Exponential Time Dependence Research Articles

Brownian motion in a viscous fluid of particles with constant and time-dependent friction

The Brownian motion (BM) of particles in a fluid under the influence of a moving harmonic potential is described analytically. It is assumed that the bath is viscous and particles constituting it experience Stokes' force with friction coefficients that can depend on time. The generalized Langevin equation (GLE) is derived from the equations of motion for the bath and Brownian particles. It has a familiar form, but its memory kernel generalizes the expressions known from the literature even in the case of constant friction. Analytical solutions of the found GLE are obtained for the mean and mean squared displacements of the Brownian particle, assuming the overdamped character of its motion and both the overdamped and underdamped dynamics of bath particles. In the case of constant friction, the model well describes the BM when the external potential does not move so that the system is in equilibrium. The results determined by the time-dependent friction of the bath particles must be specified for concrete fluids. An example of Lennard–Jones atomic liquid is considered with the recently proposed exponential time dependence of the friction coefficient.

Fiber spinning from polymer solutions

The thinning of a cylinder of a polymer solution in a volatile solvent is argued to be controlled by solvent diffusion through a dense polymer layer at the cylinder surface. This naturally leads to the exponential time dependence of cylinder radius that is observed in experiments using a fast camera, such as capillary breakup extensional rheometry (CaBER). The relaxation time is controlled by the thickness of the dense (and often glassy) polymer layer and the diffusion coefficient of solvent through that layer. If correct, this means that while CaBER is very useful for understanding fiber spinning, the relaxation time does not yield a measure of the extensional viscosity of polymer solutions in volatile solvents.

Koopman analysis of the periodic Korteweg-de Vries equation.

The eigenspectrum of the Koopman operator enables the decomposition of nonlinear dynamics into a sum of nonlinear functions of the state space with purely exponential and sinusoidal time dependence. For a limited number of dynamical systems, it is possible to find these Koopman eigenfunctions exactly and analytically. Here, this is done for the Korteweg-de Vries equation on a periodic interval using the periodic inverse scattering transform and some concepts of algebraic geometry. To the authors' knowledge, this is the first complete Koopman analysis of a partial differential equation, which does not have a trivial global attractor. The results are shown to match the frequencies computed by the data-driven method of dynamic mode decomposition (DMD). We demonstrate that in general, DMD gives a large number of eigenvalues near the imaginary axis and show how these should be interpreted in this setting.

Loop corrections in Minkowski spacetime away from equilibrium. Part I. Late-time resummations

Loop corrections to unequal-time correlation functions in Minkowski spacetime exhibit secular growth due to a breakdown of time-dependent perturbation theory. This is analogous to secular growth in equal-time correlators on time-dependent backgrounds, except that in Minkowski the divergences must not signal a real IR issue. In this paper, we calculate the late-time limit of the two-point correlator for different massless self-interacting scalar quantum field theories on a Minkowski background. We first use a late-time version of the in-in path integral starting in the vacuum of the free theory; in this limit, the calculation, including UV renormalization, reduces to that in in-out. We find linear or logarithmic growth in time, depending on whether the interaction strength is dimension-one or dimensionless, respectively. We next develop the Weisskopf-Wigner resummation method, that proceeds by demanding unitarity within a truncated Hilbert space, to calculate the resummed correlator and find that it gives an exact exponentiation of the late-time perturbative result. The resummed (unequal-time) correlator thus decays with an exponential or polynomial time-dependence, which is suggestive of ‘universal’ behavior that depends on the dimensions of the interaction strength.

Determining the proton diffusion coefficient in highly hydrated iridium oxide films by energy dispersive X-ray absorption spectroscopy

