scholarly journals Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 231
Author(s):  
M. Hidalgo-Soria ◽  
E. Barkai ◽  
S. Burov
Keyword(s):  
Diffusion Coefficient ◽  
Initial Conditions ◽  
Waiting Times ◽  
Effective Diffusion ◽  
Time Behavior ◽  
Mean Values ◽  
Gaussian Density ◽  
Long Time ◽  
Non Gaussian ◽  
Short Time

We study a two state “jumping diffusivity” model for a Brownian process alternating between two different diffusion constants, D+>D−, with random waiting times in both states whose distribution is rather general. In the limit of long measurement times, Gaussian behavior with an effective diffusion coefficient is recovered. We show that, for equilibrium initial conditions and when the limit of the diffusion coefficient D−⟶0 is taken, the short time behavior leads to a cusp, namely a non-analytical behavior, in the distribution of the displacements P(x,t) for x⟶0. Visually this cusp, or tent-like shape, resembles similar behavior found in many experiments of diffusing particles in disordered environments, such as glassy systems and intracellular media. This general result depends only on the existence of finite mean values of the waiting times at the different states of the model. Gaussian statistics in the long time limit is achieved due to ergodicity and convergence of the distribution of the temporal occupation fraction in state D+ to a δ-function. The short time behavior of the same quantity converges to a uniform distribution, which leads to the non-analyticity in P(x,t). We demonstrate how super-statistical framework is a zeroth order short time expansion of P(x,t), in the number of transitions, that does not yield the cusp like shape. The latter, considered as the key feature of experiments in the field, is found with the first correction in perturbation theory.

Energy-Aware Chaotic Communication in Wireless Sensor Network

2013 ◽  
Vol 367 ◽  
pp. 536-540 ◽  
Author(s):  
Raju Dutta ◽  
Shishir Gupta ◽  
Mukul K. Das
Keyword(s):  
Wireless Sensor Network ◽  
Sensor Network ◽  
Initial Conditions ◽  
Wireless Sensor ◽  
Time Behavior ◽  
Nonlinear Mathematical Model ◽  
Energy Aware ◽  
Long Time ◽  
Small Change ◽  
Authentic Data

A challenging task in wireless sensor network (WSN) is to deliver authentic data between source nodes and sink nodes. The collision or dead lock occurs when two or more close nodes are attempted to send data at the same time to the others node. To avoid such dead lock situation in the network we propose a nonlinear mathematical model. The effect of nonlinearity often renders a periodic solution unstable for certain parametric choices even a very small change in initial conditions can lead to different result in chaotic systems which appears to exhibit chaos for a range of parametric values when long time behavior studied. The local stability conditions for the system have been discussed and analyzed. Numerically simulations have been carried out to study the complex behavior of the system for reasonable ranges of parameters in WSN.

scholarly journals Small-data shock formation in solutions to 3D quasilinear wave equations: An overview

2016 ◽  
Vol 13 (01) ◽  
pp. 1-105 ◽  
Author(s):  
Gustav Holzegel ◽  
Sergiu Klainerman ◽  
Jared Speck ◽  
Willie Wai-Yeung Wong
Keyword(s):  
Wave Equations ◽  
Initial Conditions ◽  
Large Body ◽  
Small Data ◽  
Shock Formation ◽  
Time Behavior ◽  
Long Time ◽  
Remarkable Result ◽  
Quasilinear Wave Equations ◽  
Three Space

In his 2007 monograph, Christodoulou proved a remarkable result giving a detailed description of shock formation, for small [Formula: see text]-initial conditions (with [Formula: see text] sufficiently large), in solutions to the relativistic Euler equations in three space dimensions. His work provided a significant advancement over a large body of prior work concerning the long-time behavior of solutions to higher-dimensional quasilinear wave equations, initiated by John in the mid 1970’s and continued by Klainerman, Sideris, Hörmander, Lindblad, Alinhac, and others. Our goal in this paper is to give an overview of his result, outline its main new ideas, and place it in the context of the above mentioned earlier work. We also introduce the recent work of Speck, which extends Christodoulou’s result to show that for two important classes of quasilinear wave equations in three space dimensions, small-data shock formation occurs precisely when the quadratic nonlinear terms fail to satisfy the classic null condition.

scholarly journals SURVIVAL PROBABILITY (HEAT CONTENT) AND THE LOWEST EIGENVALUE OF DIRICHLET LAPLACIAN

2011 ◽  
Vol 25 (15) ◽  
pp. 1993-2007
Author(s):  
PAVOL KALINAY ◽  
LADISLAV ŠAMAJ ◽  
IGOR TRAVĚNEC
Keyword(s):  
Survival Probability ◽  
Heat Content ◽  
Simple Algorithm ◽  
Time Behavior ◽  
Dirichlet Laplacian ◽  
Geometric Characteristics ◽  
Long Time ◽  
Time Expansion ◽  
Short Time ◽  
Lowest Eigenvalue

We study the survival probability of a particle diffusing in a two-dimensional domain, bounded by a smooth absorbing boundary. The short-time expansion of this quantity depends on the geometric characteristics of the boundary, whilst its long-time asymptotics is governed by the lowest eigenvalue of the Dirichlet Laplacian defined on the domain. We present a simple algorithm for calculation of the short-time expansion for an arbitrary "star-shaped" domain. The coefficients are expressed in terms of powers of boundary curvature, integrated around the circumference of the domain. Based on this expansion, we look for a Padé interpolation between the short-time and the long-time behavior of the survival probability, i.e., between geometric characteristics of the boundary and the lowest eigenvalue of the Dirichlet Laplacian.

