Finite Speed Research Articles

Interfacial fatigue fracture of pressure sensitive adhesives

Pressure sensitive adhesives (PSAs) are viscoelastic polymers that can form fast and robust adhesion with various adherends under fingertip pressure. The rapidly expanding application domain of PSAs, such as healthcare, wearable electronics, and flexible displays, requires PSAs to sustain prolonged loads throughout their lifetime, calling for fundamental studies on their fatigue behaviors. However, fatigue of PSAs has remained poorly investigated. Here we study interfacial fatigue fracture of PSAs, focusing on the cyclic interfacial crack propagation due to the gradual rupture of noncovalent bonds between a PSA and an adherend. We fabricate a model PSA made of a hysteresis-free poly(butyl acrylate) bulk elastomer dip-coated with a viscoelastic poly(butyl acrylate-co-isobornyl acrylate) sticky surface, both crosslinked by poly(ethylene glycol) diacrylate. We adhere the fabricated PSA to a polyester strip to form a bilayer. The bilayer is covered by another polyester film as an inextensible backing layer. Using cyclic and monotonic peeling tests, we characterize the interfacial fatigue and fracture behaviors of the bilayer. From the experimental data, we obtain the interfacial fatigue threshold (4.6J/m2) under cyclic peeling, the slow crack threshold (33.9J/m2) under monotonic peeling, and the adhesion toughness (~ 400J/m2) at a finite peeling speed. We develop a modified Lake-Thomas model to describe the interfacial fatigue threshold due to noncovalent bond breaking. The theoretical prediction (2.6J/m2) agrees well with the experimental measurement (4.6J/m2). Finally, we discuss possible additional dissipation mechanisms involved in the larger slow crack threshold and much larger adhesion toughness. It is hoped that this study will provide new fundamental knowledge for fracture mechanics of PSAs, as well as guidance for future tough and durable PSAs.

Wasserstein speed limits for Langevin systems

Physical systems transition between states with finite speed that is limited by energetic costs. In this work, we derive bounds on transition times for general Langevin systems that admit a decomposition into reversible and irreversible dynamics, in terms of the Wasserstein distance between states and the energetic costs associated with respective reversible and irreversible currents. For illustration we discuss Brownian particles subject to arbitrary forcing and an RLC circuit with time-varying inductor. Published by the American Physical Society 2024

Formation control and lag formation control for human–machine interaction second-order multiagent systems

In this paper, the formation control and lag formation control problems for second-order multiagent systems (MASs) with and without nonlinear intrinsic dynamics are discussed, in which a fraction of agents are controlled by humans and the others are free from human intervention. By selecting suitable control protocol and Lyapunov functional, a formation criterion for the second-order nonlinear MAS is derived. Furthermore, considering that time delay always exists between agent and virtual leader due to finite transmission speed, a lag formation criterion for the second-order nonlinear MAS is also obtained by selecting appropriate control strategy. In addition, the above obtained results are further extend to the double-integrator MAS. Finally, the devised control strategies are verified by exploiting two numerical examples.

The high dimensional Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry, part 2: Sharp estimates

This is the second part of a two-part series devoted to an in depth understanding of the dynamical behaviour of the radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary |x|=h(t). In Part 1 [19], we have shown that the long-time dynamics of this problem is characterised by a spreading-vanishing dichotomy, and there exists a threshold condition on the diffusion kernel J(|x|) such that the spreading speed is ∞ when this condition is not satisfied, and when it is satisfied, the finite spreading speed c0 is determined by an associated semi-wave problem established in [15]. In Part 2 here, we obtain more precise description of the spreading profile by focusing on some natural classes of kernel functions, including those satisfying J(|x|)∼|x|−β for |x|≫1 in RN. Our results for such kernels reveal a striking difference of behaviour from the pattern exhibited in the one dimension case [18] when β crosses the value N+2. More precisely, (a) when β∈(N,N+1], we show that for t≫1, h(t)∼t1/(β−N) if β∈(N,N+1), and h(t)∼tln⁡t if β=N+1, which is of the same pattern as in dimension one, namely we recover the result in [18] by letting N=1 in the above statements; (b) when β∈(N+1,N+2], the front has a finite spreading speed c0=c0(β) in the sense that limt→∞⁡h(t)/t=c0, and our results here on the order of shift c0t−h(t) are again of the same pattern as in dimension one; (c) when β>N+2, the front still has a finite spreading speed c0, but a significant change happens to the order of shift c0t−h(t) between N≥2 and N=1: for t≫1, c0t−h(t)∼ln⁡t when N≥2, but c0t−h(t)∼1 when N=1.