Charge transfer reactions in electrodeposited iridium oxide films (EIROF) are investigated by means of operando energy dispersive X-ray absorption spectroscopy (EDXAS), where oxidation and reduction conditions are selected to drive the Ir(III)/Ir(IV) and Ir(IV)/Ir(V) reactions in acidic solutions. The Ir(III)/Ir(IV) couple is related to a well-known electrochromic phenomenon, while the Ir(IV)/Ir(V) couple might play an important role in the catalysis of the oxygen evolution reactions (OER). In the experiments, current intensity and time-resolved X-ray absorption spectroscopy (XAS) are simultaneously recorded upon application of appropriate potential steps, leading to the independent determination of both the relevant reaction rates and the rate-determining steps. This is allowed by the fast acquisition time (∼10−2 s) at the ESRF Energy Dispersive XAS (EDXAS) ID24 beam-line, in combination with the highly hydrated amorphous iridium oxide electrode material, which in turn allows to maximize the fraction of Ir sites participating in the electrochemical processes. If the experimental conditions exclude the possibility of having either oxygen evolution (or reduction), the Degree of Reaction (DoR), determined by both electrochemistry and XAS, exhibits exponential time dependence, clearly pointing to diffusion-controlled processes. Vice versa, under concomitant OER + oxidation of iridium centers or ORR + iridium reduction, the electrochemical and XAS DoRs highlight different phenomena, providing fully complementary information of the ongoing electrode reactions. In all cases, data elaboration allows to determine the diffusion coefficient of H+ ions within the catalyst layer, that is compared and confirmed by data obtained by electrochemical impedance spectroscopy (EIS). The high values of D obtained for EIROF is compared to values obtained on other IrO2 materials can help in explaining the relevant high electrocatalytic activity.

Charging a supercapacitor through a lamp: A power-law RC decay

A circuit involving a charging supercapacitor in series with a non-Ohmic tungsten lamp displays a wealth of interesting behavior. Most notably, the current through the lamp decreases in time according to a power-law function as opposed to the exponential time dependence observed in RC circuits with Ohmic resistors. We use a combination of computational and analytical techniques to model this power-law behavior as well as the behavior of the filament's temperature and resistance as the supercapacitor charges. Our results agree well with experiment, and the experiment described here can be modified to be appropriate for physics courses at a wide range of levels.

Improved bounds for discretization of Langevin diffusions: Near-optimal rates without convexity

Discretizations of the Langevin diffusion have been proven very useful for developing and analyzing algorithms for sampling and stochastic optimization. We present an improved non-asymptotic analysis of the Euler-Maruyama discretization of the Langevin diffusion. Our analysis does not require global contractivity, and yields polynomial dependence on the time horizon. Compared to existing approaches, we make an additional smoothness assumption, and improve the existing rate in discretization step size from O(η) to O(η2) in terms of the KL divergence. This result matches the correct order for numerical SDEs, without suffering from exponential time dependence. When applied to MCMC, this result simultaneously improves on the analyses of a range of sampling algorithms that are based on Dalalyan’s approach.

The synergistic effect of microwave drying and plasma surface treatments on the wettability of green wood

In spite of being both a one-step solution to several problems associated with woodworking and also energy efficient, the application of microwave (MW) modification in wood research has remained very limited and this promising method has practically no use in wood industries across the globe. Research done so far in this field primarily sheds light on its potential in enhancing wood permeability, treatability and uniform wood drying. While MW treatments are mostly used on wet or green wood, another modification technique, plasma, has potential benefits to synergistically enhance the effects of MW treatment, but so far has not been applied on wet or green wood specimens. This study takes a first step to investigate the effects of plasma treatments (PT) on green wood specimens, as well as combinations of MW and plasma treatments. As a preliminary study, the methodology focuses on water contact angle measurements, since these are most commonly used as indicators for surface modifications in industrial applications. An exponential time dependence was found for the contact angle on the investigated samples of Norway spruce (Picea abies Karst.). Initial contact angles after droplet deposition increased due to drying and migration of organic molecules during treatments. In comparison with the literature, the effect of plasma was significantly less pronounced on wet wood specimens. The initial contact angles showed the lowest statistical variations after MW treatment, whereas plasma increased inhomogeneities. The final contact angles on treated specimens was lowest for PT-only specimens as well as specimens treated with plasma after MW. In contrast to the initial contact angles, the final contact angles showed the lowest variations after PT. Wetting rates were insignificantly improved by plasma, with reduced statistical variations after all treatments.

On stable exponential cosmological solutions with two factor spaces in (1+ m + 2)-dimensional Einstein-Gauss-Bonnet model with Λ-term.

A [Formula: see text]-dimensional Einstein-Gauss-Bonnet gravitational model including the Gauss-Bonnet term and the cosmological term [Formula: see text] is considered. Exact solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters [Formula: see text] and [Formula: see text], corresponding to factor spaces of dimensions [Formula: see text] and [Formula: see text], respectively, are found. Under certain restrictions on [Formula: see text], the stability of the solutions in a class of cosmological solutions with diagonal metrics is proved. A subclass of solutions with small enough variation of the effective gravitational constant [Formula: see text] is considered and the stability of all solutions from this subclass is shown. This article is part of the theme issue 'The future of mathematical cosmology, Volume 1'.

