arXiv subjects

Carl P. Dettmann

Publications and source records attributed to Carl P. Dettmann.

At least 73 records · Page 4Linked to original sources

arXiv2014

Connectivity in Dense Networks Confined within Right Prisms

We consider the probability that a dense wireless network confined within a given convex geometry is fully connected. We exploit a recently reported theory to develop a systematic methodology for analytically characterizing the connectivity probability when the network resides within a convex right prism, a polyhedron that accurately models many geometries that can be found in practice. To maximize practicality and applicability, we adopt a general point-to-point link model based on outage probability, and present example analytical and numerical results for a network employing $2 \times 2$ multiple-input multiple-output (MIMO) maximum ratio combining (MRC) link level transmission confined within particular bounding geometries. Furthermore, we provide suggestions for extending the approach detailed herein to more general convex geometries.

cs.NI↗

arXiv2014

Keyhole and Reflection Effects in Network Connectivity Analysis

Recent research has demonstrated the importance of boundary effects on the overall connection probability of wireless networks, but has largely focused on convex domains. We consider two generic scenarios of practical importance to wireless communications, in which one or more nodes are located outside the convex space where the remaining nodes reside. Consequently, conventional approaches with the underlying assumption that only line-of-sight (LOS) or direct connections between nodes are possible, fail to provide the correct analysis for the connectivity. We present an analytical framework that explicitly considers the effects of reflections from the system boundaries on the full connection probability. This study provides a different strategy to ray tracing tools for predicting the wireless propagation environment. A simple two-dimensional geometry is first considered, followed by a more practical three-dimensional system. We investigate the effects of different system parameters on the connectivity of the network though analysis corroborated by numerical simulations, and highlight the potential of our approach for more general non-convex geometries.t system parameters on the connectivity of the network through simulation and analysis.

cs.IT↗

arXiv2014

Network Connectivity: Stochastic vs. Deterministic Wireless Channels

We study the effect of stochastic wireless channel models on the connectivity of ad hoc networks. Unlike in the deterministic geometric disk model where nodes connect if they are within a certain distance from each other, stochastic models attempt to capture small-scale fading effects due to shadowing and multipath received signals. Through analysis of local and global network observables, we present conclusive evidence suggesting that network behaviour is highly dependent upon whether a stochastic or deterministic connection model is employed. Specifically we show that the network mean degree is lower (higher) for stochastic wireless channels than for deterministic ones, if the path loss exponent is greater (lesser) than the spatial dimension. Similarly, the probability of forming isolated pairs of nodes in an otherwise dense random network is much less for stochastic wireless channels than for deterministic ones. The latter realisation explains why the upper bound of $k$-connectivity is tighter for stochastic wireless channels. We obtain closed form analytic results and compare to extensive numerical simulations.

cs.NI↗

arXiv2013

Connectivity of confined 3D Networks with Anisotropically Radiating Nodes

Nodes in ad hoc networks with randomly oriented directional antenna patterns typically have fewer short links and more long links which can bridge together otherwise isolated subnetworks. This network feature is known to improve overall connectivity in 2D random networks operating at low channel path loss. To this end, we advance recently established results to obtain analytic expressions for the mean degree of 3D networks for simple but practical anisotropic gain profiles, including those of patch, dipole and end-fire array antennas. Our analysis reveals that for homogeneous systems (i.e. neglecting boundary effects) directional radiation patterns are superior to the isotropic case only when the path loss exponent is less than the spatial dimension. Moreover, we establish that ad hoc networks utilizing directional transmit and isotropic receive antennas (or vice versa) are always sub-optimally connected regardless of the environment path loss. We extend our analysis to investigate boundary effects in inhomogeneous systems, and study the geometrical reasons why directional radiating nodes are at a disadvantage to isotropic ones. Finally, we discuss multi-directional gain patterns consisting of many equally spaced lobes which could be used to mitigate boundary effects and improve overall network connectivity.

cs.IT↗

arXiv2013

Periodic compression of an adiabatic gas: Intermittency enhanced Fermi acceleration

A gas of noninteracting particles diffuses in a lattice of pulsating scatterers. In the finite horizon case with bounded distance between collisions and strongly chaotic dynamics, the velocity growth (Fermi acceleration) is well described by a master equation, leading to an asymptotic universal non-Maxwellian velocity distribution scaling as v ~ t. The infinite horizon case has intermittent dynamics which enhances the acceleration, leading to v ~ t ln t and a non-universal distribution.

nlin.CD↗

arXiv2013

k-connectivity for confined random networks

k-connectivity is an important measure of network robustness and resilience to random faults and disruptions. We undertake both local and global approaches to k-connectivity and calculate closed form analytic formulas for the probability that a confined random network remains fully connected after the removal of k-1 nodes. Our analysis reveals that k-connectivity is governed by microscopic details of the network domain such as sharp corners rather than the macroscopic total volume. Hence, our results can aid in the design of reliable networks, an important problem in e.g. wireless ad hoc and sensor networks.