A new local and explicit kinetic method for linear and non-linear convection-diffusion problems with finite kinetic speeds: II. Multi-dimensional case

We extend to multi-dimensions the work of [1], where new fully explicit kinetic methods were built for the approximation of linear and non-linear convection-diffusion problems. The fundamental principles from the earlier work are retained: (1) rather than aiming for the desired equations in the strict limit of a vanishing relaxation parameter, as is commonly done in the diffusion limit of kinetic methods, diffusion terms are sought as a first-order correction of this limit in a Chapman-Enskog expansion, (2) introducing a coupling between the conserved variables within the relaxation process by a specifically designed collision matrix makes it possible to systematically match a desired diffusion. Extending this strategy to multi-dimensions cannot, however, be achieved through simple directional splitting, as diffusion is likely to couple space directions with each other, such as with shear viscosity in the Navier-Stokes equations. In this work, we show how rewriting the collision matrix in terms of moments can address this issue, regardless of the number of kinetic waves, while ensuring conservation systematically. This rewriting allows for introducing a new class of kinetic models called regularized models, simplifying the numerical methods and establishing connections with Jin-Xin models. Subsequently, new explicit arbitrary high-order kinetic schemes are formulated and validated on standard two-dimensional cases from the literature. Excellent results are obtained in the simulation of a shock-boundary layer interaction, validating their ability to approximate the Navier-Stokes equations with kinetic speeds obeying nothing but a subcharacteristic condition along with a hyperbolic constraint on the time step.

Quantum Retarded Field Engine

Recent efforts to conceptually design a technologically meaningful electromagnetic retarded engine indicated that this can only be done using the immense charge and current densities which exist in the atomic scale. However, this scale cannot be described by Newtonian physics, and only a quantum description will suffice to describe the dynamics of an electron on this scale properly. Here we study the retarded field quantum engine and highlight the differences between the quantum and the classical retarded engines. It is emphasized that the constituents of the retarded engine studied in the current paper do not move in relativistic speeds, hence they are analyzed using un-relativistic classical mechanics and un-relativistic quantum mechanics (Schrödinger’s equations). The retardation effect is due to the finite propagation speed of the field and not the relativistic motion of the particles.

A new local and explicit kinetic method for linear and non-linear convection-diffusion problems with finite kinetic speeds: I. One-dimensional case

We propose a numerical approach, of the BGK kinetic type, that is able to approximate with a given, but arbitrary, order of accuracy the solution of linear and non-linear convection-diffusion type problems: scalar advection-diffusion, non-linear scalar problems of this type and the compressible Navier-Stokes equations. Our kinetic model can use finite advection speeds that are independent of the relaxation parameter, and the time step does not suffer from a parabolic constraint. Having finite speeds is in contrast with many of the previous works about this kind of approach, and we explain why this is possible: paraphrasing more or less [1], the convection-diffusion like PDE is not a limit of the BGK equation, but a correction of the same PDE without the parabolic term at the second order in the relaxation parameter that is interpreted as Knudsen number. We then show that introducing a relaxation matrix instead of the well-known BGK relaxation makes it possible to target a desired convection-diffusion system.Several numerical examples, ranging from a simple pure diffusion model to the compressible Navier-Stokes equations illustrate our approach.

Free boundary regularity and support propagation in mean field games and optimal transport

We study the behavior of solutions to the first-order mean field games system with a local coupling, when the initial density is a compactly supported function on the real line. Our results show that the solution is smooth in regions where the density is strictly positive, and that the density itself is globally continuous. Additionally, the speed of propagation is determined by the behavior of the cost function near small values of the density. When the coupling is entropic, we demonstrate that the support of the density propagates with infinite speed. On the other hand, for a power-type coupling, we establish finite speed of propagation, leading to the formation of a free boundary. We prove that under a natural non-degeneracy assumption, the free boundary is strictly convex and enjoys C1,1 regularity. We also establish sharp estimates on the speed of support propagation and the rate of long time decay for the density. Moreover, the density and the gradient of the value function are both shown to be Hölder continuous up to the free boundary. Our methods are based on the analysis of a new elliptic equation satisfied by the flow of optimal trajectories. The results also apply to mean field planning problems, characterizing the structure of minimizers of a class of optimal transport problems with congestion.