Conserved quantities in non-hermitian systems via vectorization method

Open classical and quantum systems have attracted great interest in the past two decades. These include systems described by non-Hermitian Hamiltonians with parity-time (PT) symmetry that are best understood as systems with balanced, separated gain and loss. Here, we present an alternative way to characterize and derive conserved quantities, or intertwining operators, in such open systems. As a consequence, we also obtain non-Hermitian or Hermitian operators whose expectations values show single exponential time dependence. By using a simple example of a PT-symmetric dimer that arises in two distinct physical realizations, we demonstrate our procedure for static Hamiltonians and generalize it to time-periodic (Floquet) cases where intertwining operators are stroboscopically conserved. Inspired by the Lindblad density matrix equation, our approach provides a useful additionto the well-established methods for characterizing time-invariants in non-Hermitian systems.

Quantitative assessment of the exposure-efficacy relationship of glucocerebrosidase using Markovian elements in Gaucher patients treated with enzyme replacement therapy.

The aims of this study are (i) to develop a population pharmacokinetic model of enzyme activity in Gaucher-type 1 (GD1) patients after intravenous administration of enzyme replacement therapy (ERT), and (ii) to establish an exposure-efficacy relationship for bone marrow infiltration to propose dose adjustments according to patient covariate values. A prospective follow-up, semi-experimental multi-centre study was conducted in four hospitals to evaluate the pharmacokinetics, efficacy and safety of ERT in GD1 patients. Twenty-five individuals with 266 glucocerebrosidase (GCase) observations in plasma and leukocytes and 14 individuals with 68 Spanish magnetic resonance imaging (S-MRI) observations were enrolled. A two concatenated compartment model with zero-order endogenous production and first-order distribution (CL1 = 3.85 × 10-1 L/d) and elimination (CL2 = 1.25 L/d) allowed GCase observations in plasma and leukocytes to be described, respectively. An exponential time dependency (kT = 6.14 × 10-1 d-1 ) effect on CL1 was incorporated. The final exposure-efficacy model was a longitudinal logistic regression model with a first-order Markov element. An Emax function (EC50 = 15.73 U/L and Emax = 2.33) linked steady-state concentrations of GCase in leukocytes to the probability of transition across the different S-MRI stages. A population pharmacokinetic model successfully characterized the leukocyte activity-time profiles of GCase following intravenous administration of ERT in GD1 patients together with an exposure-efficacy relationship in bone marrow using Markovian elements. The information obtained from this study could be of high clinical relevance in individualization of ERT in GD1 patients, as this could lead to anticipative decision-making regarding clinical response in bone and optimal dosing strategy.

Momentum conservation in the Biot–Savart law

We study a simple system of charges and currents, consisting of two electric dipoles that are driven by exponentially growing currents running between the ends of each dipole. The exponential time dependence allows the electric and magnetic fields to be calculated exactly. The forces between the dipoles must be evaluated carefully when the exponential growth is slow and are shown to not be equal and opposite. The imbalance is accounted for by the rate of change of momentum in the electromagnetic field, so that the total momentum is conserved even though Newton's third law fails. Implications for momentum conservation for “semistatic” slowly varying currents, for which the Biot–Savart law is applicable, are discussed.

Shape recovery of deformed biomolecular droplets: Dependence on condensate viscoelasticity.

A theoretical study on the shape dynamics of phase-separated biomolecular droplets is presented, highlighting the importance of condensate viscoelasticity. Previous studies on shape dynamics have modeled biomolecular condensates as purely viscous, but recent data have shown them to be viscoelastic. Here, we present an exact analytical solution for the shape recovery dynamics of deformed biomolecular droplets. The shape recovery of viscous droplets has an exponential time dependence, with the time constant given by the "viscocapillary" ratio, i.e., viscosity over interfacial tension. In contrast, the shape recovery dynamics of viscoelastic droplets is multi-exponential, with shear relaxation yielding additional time constants. During shape recovery, viscoelastic droplets exhibit shear thickening (increase in apparent viscosity) at fast shear relaxation rates but shear thinning (decrease in apparent viscosity) at slow shear relaxation rates. These results highlight the importance of viscoelasticity and expand our understanding of how material properties affect condensate dynamics in general, including aging.

