Exponential Regime Research Articles

An almost sure limit theorem for the maximum interpoint distance of random vectors in spaces of growing dimension

Let { p n ; n ≥ 1 } be a sequence of positive integers. Let X 1 , … , X p n be i.i.d. random vectors in ℝ n with i.i.d. components that are centered and have variance 1. Consider the maximum interpoint distance M n = max 1 ≤ i < j ≤ p n ‖ X i − X j ‖ 2 . This article presents the almost sure limit theorem for M n under the polynomial and the exponential growth regimes for the dimension. The proofs rely on the moderate deviation principle of the partial sum of i.i.d. random variables, the Chen–Stein Poisson approximation method, and Gaussian approximation techniques.

Theoretical analysis of the structure, thermodynamics, and shear elasticity of deeply metastable hard sphere fluids.

The structure, thermodynamics, and slow activated dynamics of the equilibrated metastable regime of glass-forming fluids remain a poorly understood problem of high theoretical and experimental interest. We apply a highly accurate microscopic equilibrium liquid state integral equation theory, in conjunction with naïve mode coupling theory of particle localization, to study in a unified manner the structural correlations, thermodynamic properties, and dynamic elastic shear modulus in deeply metastable hard sphere fluids. Distinctive behaviors are predicted including divergent inverse critical power laws for the contact value of the pair correlation function, pressure, and inverse dimensionless compressibility, and a splitting of the second peak and large suppression of interstitial configurations of the pair correlation function. The dynamic elastic modulus is predicted to exhibit two distinct exponential growth regimes with packing fraction that have strongly different slopes. These thermodynamic, structural, and elastic modulus results are consistent with simulations and experiments. Perhaps most unexpectedly, connections between the amplitude of long wavelength density fluctuations, dimensionless compressibility, local structure, and the dynamic elastic shear modulus have been theoretically elucidated. These connections are more broadly relevant to understanding the slow activated relaxation and mechanical response of colloidal suspensions in the ultradense metastable region and deeply supercooled thermal liquids in equilibrium.

Gravitational disturbance on asteroidal ring systems by close encounter with a small object

To date, rings are found around a Centaur (10199) Chariklo, trans-neptunian objects (TNOs) (136108) Haumea, and (50000) Quaoar. These discoveries suggest that asteroidal ring systems may be common, particularly in the outer solar system. Since collisions are a ubiquitous and fundamental evolutionary process throughout the solar system, we conjecture that asteroidal ring systems must have experienced close encounters with small objects as part of their evolutionary process. Here, we investigate the response of ring systems when they experience gravitational disturbance by a close encounter with another small object, by calculating the change in eccentricity and the fraction of lost ring particles. We find that a perturber needs to be as massive as or more massive than the ringed object, and needs to pass in the immediate vicinity of the ring in order to cause significant disruption. The change in eccentricity expected for Chariklo's inner ring and Quaoar's outer ring agrees with the analytical expression derived from the impulse approximation, while that for Haumea's ring agrees with the analytical expression called “exponential regime”. If we define a lifetime of a ring as the mean time to experience disruptive close encounters that can raise the eccentricity of ring particles >0.1, the lifetime for ring systems around Chariklo, Haumea, and Quaoar are >104 Gyr. We conclude that ring systems around Chariklo, Haumea, and Quaoar are highly unlikely to suffer from close encounters with another small object even if those systems are as old as 4 Gyr. Close encounter with a small object may not be responsible for the finite eccentricity observed for Chariklo's ring system, although eccentricity of ring particles could be increased to ∼10−4 in 4 Gyr. We also find that the lifetime is shorter for smaller ringed objects, and it still exceeds 4 Gyr for km-sized ringed objects in the outer solar system. Therefore, regardless of the size of ringed objects, asteroidal ring systems in the outer solar system are unlikely to suffer severe damage by close encounter with a small object. On the other hand, the lifetime is estimated to be on the order of 100 Myr for km-sized asteroids and 10 Myr for 100 m-sized asteroids in the near-Earth region and the main belt. This finding offers a dynamical explanation why ring systems have not been found in the inner solar system.