Perturbation dynamics in unsteady pipe flows

2007 ◽  
Vol 570 ◽  
pp. 129-154 ◽  
Author(s):  
M. ZHAO ◽  
M. S. GHIDAOUI ◽  
A. A. KOLYSHKIN
Keyword(s):  
Growth Mechanism ◽  
Flow Instability ◽  
Initial Conditions ◽  
Laminar Flows ◽  
Base Flow ◽  
Linear Stability Theory ◽  
Transient Growth ◽  
Energy Growth ◽  
Long Time ◽  
Short Time

This paper deals with perturbed unsteady laminar flows in a pipe. Three types of flows are considered: a flow accelerated from rest; a flow in a pipe generated by the controlled motion of a piston; and a water hammer flow where the transient is generated by the instantaneous closure of a valve. Methods of linear stability theory are used to analyse the behaviour of small perturbations in the flow. Since the base flow is unsteady, the linearized problem is formulated as an initial-value problem. This allows us to consider arbitrary initial conditions and describe both short-time and long-time evolution of the flow. The role of initial conditions on short-time transients is investigated. It is shown that the phenomenon of transient growth is not associated with a certain type of initial conditions. Perturbation dynamics is also studied for long times. In addition, optimal perturbations, i.e. initial perturbations that maximize the energy growth, are determined for all three types of flow discussed. Despite the fact that these optimal perturbations, most probably, will not occur in practice, they do provide an upper bound for energy growth and can be used as a point of reference. Results of numerical simulation are compared with previous experimental data. The comparison with data for accelerated flows shows that the instability cannot be explained by long-time asymptotics. In particular, the method of normal modes applied with the quasi-steady assumption will fail to predict the flow instability. In contrast, the transient growth mechanism may be used to explain transition since experimental transition time is found to be in the interval where the energy of perturbation experiences substantial growth. Instability of rapidly decelerated flows is found to be associated with asymptotic growth mechanism. Energy growth of perturbations is used in an attempt to explain previous experimental results. Numerical results show satisfactory agreement with the experimental features such as the wavelength of the most unstable mode and the structure of the most unstable disturbance. The validity of the quasi-steady assumption for stability studies of unsteady non-periodic laminar flows is discussed.

Response of a Tethered Aerostat to Simulated Turbulence

2004 ◽  
Author(s):  
Keith A. Stanney ◽  
Christopher D. Rahn
Keyword(s):  
Atmospheric Turbulence ◽  
Initial Conditions ◽  
Probability Distributions ◽  
Decomposition Theorem ◽  
Worst Case ◽  
Gaussian Density ◽  
Time Histories ◽  
Wind Input ◽  
Non Gaussian ◽  
Proper Orthogonal

Aerostats are lighter-than-air vehicles tethered to the ground by a cable and used for broadcasting, communications, surveillance, and drug interdiction. The dynamic response of tethered aerostats subject to extreme atmospheric turbulence often dictates survivability. This paper develops a theoretical model that predicts the planar response of a tethered aerostat subject to atmospheric turbulence and simulates the response to 1000 simulated hurricane scale turbulent time histories. The aerostat dynamic model assumes the aerostat hull to be a rigid body with nonlinear fluid loading, instantaneous weathervaning for planar response, and a continuous tether. Galerkin’s method discretizes the coupled aerostat and tether partial differential equations to produce a nonlinear initial value problem that is integrated numerically given initial conditions and wind inputs. The proper orthogonal decomposition theorem generates, based on Hurricane Georges wind data, turbulent time histories that possess the sequential behavior of actual turbulence, are spectrally accurate, and have non-Gaussian density functions. The generated turbulent time histories are simulated to predict the aerostat response to severe turbulence. The resulting probability distributions for the aerostat position, pitch angle, and confluence point tension predict the aerostat behavior in high gust environments. The results uncover a worst case wind input consisting of a two-pulse vertical gust.