Non-monotonic threshold voltage variation in 4H-SiC metal–oxide–semiconductor field-effect transistor: Investigation and modeling

We propose an analytical model to reproduce the non-monotonic instability of the threshold voltage in 4H-SiC MOSFETs submitted to a positive gate stress bias. Experimental analysis of the threshold voltage transients indicates that both electron and hole trappings take place in the gate dielectric or at the dielectric/semiconductor interface, responsible for a VTH increasing–decreasing–increasing pattern. At low/moderate stress fields (<7 MV/cm), the electron trapping kinetics responsible for a positive VTH shift are modeled by a rate equation considering a trapping-inhibition model, which explains the logarithmic degradation kinetics. In the high field regime (>8 MV/cm), we propose that electrons can tunnel through the SiO2, be accelerated by the high field, and generate holes through impact ionization (II) or anode hole injection. These holes are then trapped in the oxide, thus generating a negative VTH shift. This second process has an exponential time-dependency, as found through the analysis of the corresponding rate equations. The time constant of the positive VTH shift is evaluated as a function of stress voltage and temperature. The results indicate that the time constant is strongly dependent on the electric field (that accelerates electrons to generate holes), and not thermally activated, in agreement with theoretical considerations.

Lens shape liquid crystals in electric fields

Experimental studies of the structure of plano-convex lens shaped liquid crystal droplets subjected to electric fields are presented. Similar to previous observations by Salamon et al. (2020) in DC magnetic fields, we found the formation of an inversion wall normal to AC electric fields. While at low frequencies the direction of the wall is stationary, at higher frequencies it turns toward the external electric field. In both cases, the defect wall is also swept toward the periphery of the drop, where it eventually disappears. The linear displacement of the electric field-induced defect wall could be described by an exponential time dependence without any fitting parameter. This combined with threshold for director deformation enables to determine both the bend elastic constant and the rotational viscosity using much less substance than existing techniques. The rotation of the defect wall at high frequencies is a result of the antiparallel orientation of the effective moment vector and the electric field due to the lower dielectric constant and higher electric conductivity of the defect wall than of those of the rest of the liquid crystal droplet. Uniform electric field-induced generation, rotation and linear movement of defect walls is a unique phenomenon in soft matters.

Streamlines in the Two-Dimensional Spreading of a Thin Fluid Film: Blowing and Suction Velocity Proportional to the Height

The two-dimensional spreading under gravity of a thin fluid film with suction (fluid leak-off) or blowing (fluid injection) at the base is considered. The thin fluid film approximation is imposed. The height of the thin film satisfies a nonlinear diffusion equation with a source/sink term. The Lie point symmetries of the nonlinear diffusion equation are derived and exist, which provided the fluid velocity at the base, vn satisfies a first order linear partial differential equation. The general form has algebraic time dependence while a special case has exponential time dependence. The solution in which vn is proportional to the height of the thin film is studied. The width of the base always increases with time even for suction while the height decreases with time for sufficiently weak blowing. The streamlines of the fluid flow inside the thin film are plotted by first solving a cubic equation. For sufficiently weak blowing there is a dividing streamline, emanating from the stagnation point on the centre line which separates the fluid flow into two regions, a lower region consisting of rising fluid and dominated by fluid injection at the base and an upper region consisting of descending fluid and dominated by spreading due to gravity. For sufficiently strong blowing the lower region expands to completely fill the whole thin film.

Stable exponential cosmological solutions with three different Hubble-like parameters in EGB model with a Lambda -term

We consider a D-dimensional Einstein-Gauss-Bonnet model with a cosmological term Lambda and two non-zero constants: alpha _1 and alpha _2. We restrict the metrics to be diagonal ones and study a class of solutions with exponential time dependence of three scale factors, governed by three non-coinciding Hubble-like parameters: H ne 0, h_1 and h_2, obeying m H + k_1 h_1 + k_2 h_2 ne 0 and corresponding to factor spaces of dimensions m > 1, k_1 > 1 and k_2 > 1, respectively (D = 1 + m + k_1 + k_2). We analyse two cases: i) m< k_1 < k_2 and ii) 1< k_1 = k_2 = k, k ne m. We show that in both cases the solutions exist if alpha = alpha _2 / alpha _1 > 0 and alpha Lambda > 0 satisfies certain restrictions, e.g. upper and lower bounds. In case ii) explicit relations for exact solutions are found. In both cases the subclasses of stable and non-stable solutions are singled out. For m > 3 the case i) contains a subclass of solutions describing an exponential expansion of 3-dimensional subspace with Hubble parameter H > 0 and zero variation of the effective gravitational constant G. The case H = 0 is also considered.

