Exponential Spatial Decay Research Articles

A Moore-Gibson-Thompson heat conduction problem with second gradient

In this work, we study, from both analytical and numerical points of view, a heat conduction model which is based on the Moore-Gibson-Thompson equation. The second gradient effects are also included. First, the existence of a unique solution is proved by using the theory of linear semigroups, and the exponential energy decay is also shown when the constitutive tensors are homogeneous. The analyticity of the semigroup is also discussed in the isotropic case, and its spatial behavior is studied. The spatial exponential decay is also proved. Then, we provide the numerical analysis of a fully discrete approximation obtained by using the finite element method and an implicit Euler scheme. A discrete stability property is shown, and some a priori error estimates are derived, from which the linear convergence is concluded under suitable regularity conditions. Finally, some one-dimensional numerical simulations are presented to demonstrate the accuracy of the approximations and the behavior of the discrete energy.

On dilation analyticity and spatial exponential decay of atomic ground states in non-relativistic QED

We consider the ground state and the ground state energy of an atom with spinless electrons in the framework of non-relativistic quantum electrodynamics. We show that the ground state energy as well as the ground state depend analytically on the parameters of the group of dilations, the parameter of a group of spatial dependent phase changes, and on the minimal coupling constant. As a corollary we obtain spatial exponential decay of the ground state as well as of its dilation analytic extension. No infrared regularization is needed for the result. Our result is based on operator theoretic renormalization.

Information Structures of the Kalman Filter for the Elastic Wave Equation

We study the Kalman Filter for the linear elastic wave equation over the real line with spatially distributed partial state measurements. The dynamics of the filter are described by a spatial convolution operator with asymptotic exponential spatial decay rate. This decay rate dictates how measurements from different spatial locations must be exchanged to implement the filter: faster spatial decay implies local measurements are more relevant and the filter is more “decentralized”; slower decay implies farther measurements also become relevant and the filter is more “centralized”. Using dimensional analysis, we demonstrate that this decay rate is a function of one dimensionless group defined from system parameters, such as wave speed and noise variances. We find a critical value of such dimensionless group for which the Kalman Filter is completely decentralized.

The determination of highly time-resolved and source-separated black carbon emission rates using radon as a tracer of atmospheric dynamics

Abstract. We present a new method for the determination of the source-specific black carbon emission rates. The methodology was applied in two different environments: an urban location in Ljubljana and a rural one in the Vipava valley (Slovenia, Europe), which differ in pollution sources and topography. The atmospheric dynamics was quantified using the atmospheric radon (222Rn) concentration to determine the mixing layer height for periods of thermally driven planetary boundary layer evolution. The black carbon emission rate was determined using an improved box model taking into account boundary layer depth and a horizontal advection term, describing the temporal and spatial exponential decay of black carbon concentration. The rural Vipava valley is impacted by a significantly higher contribution to black carbon concentration from biomass burning during winter (60 %) in comparison to Ljubljana (27 %). Daily averaged black carbon emission rates in Ljubljana were 210 ± 110 and 260 ± 110 µgm-2h-1 in spring and winter, respectively. Overall black carbon emission rates in Vipava valley were only slightly lower compared to Ljubljana: 150 ± 60 and 250 ± 160 µgm-2h-1 in spring and winter, respectively. Different daily dynamics of biomass burning and traffic emissions was responsible for slightly higher contribution of biomass burning to measured black carbon concentration, compared to the fraction of its emission rate. Coupling the high-time-resolution measurements of black carbon concentration with atmospheric radon concentration measurements can provide a useful tool for direct, highly time-resolved measurements of the intensity of emission sources. Source-specific emission rates can be used to assess the efficiency of pollution mitigation measures over longer time periods, thereby avoiding the influence of variable meteorology.

Large Deviations for Continuous Time Random Walks.

Recently observation of random walks in complex environments like the cell and other glassy systems revealed that the spreading of particles, at its tails, follows a spatial exponential decay instead of the canonical Gaussian. We use the widely applicable continuous time random walk model and obtain the large deviation description of the propagator. Under mild conditions that the microscopic jump lengths distribution is decaying exponentially or faster i.e., Lévy like power law distributed jump lengths are excluded, and that the distribution of the waiting times is analytical for short waiting times, the spreading of particles follows an exponential decay at large distances, with a logarithmic correction. Here we show how anti-bunching of jump events reduces the effect, while bunching and intermittency enhances it. We employ exact solutions of the continuous time random walk model to test the large deviation theory.

Analytical description of the topological interaction between magnetic domain walls in nanowires

Magnetic domain walls in nanowires behave as particles interacting through the exchange field. As topological objects, their interaction is determined by their chirality or topological charge. We investigate analytically the topological repulsion between magnetic domain walls with the same topological charge in nanostripes (with easy-plane magnetization) and show that it decays algebraically as ${r}^{\ensuremath{-}2}$, being part of a larger class of interactions that produce topological long-range order in two dimensions. We compare the topological repulsion between the walls with other types of fundamental interactions with exponential spatial decay, such as the Yukawa-Reid potential, and with micromagnetic simulations. We determine that trains of such walls can be well described analytically and can be displaced regularly in nanowires leading to practical applications.

