing on the framework of special relativity, we review, here, recent progress in the phenomenological description of relativistic diffusion processes. After a brief historical overview, we will summarize basic concepts from the Langevin theory of nonrelativistic Brownian motions and discuss relevant aspects of relativistic equilibrium thermostatistics. The introductory parts are followed by a detailed discussion of relativistic Langevin equations in phase space. We address the choice of time parameters, discretization rules, relativistic fluctuation-dissipation theorems, and Lorentz transformations of stochastic differential equations. The general theory is illustrated through analytical and numerical results for the diffusion of free relativistic Brownian particles. Subsequently, we discuss how Langevin-type equations can be obtained as approximations to microscopic models. The final part of the article is dedicated to relativistic diffusion processes in Minkowski spacetime. Due to the finiteness of velocities in relativity, nontrivial relativistic Markov processes in spacetime do not exist; i.e., relativistic generalizations of the nonrelativistic diffusion equation and its Gaussian solutions must necessarily be non-Markovian. We compare different proposals that were made in the literature and discuss their respective benefits and drawbacks. The review concludes with a summary of open questions, which may serve as a starting point for future investigations and extensions of the theory.Comment: review article, 159 pages, references updated, misprints corrected, App. A.4. correcte

Extended Poisson-Kac theory: A unifying framework for stochastic processes with finite propagation velocity

Stochastic processes play a key role for mathematically modeling a huge variety of transport problems out of equilibrium. To formulate models of stochastic dynamics the mainstream approach consists in superimposing random fluctuations on a suitable deterministic evolution. These fluctuations are sampled from probability distributions that are prescribed a priori, most commonly as Gaussian or Levy. While these distributions are motivated by (generalised) central limit theorems they are nevertheless unbounded. This property implies the violation of fundamental physical principles such as special relativity and may yield divergencies for basic physical quantities like energy. It is thus clearly never valid in real-world systems by rendering all these stochastic models ontologically unphysical. Here we solve the fundamental problem of unbounded random fluctuations by constructing a comprehensive theoretical framework of stochastic processes possessing finite propagation velocity. Our approach is motivated by the theory of Levy walks, which we embed into an extension of conventional Poisson-Kac processes. Our new theory possesses an intrinsic flexibility that enables the modelling of many different kinds of dynamical features, as we demonstrate by three examples. The corresponding stochastic models capture the whole spectrum of diffusive dynamics from normal to anomalous diffusion, including the striking Brownian yet non Gaussian diffusion, and more sophisticated phenomena such as senescence. Extended Poisson-Kac theory thus not only ensures by construction a mathematical representation of physical reality that is ontologically valid at all time and length scales. It also provides a toolbox of stochastic processes that can be used to model potentially any kind of finite velocity dynamical phenomena observed experimentally.Comment: 25 pages, 5 figure

Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

From car traffic to production flows:a guided tour through solvable stochastic transport processes