Short-time behavior of the diffusion coefficient as a geometrical probe of porous media

1993 ◽  
Vol 47 (14) ◽  
pp. 8565-8574 ◽  
Author(s):  
Partha P. Mitra ◽  
Pabitra N. Sen ◽  
Lawrence M. Schwartz
Keyword(s):  
Porous Media ◽  
Diffusion Coefficient ◽  
Time Behavior ◽  
Short Time

Stress Distribution in Anisotropic Viscoelastic Hollow Sphere

1965 ◽  
Vol 32 (1) ◽  
pp. 31-36 ◽  
Author(s):  
S. R. Moghe ◽  
C. C. Hsiao
Keyword(s):  
Hollow Sphere ◽  
Homogeneous Medium ◽  
Small Strain ◽  
Deformation Analysis ◽  
Isotropic Case ◽  
Time Behavior ◽  
Long Time ◽  
Linear Viscoelastic ◽  
Stress And Deformation ◽  
Short Time

The stress and deformation analysis of an anisotropic linear viscoelastic hollow sphere under a symmetrical loading on the boundaries is considered. In order to reduce some of the mathematical difficulties, assumptions of small strain with quasistatic stress-strain conditions are consistently made. The solution is found for a system of non-homogeneous anisotropic viscoelastic media, and subsequently reduced to that for a homogeneous medium. Numerical results for both short-time and long-time behavior are presented for a number of simple model media in the anisotropic viscoelastic case and compared with the result for the elastic or viscoelastic medium in the isotropic case.

scholarly journals Transition of Liesegang Precipitation Systems: Simulations with an Adaptive Grid PDE Method

2011 ◽  
Vol 10 (4) ◽  
pp. 867-881 ◽  
Author(s):  
Paul A. Zegeling ◽  
István Lagzi ◽  
Ferenc Izsák
Keyword(s):  
Pattern Formation ◽  
Numerical Simulations ◽  
Initial Conditions ◽  
Qualitative Change ◽  
Adaptive Grid ◽  
Time Behavior ◽  
Centrally Symmetric ◽  
Type Pattern ◽  
Long Time ◽  
Pde Method

AbstractThe dynamics of the Liesegang type pattern formation is investigated in a centrally symmetric two-dimensional setup. According to the observations in real experiments, the qualitative change of the dynamics is exhibited for slightly different initial conditions. Two kinds of chemical mechanisms are studied; in both cases the pattern formation is described using a phase separation model including the Cahn-Hilliard equations. For the numerical simulations we make use of an adaptive grid PDE method, which successfully deals with the computationally critical cases such as steep gradients in the concentration distribution and investigation of long time behavior. The numerical simulations show a good agreement with the real experiments.

Understanding evolution of vortex rings in viscous fluids

2017 ◽  
Vol 836 ◽  
pp. 873-909 ◽  
Author(s):  
Aashay Tinaikar ◽  
S. Advaith ◽  
S. Basu
Keyword(s):  
High Speed ◽  
Initial Conditions ◽  
Operating Conditions ◽  
Force Balance ◽  
Vortex Rings ◽  
Time Behaviour ◽  
Theoretical Predictions ◽  
Long Time ◽  
Vortex Size ◽  
Short Time

The evolution of vortex rings in isodensity and isoviscosity fluid has been studied analytically using a novel mathematical model. The model predicts the spatiotemporal variation in peak vorticity, circulation, vortex size and spacing based on instantaneous vortex parameters. This proposed model is quantitatively verified using experimental measurements. Experiments are conducted using high-speed particle image velocimetry (PIV) and laser induced fluorescence (LIF) techniques. Non-buoyant vortex rings are generated from a nozzle using a constant hydrostatic pressure tank. The vortex Reynolds number based on circulation $(\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D708})$ is varied in the range 100–1500 to account for a large range of operating conditions. Experimental results show good agreement with theoretical predictions. However, it is observed that neither Saffman’s thin-core model nor the thick-core equations could correctly explain vortex evolution for all initial conditions. Therefore, a transitional theory is framed using force balance equations which seamlessly integrate short- and long-time asymptotic theories. It is found that the parameter $A=(a/\unicode[STIX]{x1D70E})^{2}$, where $a$ is the vortex half-spacing and $\unicode[STIX]{x1D70E}$ denotes the standard deviation of the Gaussian vorticity profile, governs the regime of vortex evolution. For higher values of $A$, evolution follows short-time behaviour, while for $A=O(1)$, long-time behaviour is prominent. Using this theory, many reported anomalous observations have been explained.

scholarly journals $m$th-order Fisher-KPP equation with free boundaries and time-aperiodic advection

2021 ◽  
Author(s):  
Changqing Ji ◽  
Dandan Zhu ◽  
Jingli Ren
Keyword(s):  
Large Amplitude ◽  
Small Amplitude ◽  
Initial Conditions ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Free Boundaries ◽  
Kpp Equation ◽  
Advection Term ◽  
Long Time ◽  
Theoretical Results

In this paper, we investigate a $m$th-order Fisher-KPP equation with free boundaries and time-aperiodic advection. Considering the influence of advection term and initial conditions on the long time behavior of solutions, we obtain spreading-vanishing dichotomy, spreading-transition-vanishing trichotomy, and vanishing happens with the coefficient of advection term in small amplitude, medium-sized amplitude and large amplitude, respectively. Then, the appropriate parameters are selected in the simulation to intuitively show the corresponding theoretical results. Moreover, the wave-spreading and wave-vanishing cases of the solutions are observed in our study.

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