Exponential Cosmological Solutions with Three Different Hubble-Like Parameters in (1 + 3 + k1 + k2)-Dimensional EGB Model with a Λ-Term

A D-dimensional Einstein–Gauss–Bonnet model with a cosmological term Λ , governed by two non-zero constants: α 1 and α 2 , is considered. By restricting the metrics to diagonal ones, we study a class of solutions with the exponential time dependence of three scale factors, governed by three non-coinciding Hubble-like parameters: H &gt; 0 , h 1 , and h 2 , obeying 3 H + k 1 h 1 + k 2 h 2 ≠ 0 and corresponding to factor spaces of dimensions: 3, k 1 &gt; 1 , and k 2 &gt; 1 , respectively, with D = 4 + k 1 + k 2 . The internal flat factor spaces of dimensions k 1 and k 2 have non-trivial symmetry groups, which depend on the number of compactified dimensions. Two cases: (i) 3 &lt; k 1 &lt; k 2 and (ii) 1 &lt; k 1 = k 2 = k , k ≠ 3 , are analyzed. It is shown that in both cases, the solutions exist if α = α 2 / α 1 &gt; 0 and α Λ &gt; 0 obey certain restrictions, e.g., upper and lower bounds. In Case (ii), explicit relations for exact solutions are found. In both cases, the subclasses of stable and non-stable solutions are singled out. Case (i) contains a subclass of solutions describing an exponential expansion of 3 d subspace with Hubble parameter H &gt; 0 and zero variation of the effective gravitational constant G.

Properties of Undoped Few-Layer Graphene-Based Transparent Heaters.

In many applications like sensors, displays, and defoggers, there is a need for transparent and efficient heater elements produced at low cost. For this reason, we evaluated the performance of graphene-based heaters with from one to five layers of graphene on flexible and transparent polyethylene terephthalate (PET) substrates in terms of their electrothermal properties like heating/cooling rates and steady-state temperatures as a function of the input power density. We found that the heating/cooling rates followed an exponential time dependence with a time constant of just below 6 s for monolayer heaters. From the relationship between the steady-state temperatures and the input power density, a convective heat-transfer coefficient of 60 W·m−2·°C−1 was found, indicating a performance much better than that of many other types of heaters like metal thin-film-based heaters and carbon nanotube-based heaters.

Koopman mode expansions between simple invariant solutions

A Koopman decomposition is a powerful method of analysis for fluid flows leading to an apparently linear description of nonlinear dynamics in which the flow is expressed as a superposition of fixed spatial structures with exponential time dependence. Attempting a Koopman decomposition is simple in practice due to a connection with dynamic mode decomposition (DMD). However, there are non-trivial requirements for the Koopman decomposition and DMD to overlap, which mean it is often difficult to establish whether the latter is truly approximating the former. Here, we focus on nonlinear systems containing multiple simple invariant solutions where it is unclear how to construct a consistent Koopman decomposition, or how DMD might be applied to locate these solutions. First, we derive a Koopman decomposition for a heteroclinic connection in a Stuart–Landau equation revealing two possible expansions. The expansions are centred about the two fixed points of the equation and extend beyond their linear subspaces before breaking down at a cross-over point in state space. Well-designed DMD can extract the two expansions provided that the time window does not contain this cross-over point. We then apply DMD to the Navier–Stokes equations near to a heteroclinic connection in low Reynolds number ($Re=O(100)$) plane Couette flow where there are multiple simple invariant solutions beyond the constant shear basic state. This reveals as many different Koopman decompositions as simple invariant solutions present and once more indicates the existence of cross-over points between the expansions in state space. Again, DMD can extract these expansions only if it does not include a cross-over point. Our results suggest that in a dynamical system possessing multiple simple invariant solutions, there are generically places in phase space – plausibly hypersurfaces delineating the boundary of a local Koopman expansion – across which the dynamics cannot be represented by a convergent Koopman expansion.