Kinetic model for stochastic heating in the INCA discharge

A novel electron heating mechanism based on periodically structured vortex fields induced in a plane was first proposed in 2014 (Czarnetzki and Tarnev 2014 Phys. Plasmas 21 123508). This theoretical concept has now been realized in an experiment which confirms efficient collisionless heating in such array structures (Ahr et al 2018 Plasma Sources Sci. Technol. 27 105010). The new concept is called ‘INductively Coupled Array’: INCA. Here, the physical mechanism behind the collisionless (stochastic) heating is investigated by two analytical models. Firstly, the electron heating rate in an array field structure with an exponential spatial decay of the field in the direction perpendicular to the plane is investigated by stochastically averaging single electron trajectories. The approach is similar to the Lieberman model for the classical stochastic heating in standard inductively coupled plasmas. This analysis shows that classical stochastic heating by thermal motion along the vertical direction makes a negligible contribution. However, there is a strong collisionless nonlocal heating effect in the plane. In conclusion, heating is nonlocal in the plane but local in the vertical direction. This insight allows a straightforward solution of the collisionless Boltzmann equation which not only confirms the results of the Lieberman model but provides also explicit expressions for the complex conductivity. Based on the conductivity an effective stochastic collision frequency, the complex damping coefficient and the related penetration of the field into the plasma is calculated. Finally, elastic collisions with neutral background atoms are included in the model and a condition for dominance of stochastic heating over Ohmic heating is derived.

On stability in the thermoelastostatics of dipolar bodies

Our study is concerned with the initial boundary value problem in the context of the thermoelastostatics of dipolar bodies. We will derive a result which describes the exponential spatial decay of solutions of this problem. We will also find a superior limit for the amplitude, which is dependent on the initial and boundary conditions.

Direct observation of Anderson localization in plasmonic terahertz devices

We present the first experimental observation of Anderson localization in the terahertz frequency range using plasmonic structures. To accomplish this goal, we designed THz waveguides consisting of a one-dimensional array of rectangular apertures that were fabricated in a freestanding metal foil. Disorder is introduced into the waveguide by offsetting the position of each aperture by a random distance within a prescribed range. For example, for a waveguide with apertures spaced by 250 μm in a periodic waveguide, 10% disorder would correspond to the apertures being shifted by a random value between ±25 μm along the waveguide axis. We find that for disorder levels below 25%, there is only an increase in the propagation loss along the device. However, for two specific waveguides with 25% disorder, we observe a spatially localized mode that lies just within the stop band of the device and exhibits a double-sided exponential spatial decay away from the maximum.

Modeling and experimental validation of an evanescent wavefield using a wavenumber integration method

When sound is incident on the boundary between two media, where the first medium has a lower sound speed than the second (such as sound in water incident on some sediments), a critical angle exists. If the incident angle is below this critical angle an evanescent wavefield is formed in the second medium. Experimental work measured the structure of such a wavefield at an oil-water interface insonified by an appropriately oriented transducer, where the region of interest is not in the transducer far-field [Osterhoudt et al., IEEE J. Ocean. Eng. 33, 397–404 (2008)]. Several features of note within the wavefield were observed. This work attempts to model those observations using wavenumber integration techniques. In the incident plane wave approximation, the evanescent wavefield is vertically phase-locked and the characteristic exponential spatial decay is constant across the wavefield. However, the contribution of many wavenumber components across the region of interest is key to understanding the wavefield structure seen in experiments. This model will be compared to the experimental results, and the details of phase, amplitude, and the overall wavefield structure discussed. An improved approximation of the transducer orientation used in experiments may also be inferred from these results. [Work supported by ONR.]

Breathers and surface modes in oscillator chains with Hertzian interactions

We study localized waves in chains of oscillators coupled by Hertzian interactions and trapped in local potentials. This problem is originally motivated by Newton's cradle, a mechanical system consisting of a chain of touching beads subject to gravity and attached to inelastic strings. We consider an unusual setting with local oscillations and collisions acting on similar time scales, a situation corresponding e.g. to a modified Newton's cradle with beads mounted on stiff cantilevers. Such systems support static and traveling breathers with unusual properties, including double exponential spatial decay, almost vanishing Peierls-Nabarro barrier and spontaneous direction-reversing motion. We prove analytically the existence of surface modes and static breathers for anharmonic on-site potentials and weak Hertzian interactions.