Balance between asymmetric allocation and repair of somatic damage in unicellular life forms as an ancient form of r/K selection

Over the course of multiple divisions, cells accumulate diverse nongenetic, somatic damage including misfolded and aggregated proteins and cell wall defects. If the rate of damage accumulation exceeds the rate of dilution through cell growth, a dedicated mitigation strategy is required to prevent eventual population collapse. Strategies for somatic damage control can be divided into two categories, asymmetric allocation and repair, which are not, in principle, mutually exclusive. We explore a mathematical model to identify the optimal strategy, maximizing the total cell number, over a wide range of environmental and physiological conditions. The optimal strategy is primarily determined by extrinsic, damage-independent mortality and the physiological model for damage accumulation that can be either independent (linear) or increasing (exponential) with respect to the prior accumulated damage. Under the linear regime, the optimal strategy is either exclusively repair or asymmetric allocation, whereas under the exponential regime, the optimal strategy is a combination of asymmetry and repair. Repair is preferred when extrinsic mortality is low, whereas at high extrinsic mortality, asymmetric damage allocation becomes the strategy of choice. We hypothesize that at an early stage of life evolution, optimization over repair and asymmetric allocation of somatic damage gave rise to r and K selection strategists.

Casimir Energy in (2 + 1)-Dimensional Field Theories

We explore the dependence of vacuum energy on the boundary conditions for massive scalar fields in (2 + 1)-dimensional spacetimes. We consider the simplest geometrical setup given by a two-dimensional space bounded by two homogeneous parallel wires in order to compare it with the non-perturbative behaviour of the Casimir energy for non-Abelian gauge theories in (2 + 1) dimensions. Our results show the existence of two types of boundary conditions which give rise to two different asymptotic exponential decay regimes of the Casimir energy at large distances. The two families are distinguished by the feature that the boundary conditions involve or not interrelations between the behaviour of the fields at the two boundaries. Non-perturbative numerical simulations and analytical arguments show such an exponential decay for Dirichlet boundary conditions of SU(2) gauge theories. The verification that this behaviour is modified for other types of boundary conditions requires further numerical work. Subdominant corrections in the low-temperature regime are very relevant for numerical simulations, and they are also analysed in this paper.

Turbulent transition in a channel with superhydrophobic walls: anisotropic slip and shear misalignment effects

Superhydrophobic surfaces dramatically reduce skin friction of overlying liquid flows. These surfaces are complex and numerical simulations usually rely on models to reduce this complexity. One of the simplest consists of finding an equivalent boundary condition through a homogenisation procedure, which in the case of channel flow over oriented riblets, leads to the presence of a small spanwise component in the homogenised base flow velocity. This work aims at investigating the influence of such a three-dimensionality of the base flow on stability and transition in a channel with walls covered by oriented riblets. Linear stability for this base flow is investigated: a new instability region, linked to cross-flow effects, is observed. Tollmien–Schlichting waves are also retrieved but the most unstable are three-dimensional. Transient growth is also affected as oblique streaks with non-zero streamwise wavenumber become the most amplified perturbations. When transition is induced by Tollmien–Schlichting waves, after an initial exponential growth regime, streaky structures with large spanwise wavenumber rapidly arise. Modal mechanisms appear to play a leading role in the development of these structures and a secondary stability analysis is performed to retrieve successfully some of their characteristics. The second scenario, initiated with cross-flow vortices, displays a strong influence of nonlinearities. The flow develops into large quasi-spanwise-invariant structures before breaking down to turbulence. Secondary stability on the saturated cross-flow vortices sheds light on this stage of transition. In both cases, cross-flow effects dominate the flow dynamics, suggesting the need to consider the anisotropicity of the wall condition when modelling superhydrophobic surfaces.

Discrete nutrient utilizations in two Bauhinia species exposed to supplemental light-emitting diode spectra and exponential fertilization

Bauhinia is a widely planted urban tree plant in tropical cities, which are frequently dwelling in a habitat exposed to inevitable streetlamp lighting. It is of practical meaning to figure out the lighting spectra that benefit nutrient utilization in corporation with a proper exponential fertilization regime. In this study, Bauhinia brachycarpa and B. variegata seedlings were cultured with exponential fertilization at the rate of 80 mg nitrogen (N) plant-1 (N-phosphorus-potassium, 10-7-9) with an unfertilized control, and both were exposed to light-emitting diode spectra of R1BG5 (13.9% red, 77.0% green, 9.2% blue), R2BG3 (26.2% red, 70.2% green, 3.5% blue), and R3BG1 (42.3% red, 57.3% green, 0.4% blue). The R3BG1 spectrum resulted in smaller growing size and dry mass but higher nutrient concentration and root water content compared to the other two spectra. Exponential fertilization increased fresh mass for two Bauhinia species but only increased dry mass in B. brachycarpa. Compared to B. variegata, B. brachycarpa was verified to have a higher capacity to accommodate exogeneous nutrient input through exponential fertilization. The R3BG1 spectrum was recommended for the illumination in streetlamps for Bauhinia because this spectrum can promote nutrient uptake without too fast rate of growth relative to other spectra with lower red-light proportions.