Global solvability of a mixed problem for a singular semilinear hyperbolic 1d system

Using the method of characteristics and the Banach fixed point theorem (for the Bielecki metric), in the paper it is established the existence and uniqueness of a global (continuous) solution of the mixed problem in the rectangle $\Pi=\{(x,t)\colon 0<x<l<\infty,\ 0<t<T<\infty\}$ for the first order hyperbolic system of semi-linear equations of the form $$ \dfrac{\partial u}{\partial t}+\Lambda(x,t) \dfrac{\partial u}{\partial x}=f(x,t,u,v,w), \dfrac{\partial v}{\partial x}=g(x,t,u,v,w), \dfrac{\partial w}{\partial t }=h(x,t,u,v,w), $$ for a singular with orthogonal (degenerate) and non-orthogonal to the coordinate axes characte\-ristics and with nonlinear boundary conditions, where $\Lambda(x,t)=\mathrm{diag}(\lambda_1(x,t),\ldots,\lambda_k(x,t)),$ $u=(u_1,\ldots,u_k),$ $v=(v_1,\ldots,v_m),$ $w=(w_1,\ldots,w_n),$ $f=(f_1,\ldots,f_k),$ $g=(g_1,\ldots,g_m),$ $h=(h_1,\ldots,h_n)$ and besides $\textrm{sign } \lambda_i(0,t)=\mathrm{const}\neq 0$, $\textrm{sign } \lambda_i(l,t)=\mathrm{const}\neq 0 $ $\text{for all} \ t \in [0, T] $ and for all $i \in \{1,\ldots,k\}$. The presence of non-orthogonal and degenerate characteristics of the hyperbolic system for physical reasons indicates that part of the oscillatory disturbances in the medium propagates with a finite speed, and part with an unlimited one. Such a singularity (degeneracy of characteristics) of the hyperbolic system allows mathematical interpretation of many physical processes, or act as auxiliary equations in the analysis of multidimensional problems.

Delayed self-feedback echo state network for long-term dynamics of hyperchaotic systems.

Analyzing the long-term behavior of hyperchaotic systems poses formidable challenges in the field of nonlinear science. This paper proposes a data-driven model called the delayed self-feedback echo state network (self-ESN) specifically designed for the evolution behavior of hyperchaotic systems. Self-ESN incorporates a delayed self-feedback term into the dynamic equationof a reservoir to reflect the finite transmission speed of neuron signals. Delayed self-feedback establishes a connection between the current and previous m time steps of the reservoir state and provides an effective means to capture the dynamic characteristics of the system, thereby significantly improving memory performance. In addition, the concept of local echo state property (ESP) is introduced to relax the conventional ESP condition, and theoretical analysis is conducted on guiding the selection of feedback delay and gain to ensure the local ESP. The judicious selection of feedback gain and delay in self-ESN improves prediction accuracy and overcomes the challenges associated with obtaining optimal parameters of the reservoir in conventional ESN models. Numerical experiments are conducted to assess the long-term prediction capabilities of the self-ESN across various scenarios, including a 4D hyperchaotic system, a hyperchaotic network, and an infinite-dimensional delayed chaotic system. The experiments involve reconstructing bifurcation diagrams, predicting the chaotic synchronization, examining spatiotemporal evolution patterns, and uncovering the hidden attractors. The results underscore the capability of the proposed self-ESN as a strategy for long-term prediction and analysis of the complex systems.

Matrix numerical method for probability densities of stochastic delay differential equations

Stochastic processes with time delay are invaluable for modeling in science and engineering when finite signal transmission and processing speeds can not be neglected. However, they can seldom be treated with sufficient precision analytically if the corresponding stochastic delay differential equations (SDDEs) are nonlinear. This work presents a numerical algorithm for calculating the probability densities of processes described by nonlinear SDDEs. The algorithm is based on Markovian embedding and solves the problem by basic matrix operations. We validate it for a broad class of parameters using exactly solvable linear SDDEs and a cubic SDDE. Besides, we show how to apply the algorithm to calculate transition rates and first passage times for a Brownian particle diffusing in a time-delayed cusp potential.

Stability of twisted states in power-law-coupled Kuramoto oscillators on a circle with and without time delay.

Other than the fully synchronized state, a twisted state can also be an equilibrium solution in the Kuramoto model and its variations. In the present work, we explore the stability of the twisted state in Kuramoto oscillators put on a rim of a planar circle in the two-dimensional space in the presence of power-law decaying interaction strength (∼r^{-α} with the distance r) and time delays due to a finite speed of information transfer. For example, our model can phenomenologically mimic a large sports stadium where many people try to sing or clap their hands in unison; the sound intensity decays with the distance and there can exist a time delay proportional to the distance due to the finiteness of sound speed. We first consider the case without the time delay effect and numerically find that stable twisted states emerge when the exponent α exceeds a critical value of α_{c}≈2. In other words, for α<α_{c}, the fully synchronized state, not the twisted state, is the only stable fixed point of the dynamics. In our analytic approach, we also derive an equation for α_{c} and discuss its solutions. In the presence of time delay, we find that it is possible that the synchronized state becomes unstable while twisted states are stable.