cond-mat.dis-nn↗

arXiv2013

Network connectivity through small openings

Network connectivity is usually addressed for convex domains where a direct line of sight exists between any two transmitting/receiving nodes. Here, we develop a general theory for the network connectivity properties across a small opening, rendering the domain essentially non-convex. Our analytic approach can go only so far as we encounter what is referred to in statistical physics as quenched disorder making the problem non-trivial. We confirm our theory through computer simulations, obtain leading order approximations and discuss possible extensions and applications.

cond-mat.dis-nn↗

arXiv2012

Faster than expected escape for a class of fully chaotic maps

We investigate the dependence of the escape rate on the position of a hole placed in uniformly hyperbolic systems admitting a finite Markov partition. We derive an exact periodic orbit formula for finite size Markov holes which differs from other periodic expansions in the literature and can account for additional distortion to maps with piecewise constant expansion rate. Using asymptotic expansions in powers of hole size we show that for systems conjugate to the binary shift, the average escape rate is always larger than the expectation based on the hole size. Moreover, we show that in the small hole limit the difference between the two decays like a known constant times the square of the hole size. Finally, we relate this problem to the random choice of hole positions and we discuss possible extensions of our results to non-Markov holes as well as applications to leaky dynamical networks.

nlin.CD↗

arXiv2012

Stickiness in a bouncer model: A slowing mechanism for Fermi acceleration

Some phase space transport properties for a conservative bouncer model are studied. The dynamics of the model is described by using a two-dimensional measure preserving mapping for the variables velocity and time. The system is characterized by a control parameter $ε$ and experiences a transition from integrable ($ε=0$) to non integrable ($ε\ne 0$). For small values of $ε$, the phase space shows a mixed structure where periodic islands, chaotic seas and invariant tori coexist. As the parameter $ε$ increases and reaches a critical value $ε_c$ all invariant tori are destroyed and the chaotic sea spreads over the phase space leading the particle to diffuse in velocity and experience Fermi acceleration (unlimited energy growth). During the dynamics the particle can be temporarily trapped near periodic and stable regions. We use the finite time Lyapunov exponent to visualize this effect. The survival probability was used to obtain some of the transport properties in the phase space. For large $ε$, the survival probability decays exponentially when it turns into a slower decay as the control parameter $ε$ is reduced. The slower decay is related to trapping dynamics, slowing the Fermi Acceleration, i.e., unbounded growth of the velocity

nlin.CD↗

arXiv2012

Dependence of chaotic diffusion on the size and position of holes

A particle driven by deterministic chaos and moving in a spatially extended environment can exhibit normal diffusion, with its mean square displacement growing proportional to the time. Here we consider the dependence of the diffusion coefficient on the size and the position of areas of phase space linking spatial regions (`holes') in a class of simple one-dimensional, periodically lifted maps. The parameter dependent diffusion coefficient can be obtained analytically via a Taylor-Green-Kubo formula in terms of a functional recursion relation. We find that the diffusion coefficient varies non-monotonically with the size of a hole and its position, which implies that a diffusion coefficient can increase by making the hole smaller. We derive analytic formulas for small holes in terms of periodic orbits covered by the holes. The asymptotic regimes that we observe show deviations from the standard stochastic random walk approximation. The escape rate of the corresponding open system is also calculated. The resulting parameter dependencies are compared with the ones for the diffusion coefficient and explained in terms of periodic orbits.

math-ph↗

arXiv2012

Recurrence of particles in static and time varying oval billiards

Dynamical properties are studied for escaping particles, injected through a hole in an oval billiard. The dynamics is considered for both static and periodically moving boundaries. For the static boundary, two different decays for the recurrence time distribution were observed after exponential decay for short times: A changeover to: (i) power law or; (ii) stretched exponential. Both slower decays are due to sticky orbits trapped near KAM islands, with the stretched exponential apparently associated with a single group of large islands. For time dependent case, survival probability leads to the conclusion that sticky orbits are less evident compared with the static case.

nlin.CD↗

arXiv2012

Connectivity of Confined Dense Networks: Boundary Effects and Scaling Laws

In this paper, we study the probability that a dense network confined within a given geometry is fully connected. We employ a cluster expansion approach often used in statistical physics to analyze the effects that the boundaries of the geometry have on connectivity. To maximize practicality and applicability, we adopt four important point-to-point link models based on outage probability in our analysis: single-input single-output (SISO), single-input multiple-output (SIMO), multiple-input single-output (MISO), and multiple-input multiple-output (MIMO). Furthermore, we derive diversity and power scaling laws that dictate how boundary effects can be mitigated (to leading order) in confined dense networks for each of these models. Finally, in order to demonstrate the versatility of our theory, we analyze boundary effects for dense networks comprising MIMO point-to-point links confined within a right prism, a polyhedron that accurately models many geometries that can be found in practice. We provide numerical results for this example, which verify our analytical results.