The purpose of this thesis is to show on explicit examples how various theoretical concepts, ranging from statistical mechanics to stochastic control and from traffic theory to queuing systems, can be transferred to transport processes, encountered in particular in manufacturing systems, with benefic implications for their dynamical understanding, optimization and control. The thesis collects several articles where such implications are exposed [38]-[43]. We start with the observation that car traffic and production flows share several common dynamical properties (chapter 3). The main reason for the similarities are the presence of non-linear interactions in both settings. In traffic theory the interactions are between competing cars and originate from a trade off between safe and fast driving. They directly influence the speed of the cars. In production flow engineering the interactions are between cooperating work-cells forming the manufacturing system. They govern the production policy and hence the throughput of the manufacturing system. We exploit this analogy in case of a serial production line where the influence on the production rate of a work-cell is determined by the contents of its adjacent buffers (fig. 0.1) and derive a dictionary between the two fields. As a first result, this analogy allows the recognition of free-flow and jamming-flow regimes —well studied in traffic theory — in the context of production lines. Fig. 0.1. Above: Sketch of a serial production line composed of N machines Mi with production rates vi and N -1 buffers Bi with buffer content yi. Below: Sketch of a one-lane traffic system composed of N cars with velocities vi and headways xi. Dynamical similarities between cars and work-cells: the production rates and the car velocities, depend both on their environment e.g., the content of the next nearest buffers vi = vi(yi-1, yi) resp. the distances to the next nearest cars vi = vi(xi-1, xi). Applying a linear stability analysis to a given stationary flow regime, we draw a flow diagram which defines the boundary between the free and the jammed regime as a function of the control parameters. The relevant conclusions include the introduction of a dimensionless performance parameter, an enlightening connection between transient and stationary performance measures for production lines, a discussion of both the bull-whip effect and the stabilizing effect of pull production controls in serial production lines. The traffic models used in the analogy with serial production lines are socalled optimal-velocity car following models which assume that the velocity of a car is adapted to a distance dependent optimal velocity which reflects the safety requirements of two neighboring cars. This optimal velocity is chosen in an ad hoc fashion by traffic engineers and is not related to a cost functional which defines "optimality" via a minimization procedure. Here we calculate in the context of serial production lines the "optimal velocity" (i.e., the optimal production control) based on a specific cost functional. We solve in chapter 4 an optimal control problem for the production rates where the cost structure penalizes the entrance of the buffer content into a boundary state. We show that the optimal control is of four thresholds type and give the optimal position of the thresholds. The optimal control problem, explicitly discussed for a serial two-stage production line, can not be solved analytically for longer lines. This forces us to look in chapter 5 for other ways to describe relations between the throughput and the work in process of production flows. The analogous quantities in traffic theory — flow of cars and car density — are related in the so-called fundamental diagram (fig. 0.2). It encodes in a single graph the functional relation between the flow of cars and the car-density. Inspired by the micro-macro paradigm of mechanical statistics, we derive from a mesoscopic level the fundamental diagram introduced by Greenshields in 1931. The study is based on the Boltzmann equations introduced by Ruijgrok and Wu, which we derive from a space discrete interacting particle system. The fundamental kinetic features of the microscopic model are migration, reaction and collisions of particles. Performing the hydrodynamic limit of the model, we have that the macroscopic density distribution ρ is governed by the Burgers equation and that the macroscopic flow J is proportional to the logistic equation. Fig. 0.2. Generic form of the density-flow relation in one-lane car traffic. Another property of production flows shared with cars in traffic is the simple fact that the circulating items have spatial extensions. This is of foremost importance especially when multiplexing structures are present in the production line and/or the traffic network. The distribution of items flowing out of a merge structure into a single collecting flow definitely depends on the physical size of the circulating items. In chapter 6 we will study a discrete materials flow merge system connected to a downstream station (fig. 0.3). The outflow process from the merge as a function of the the items extensions is given. Fig. 0.3. Merging of N streams of items into a buffer B. A conveyor transports the items from B to M. The spatial extensions of the items are crucial for the outflow. The mentioned discrete velocities Boltzmann equations of Ruijgrok and Wu are related to random evolutions. They are particularly well adapted to model the dynamics of failure prone machines switching between their states (e.g., between "on" and "off"). For the inhomogeneous two-states case (i.e., when the switching rates depend on the environment), we show in chapter 7 that the probability density and the associated probability current are in a supersymmetric relation — a algebraic structure well known in quantum mechanics. The quest to optimize throughput in stochastic manufacturing systems and vehicles flow in traffic systems can be unified through the following question: Given the initial distribution of items (of workload or cars) how do I have to influence the noisy dynamics in order to efficiently transport the items involved (workpieces or cars) to a given final distribution? This point of view seems natural to us and is directly related to a problem addressed by E. Schrödinger in 1931. He asks for a Markov diffusion process satisfying given initial and final conditions and which minimizes some energy functional. Based on this, we propose in chapter 8 an efficiency measure relevant for a large class of diffusion-mediated transport processes

Abstracts on Radio Direction Finding (1899 - 1995)

The files on this record represent the various databases that originally composed the CD-ROM issue of "Abstracts on Radio Direction Finding" database, which is now part of the Dudley Knox Library's Abstracts and Selected Full Text Documents on Radio Direction Finding (1899 - 1995) Collection. (See Calhoun record https://calhoun.nps.edu/handle/10945/57364 for further information on this collection and the bibliography). Due to issues of technological obsolescence preventing current and future audiences from accessing the bibliography, DKL exported and converted into the three files on this record the various databases contained in the CD-ROM. The contents of these files are: 1) RDFA_CompleteBibliography_xls.zip [RDFA_CompleteBibliography.xls: Metadata for the complete bibliography, in Excel 97-2003 Workbook format; RDFA_Glossary.xls: Glossary of terms, in Excel 97-2003 Workbookformat; RDFA_Biographies.xls: Biographies of leading figures, in Excel 97-2003 Workbook format]; 2) RDFA_CompleteBibliography_csv.zip [RDFA_CompleteBibliography.TXT: Metadata for the complete bibliography, in CSV format; RDFA_Glossary.TXT: Glossary of terms, in CSV format; RDFA_Biographies.TXT: Biographies of leading figures, in CSV format]; 3) RDFA_CompleteBibliography.pdf: A human readable display of the bibliographic data, as a means of double-checking any possible deviations due to conversion