Radiative transfer of 21-cm line through ionised cavities in an expanding universe

Abstract The optical depth parameterisation is typically used to study the 21-cm signals associated with the properties of the neutral hydrogen (H i) gas and the ionisation morphology during the Epoch of Reionisation (EoR), without solving the radiative transfer equation. To assess the uncertainties resulting from this simplification, we conduct explicit radiative transfer calculations using the cosmological 21-cm radiative transfer (C21LRT) code and examine the imprints of ionisation structures on the 21-cm spectrum. We consider a globally averaged reionisation history and implement fully ionised cavities (H ii bubbles) of diameters d ranging from 0.01 Mpc to 10 Mpc at epochs within the emission and the absorption regimes of the 21-cm global signal. The single-ray C21LRT calculations show that the shape of the imprinted spectral features are primarily determined by d and the 21-cm line profile, which is parametrised by the turbulent velocity of the H i gas. It reveals the spectral features tied to the transition from ionised to neutral regions that calculations based on the optical depth parametrisation were unable to capture. We also present analytical approximations of the calculated spectral features of the H ii bubbles. The multiple-ray calculations show that the apparent shape of a H ii bubble (of d = 5 Mpc at z = 8), because of the finite speed of light, differs depending on whether the bubble’s ionisation front is stationary or expanding. Our study shows the necessity of properly accounting for the effects of line-continuum interaction, line broadening and cosmological expansion to correctly predict the EoR 21-cm signals.

Three-Body 3D-Kepler Electromagnetic Problem—Existence of Periodic Solutions

The main purpose of the present paper is to prove the existence of periodic solutions of the three-body problem in the 3D Kepler formulation. We have solved the same problem in the case when the three particles are considered in an external inertial system. We start with the three-body equations of motion, which are a subset of the equations of motion (previously derived by us) for any number of bodies. In the Minkowski space, there are 12 equations of motion. It is proved that three of them are consequences of the other nine, so their number becomes nine, as much as the unknown trajectories are. The Kepler formulation assumes that one particle (the nucleus) is placed at the coordinate origin. The motion of the other two particles is described by a neutral system with respect to the unknown velocities. The state-dependent delays arise as a consequence of the finite vacuum speed of light. We obtain the equations of motion in spherical coordinates and split them into two groups. In the first group all arguments of the unknown functions are delays. We take their solutions as initial functions. Then, the equations of motion for the remaining two particles must be solved to the right of the initial point. To prove the existence–uniqueness of a periodic solution, we choose a space consisting of periodic infinitely smooth functions satisfying some supplementary conditions. Then, we use a suitable operator which acts on these spaces and whose fixed points are periodic solutions. We apply the fixed point theorem for the operators acting on the spaces of periodic functions. In this manner, we show the stability of the He atom in the frame of classical electrodynamics. In a previous paper of ours, we proved the existence of spin functions for plane motion. Thus, we confirm the Bohr and Sommerfeld’s hypothesis for the He atom.

Analysis of different wave characteristics in a fiber-reinforced thermo-piezoelectric rotating medium influenced by gravity

Fiber-reinforced polymer composites possess a lightweight nature, corrosion resistance, high strength, and specific stiffness. As a result, they play a crucial role in the construction and rehabilitation of structures, providing reinforcement with their versatile and beneficial properties. Owing to this, proposed mathematical model is intended to give a detailed analysis of the characteristics of surface waves in a homogeneous, transversely isotropic fiber-reinforced thermo-piezoelectric (FRTPC) composite half-space rotating with uniform angular frequency including Coriolis and centrifugal forces. Recently proposed Moore–Gibson–Thompson thermoelasticity theory with two-temperature is employed to describe the mathematical framework. The free surface is considered to be stress free, electrically shorted/charge free, and thermally insulated/isothermal. The FRTPC structure is analytically modeled by the Strength of Materials technique with the Rule of Mixtures approach. Normal mode technique is used to derive the expression for mechanical stresses, electrical displacement, and temperature gradient. Depending upon four different boundary conditions, corresponding secular equations are derived for the considered half-space. The influence of rotation, gravity, fiber volume fraction, and two-temperature on different wave characteristics (phase velocity, attenuation coefficient, specific loss, and frequency shift) is demonstrated graphically considering PZT-5H epoxy material. Numerical simulation indicates that wave propagates at a finite speed in the FRTPC medium in the case of MGT thermoelastic theory, rather than the infinite speed in the classical thermoelasticity model. The present mathematical study may find its application in designing of gyroscopes, rotation sensors, temperature sensors, and other pyro/piezoelectric surface acoustic wave devices.