cs.NI↗

arXiv2012

Full Connectivity: Corners, edges and faces

We develop a cluster expansion for the probability of full connectivity of high density random networks in confined geometries. In contrast to percolation phenomena at lower densities, boundary effects, which have previously been largely neglected, are not only relevant but dominant. We derive general analytical formulas that show a persistence of universality in a different form to percolation theory, and provide numerical confirmation. We also demonstrate the simplicity of our approach in three simple but instructive examples and discuss the practical benefits of its application to different models.

cond-mat.dis-nn↗

arXiv2011

Impact of boundaries on fully connected random geometric networks

Many complex networks exhibit a percolation transition involving a macroscopic connected component, with universal features largely independent of the microscopic model and the macroscopic domain geometry. In contrast, we show that the transition to full connectivity is strongly influenced by details of the boundary, but observe an alternative form of universality. Our approach correctly distinguishes connectivity properties of networks in domains with equal bulk contributions. It also facilitates system design to promote or avoid full connectivity for diverse geometries in arbitrary dimension.

cond-mat.dis-nn↗

arXiv2011

Quantifying intermittency in the open drivebelt billiard

A "drivebelt" stadium billiard with boundary consisting of circular arcs of differing radius connected by their common tangents shares many properties with the conventional "straight" stadium, including hyperbolicity and mixing, as well as intermittency due to marginally unstable periodic orbits (MUPOs). Interestingly, the roles of the straight and curved sides are reversed. Here we discuss intermittent properties of the chaotic trajectories from the point of view of escape through a hole in the billiard, giving the exact leading order coefficient $\lim_{t\to\infty} t P(t)$ of the survival probability $P(t)$ which is algebraic for fixed hole size. However, in the natural scaling limit of small hole size inversely proportional to time, the decay remains exponential. The big distinction between the straight and drivebelt stadia is that in the drivebelt case there are multiple families of MUPOs leading to qualitatively new effects. A further difference is that most marginal periodic orbits in this system are oblique to the boundary, thus permitting applications that utilise total internal reflection such as microlasers.

nlin.CD↗

arXiv2011

New horizons in multidimensional diffusion: The Lorentz gas and the Riemann Hypothesis

The Lorentz gas is a billiard model involving a point particle diffusing deterministically in a periodic array of convex scatterers. In the two dimensional finite horizon case, in which all trajectories involve collisions with the scatterers, displacements scaled by the usual diffusive factor $\sqrt{t}$ are normally distributed, as shown by Bunimovich and Sinai in 1981. In the infinite horizon case, motion is superdiffusive, however the normal distribution is recovered when scaling by $\sqrt{t\ln t}$, with an explicit formula for its variance. Here we explore the infinite horizon case in arbitrary dimensions, giving explicit formulas for the mean square displacement, arguing that it differs from the variance of the limiting distribution, making connections with the Riemann Hypothesis in the small scatterer limit, and providing evidence for a critical dimension $d=6$ beyond which correlation decay exhibits fractional powers. The results are conditional on a number of conjectures, and are corroborated by numerical simulations in up to ten dimensions.

math.DS↗

arXiv2010

Open Mushrooms: Stickiness revisited

We investigate mushroom billiards, a class of dynamical systems with sharply divided phase space. For typical values of the control parameter of the system $ρ$, an infinite number of marginally unstable periodic orbits (MUPOs) exist making the system sticky in the sense that unstable orbits approach regular regions in phase space and thus exhibit regular behaviour for long periods of time. The problem of finding these MUPOs is expressed as the well known problem of finding optimal rational approximations of a real number, subject to some system-specific constraints. By introducing a generalized mushroom and using properties of continued fractions, we describe a zero measure set of control parameter values $ρ\in(0,1)$ for which all MUPOs are destroyed and therefore the system is less sticky. The open mushroom (billiard with a hole) is then considered in order to quantify the stickiness exhibited and exact leading order expressions for the algebraic decay of the survival probability function $P(t)$ are calculated for mushrooms with triangular and rectangular stems.

math.DS↗

arXiv2010

Asymmetric transport in the bouncer model: Mixed, time dependent, noncompact dynamics

We consider time-dependence of dynamical transport, following a recent study of the stadium billiard in which classical transmission and reflection probabilities were shown to exhibit exponential or algebraic decay depending on the choice of the lead or "hole". The system considered here is much more general, having a generic mixed phase space structure, time-dependence of the dynamics, and Fermi acceleration (trajectories with unbounded velocity). We propose an efficient numerical scheme for this model, observe the asymmetric transport effect, and discuss observed stretched exponential decays.

nlin.CD↗