Impact of Zr substitution on the electronic structure of ferroelectric hafnia

HfO2-based dielectrics are promising for nanoscale ferroelectric applications, and the most favorable material within the family is Zr-substituted hafnia, i.e., Hf1−xZrxO2 (HZO). The extent of Zr substitution can be great, and x is commonly set to 0.5. However, the bandgap of ZrO2 is lower than HfO2, thus it is uncertain how the Zr content should influence the electronic band structure of HZO. A reduced bandgap is detrimental to the cycling endurance as charge injection and dielectric breakdown would become easier. Another issue is regarding the comparison on the bandgaps between HfO2/ZrO2 superlattices and HZO solid-state solutions. In this work, we systematically investigated the electronic structures of HfO2, ZrO2, and HZO using self-energy corrected density functional theory. In particular, the conduction band minimum of Pca21-HfO2 is found to lie at an ordinary k-point on the Brillouin zone border, not related to any interlines between high-symmetry k-points. Moreover, the rule of HZO bandgap variation with respect to x has been extracted. The physical mechanisms for the exponential reduction regime and linear decay regime have been revealed. The bandgaps of HfO2/ZrO2 ferroelectric superlattices are investigated in a systematic manner, and the reason why the superlattice could possess a bandgap lower than that of ZrO2 is revealed through comprehensive analysis.

Self-Patterning of Polyelectrolyte Multilayer Films: The Roles of PSS Molecular Weight, the Top Layer, and Post-Preparation Treatment

The self-patterning of thin films is relevant for both fundamental research and applications. We investigate polyelectrolyte multilayer films made from poly(diallyldimethylammonium) chloride and poly(styrene sulfonate) sodium salt (PDADMA/PSS). Various PSS with low molecular weight were used. First, the film thickness increases exponentially with the number of deposited PDADMA/PSS bilayers. The separation and height of the pillar-like domains increase significantly with each deposited PDADMA/PSS bilayer, as AFM images show. After the exponential growth regime, either a parabolic (and then a linear) or a linear growth regime follows, depending on the PSS molecular weight. The domain separation changes less and correlates with the vertical growth regimes. The domain separation varies between 70 and 750 nm and always exceeds the domain height. PSS-terminated films show the same domain distance in water and air. However, when PDADMA-terminated films are dried, the domain distance in air increases while the domain height decreases, causing a reduction in total area. In the air, the surface energy is greater than in water and a highly textured surface costs a lot of energy. We propose that the changed surface pattern is attributable to energy minimization. Furthermore, the domains are stable when exposed to 1 M NaCl solution but shrink enormously in 2 M NaCl, while their separation increases slightly. Under certain conditions, 50 nm broad filaments consisting of PDADMA/PSS complexes are observed. We suggest that these complexes diffuse between the domains and adjust their separation.

Quantitative Relation between Ionic Diffusivity and Ionic Association in Ionic Liquid–Water Mixtures

Molecular dynamic simulations of aqueous mixtures of imidazolium ionic liquids (ILs) were performed to elucidate the dependence of the ionic diffusivity on the microscopic structures changed by water. Two distinct regimes of the average ionic diffusivity (Dave) were identified with the increased water concentrations: the jam regime with slowly increased Dave and the exponential regime with rapidly increased Dave, which are found to be directly correlated to the ionic association. Further analysis leads to two general relationships independent of IL species between Dave and the degree of ionic association: (i) a consistent linear relationship between Dave and the inverse of ion-pair lifetimes (1/τIP) in the two regimes and (ii) an exponential relationship between normalized diffusivities (D̃ave) and short-ranged interactions between cations and anions (Ẽions), with different interdependent strengths in the two regimes. These findings revealed and quantified the direct correlation between dynamic properties and ionic association in IL-water mixtures.

Anti-plane shear waves in an elastic strip rigidly attached to an elastic half-space

We consider the anti-plane shear waves in a domain consisting of an infinite layer with a thin coating lying on an elastic half-space. The elastic properties of the coating, layer, and half-space are assumed to be different. On the free upper surface we assume the compatibility condition within the Gurtin–Murdoch surface elasticity, whereas at the plane interface we consider perfect contact. For this problem there exist two possible regimes related to waves exponentially decaying in the half-space. The first one, called transversely exponential–transversely exponential (TE–TE) regime, is related to waves described by exponential in transverse direction functions; the second, transversely harmonic–transversely exponential (TH–TE) regime, corresponds to waves in the upper layer which have the harmonic behaviour in the transverse direction. Detailed analysis of the derived dispersion equations for both regimes is provided. In particular, the effects of surface stresses, the layer thickness as well as of the ratio of shear moduli of the upper layer and half-space on the dispersion curves is analysed.