Mathematical analysis and numerical simulations for a nonlinear Klein Gordon equation in an exterior domain

In this study, the finite propogation speed properties investigated for a two dimensional exterior problem defined by nonlinear Klein-Gordon equation. Under some assumptions on the initial data and the nonlinearity, the solution is shown to have a finite propogation speed. Furthermore, it is demonstrated that the problem has a unique solution, and accurate numerical solutions have been produced by the use of the dual reciprocity boundary element approach with linear radial basis functions.

Wave propagation in three-dimensional fractional viscoelastic infinite solid body

In order to analyze the wave propagation in three-dimensional isotropic and viscoelastic body, the Cauchy initial value problem on unbounded domain is considered for the wave equation written as a system of fractional partial differential equations consisting of equation of motion of three-dimensional solid body, equation of strain, as well as of the constitutive equation, obtained by generalizing the classical Hooke’s law of three-dimensional isotropic and elastic body by replacing Lamé coefficients with the relaxation moduli to account for different memory kernels corresponding to the propagation of compressive and shear waves. By the use of the method of integral transforms, namely Laplace and Fourier transforms, the displacement field, as a solution of the Cauchy initial value problem, is expressed through Green’s functions corresponding to both compressive and shear wave propagation. Two approaches are adopted in solving the Cauchy problem: in the first one the displacement field is expressed through the scalar and vector fields, obtained as solutions of wave equations which are consequences of decoupling the wave equation for displacement field, while in the second approach the displacement field is obtained by the action of the resolvent tensor on the initial conditions. Fractional anti-Zener and Zener model I+ID.ID, as well as fractional Burgers model VII, as representatives of models yielding infinite and finite wave propagation speed, are used in numerical calculations and graphical representations of Green’s functions and its short and large time asymptotic behavior.

Newtonian gravitational waves from a continuum

Gravitational waves are being shown to derive directly from Newtonian dynamics for a continuous mass distribution, e.g. compressible fluids or equivalent. It is shown that the equations governing a continuous mass distribution, i.e. the inviscid Navier–Stokes equations for a general variable gravitational field g ( t , x ) , are equivalent to a form identical to the Maxwell equations from electromagnetism, subject to a specified condition. The consequence of this equivalence is the creation of gravity waves that propagate at a finite speed. The latter implies that Newtonian gravitation as presented in this paper is not ‘spooky action at a distance’ but rather is similar to electromagnetic waves propagating at finite speed, despite the apparent form appearing in the integrated field formula. In addition, this proves that, in analogy to the Maxwell equations, the Newtonian gravitation equations are Lorentz invariant for waves propagating at the speed of light. Since gravitational waves were so far derived only from Einstein’s general relativity theory, it becomes appealing to check if there is a connection between the Newtonian waves presented in this paper and the general relativity type of waves at least in a certain limit of overlapping validity, i.e. as a flat-space approximation. The latter is left for follow-up research.

Precise rates of propagation in reaction–diffusion equations with logarithmic Allee effect

This paper focuses on propagation phenomena in reaction–diffusion equations with a weakly degenerate monostable nonlinearity. The kind of reaction term we consider can be seen as an intermediate between the classical logistic one (or Fisher–KPP) and more usual power laws that usually model Allee effects. We investigate the effect of the decay rate of the initial data on the propagation rate. When the tail of the initial data is sub-exponential, both finite speed propagation and acceleration may happen. We derive the exact separation between the two situations. When the initial data is sub-exponentially unbounded, acceleration unconditionally occurs. Estimates for the locations of the level sets are expressed in terms of the decay of the initial data. In addition, sharp exponents of acceleration for initial data with sub-exponential and algebraic tails are given. Numerical simulations are presented to illustrate the above findings.

Light cones for open quantum systems in the continuum

We consider Markovian open quantum dynamics (MOQD) in the continuum. We show that, up to small-probability tails, the supports of quantum states evolving under such dynamics propagate with finite speed in any finite-energy subspace. More precisely, we prove that if the initial quantum state is localized in space, then any finite-energy part of the solution of the von Neumann–Lindblad equation is approximately localized inside an energy-dependent light cone. We also obtain an explicit upper bound for the slope of this light cone.