Existence and Stability of Travelling Front Solutions for General Auto-catalytic Chemical Reaction Systems

This paper is concerned with the existence and stability of travelling front solutions for more general autocatalytic chemical reaction systems u t = d u xx − uf (v ), v t = v xx + uf (v ) with d > 0 and d ≠ 1, where f (v ) has super-linear or linear degeneracy at v = 0. By applying Lyapunov-Schmidt decomposition method in some appropriate exponentially weighted spaces, we obtain the existence and continuous dependence of wave fronts with some critical speeds and with exponential spatial decay for d near 1. By applying special phase plane analysis and approximate center manifold theorem, the existence of traveling waves with algebraic spatial decay or with some lower exponential decay is also obtained for d > 0. Further, by spectral estimates and Evans function method, the wave fronts with exponential spatial decay are proved to be spectrally or linearly stable in some suitable exponentially weighted spaces. Finally, by adopting the main idea of proof in [12] and some similar arguments as in [21], the waves with critical speeds or with non-critical speeds are proved to be locally exponentially stable in some exponentially weighted spaces and Lyapunov stable in C unif (ℝ) space, if the initial perturbation of the waves is small in both the weighted and unweighted norms; the perturbation of the waves also stays small in L 1 (ℝ) norm and decays algebraically in C unif (ℝ) norm, if the initial perturbation is in addition small in L 1 norm.

Direct Imaging of Anisotropic Exciton Diffusion and Triplet Diffusion Length in Rubrene Single Crystals

We visualize exciton diffusion in rubrene single crystals using localized photoexcitation and spatially resolved detection of excitonic luminescence. We show that the exciton mobility in this material is strongly anisotropic with long-range diffusion by several micrometers associated only with the direction of molecular stacking in the crystal, along the b axis. We determine a triplet exciton diffusion length of 4.0 ± 0.4 μm from the spatial exponential decay of the photoluminescence that originates from singlet excitons formed by triplet-triplet fusion.

Direct Measurement of Surface-Induced Orientational Order Parameter Profile above the Nematic-Isotropic Phase Transition Temperature

The spatial and temperature dependence of the surface-induced orientational order parameter S(z,T) was determined in the isotropic phase. An optical fiber was immersed in a thin liquid crystal layer and the retardation was measured as a function of the fiber's height above the surface, from which the model-independent S(z,T) was deduced with resolution <or=2 nm. It was found that (i) S(z=0) <or=0.12 close to the nematic transition temperature, (ii) the susceptibility is mean-field-like, and (iii) S(z,T) deviates significantly from exponential spatial decay. The results are discussed in terms of a nonlocal potential.

Spatial Stability for the Quasi-Static Problem in Thermoelastic Diffusion Theory

In this paper we derive some spatial stability results for the quasi-static problem in thermoelastic diffusion theory for anisotropic media. The coupled system of equations of thermoelasticity with diffusion is a coupling of an elliptic equation with two parabolic equations. It poses some new mathematical difficulties. Here we study the exponential spatial decay of solutions. An upper bound for the amplitude in terms of the boundary and initial conditions is obtained. The extension of the spatial stability results is also treated.

Exponential spatial decay of spin-spin correlations in translation invariant quasifree states

We study the spatial decay of the transversal spin-spin correlations in translation invariant quasifree states on the canonical anticommutation relation algebra on the discrete line. We establish a simple criterion on the spectrum of the density of the quasifree state which ensures the decay to be exponential. Moreover, we illustrate exponential and slower decay with the example of the XY spin chain out of equilibrium, in thermal equilibrium at positive temperature, in the ground state, and in the chaotic state.

Out of Equilibrium Correlations in the XY Chain

We study the transversal XY spin-spin correlations in the non-equilibrium steady state constructed in \cite{AP03} and prove their spatial exponential decay close to equilibrium.

Green's Function for Magnetically Incoherent Interacting Electrons in One Dimension

Using a path integral approach and bosonization, we calculate the low-energy asymptotics of the one particle Green's function for a "magnetically incoherent" one dimensional strongly interacting electron gas at temperatures much greater than the typical exchange energy but much lower than the Fermi energy. The Green's function exhibits features reminiscent of spin-charge separation, with exponential spatial decay and scaling behavior with interaction dependent anomalous exponents inconsistent with any unitary conformal field theory. We compute the tunneling density of states at low energies and find that it is a power law in energy with exponent 1/(4g)-1, where g is the Luttinger interaction parameter in the charge sector. The underlying physics is made transparent by the simplicity of the approach. Our results generalize those of Cheianov and Zvonarev [Phys. Rev. Lett. 92, 176401 (2004)]].

Spatial Stability for the Quasi-static Problem of Thermoelasticity

This paper is concerned with the quasi-static problem of thermoelasticity. The classical system of equations of thermoelasticity is a coupling of an elliptic equation with a parabolic equation. It poses some new mathematical difficulties. Here we study the exponential spatial decay of solutions. An upper bound for the amplitude in terms of the boundary and initial conditions is obtained. The extension of the spatial stability results to thermoelasticity of type III is also treated.

SPATIAL BEHAVIOR IN DYNAMICAL THERMOELASTICITY BACKWARD IN TIME FOR POROUS MEDIA

The aim of this paper is to study the spatial behavior of the solutions to the final-boundary-value problems associated with the linear theory of elastic materials with voids. More precisely, the present study is devoted to porous materials with a memory effect for the intrinsic equilibrated body forces. An appropriate time-weighted volume measure is associated with the backward-in-time thermoelastic processes. Then a first-order partial differential inequality in terms of such measure is established and it is further shown how the inequality implies the spatial exponential decay of the thermoelastic process in question.