A principal stress cap model for stresses in square silos, with some examples of exponential stress

The principal stress cap approach has been used to model stresses in square hoppers, based upon an orthogonal grid upon a spherical surface sector, bounded by “great circles”. The model agrees with Rotter's correlation for lateral variation of wall normal stress for concave, active systems at uniform state of stress. However, systems of convex, passive stress do not conform to Rotter. The data of Jarrett{Geotechnique, 1995, 45, 95–104} and Brown{Chemical Engineering Science, 2000, 55, 4399–4413} have been modelled successfully, both qualitatively and quantitatively. Regions of the exponential stress regime have been found in the corner regions of the silo, alongside conventional Janssenian stress at the wall mid-points. Equivalent data of Jarrett and Brown are in reasonable agreement when plotted in dimensionless form. • Stresses in square hoppers have been modelled. • Principal stress cap approach is easy-to-use and visual model. • The model has been fitted to 2 sources of experimental data. • The data and the model show regions of exponential stress. • Horizontal variations in wall normal stress are seen.

Fire retainment on Cayley graphs

We study the fire-retaining problem on groups, a quasi-isometry invariant11See Section 2 for some caveats. introduced by Martínez-Pedroza and Prytuła [8], related to the firefighter problem. We prove that any Cayley graph with degree-d polynomial growth does not satisfy {f(n)}-retainment, for any f(n)=o(nd−2), matching the upper bound given for the firefighter problem for these graphs. In the exponential growth regime we prove general lower bounds for direct products and wreath products. These bounds are tight, and show that for exponential-growth groups a wide variety of behaviors is possible. In particular, we construct, for any d≥1, groups that satisfy {nd}-retainment but not o(nd)-retainment, as well as groups that do not satisfy sub-exponential retainment.

Post-extinction recovery of the Phanerozoic oceans and biodiversity hotspots

The fossil record of marine invertebrates has long fuelled the debate as to whether or not there are limits to global diversity in the sea1–5. Ecological theory states that, as diversity grows and ecological niches are filled, the strengthening of biological interactions imposes limits on diversity6,7. However, the extent to which biological interactions have constrained the growth of diversity over evolutionary time remains an open question1–5,8–11. Here we present a regional diversification model that reproduces the main Phanerozoic eon trends in the global diversity of marine invertebrates after imposing mass extinctions. We find that the dynamics of global diversity are best described by a diversification model that operates widely within the exponential growth regime of a logistic function. A spatially resolved analysis of the ratio of diversity to carrying capacity reveals that less than 2% of the global flooded continental area throughout the Phanerozoic exhibits diversity levels approaching ecological saturation. We attribute the overall increase in global diversity during the Late Mesozoic and Cenozoic eras to the development of diversity hotspots under prolonged conditions of Earth system stability and maximum continental fragmentation. We call this the ‘diversity hotspots hypothesis’, which we propose as a non-mutually exclusive alternative to the hypothesis that the Mesozoic marine revolution led this macroevolutionary trend12,13.

Detecting few-body quantum chaos: out-of-time ordered correlators at saturation

We study numerically and analytically the time dependence and saturation of out-of-time ordered correlators (OTOC) in chaotic few-body quantum-mechanical systems: quantum Henon-Heiles system (weakly chaotic), BMN matrix quantum mechanics (strongly chaotic) and Gaussian random matrix ensembles. The growth pattern of quantum-mechanical OTOC is complex and nonuniversal, with no clear exponential regime at relevant timescales in any of the examples studied (which is not in contradiction to the exponential growth found in the literature for many-body systems, i.e. fields). On the other hand, the plateau (saturated) value of OTOC reached at long times decreases with temperature in a simple and universal way: exp(const./T2) for strong chaos (including random matrices) and exp(const./T) for weak chaos. For small matrices and sufficiently complex operators, there is also another, high-temperature regime where the saturated OTOC grows with temperature. Therefore, the plateau OTOC value is a meaningful indicator of few-body quantum chaos. We also discuss some general consequences of our findings for the AdS/CFT duality.

Electroosmotic pressure-driven oscillatory flow and mass transport of Oldroyd-B fluid under high zeta potential and slippage conditions in microchannels

The appropriate analysis for microscale transport of complex fluids is essential in many scientific and technological devices to establish the accurate theoretical model. For example, the viscoelastic fluid plays a vital role in medical diagnostics to separate biomolecules using the concept of a microchannel flow under time-periodic oscillation effects. The available literature demonstrates no such analysis existed for the time-periodic oscillation of Oldroyd-B fluid subjected to the electroosmotic and slippage flow conditions. The present paper develops a new mathematical approach for an electroosmotic pressure-driven pulsatile-Oldroyd-B fluid flow between two parallel plates in microchannels under slipping actions on the wall surface. The flow takes place in an exponential electric field regime by imposing an exponential pressure gradient with time to simulate the elastic and viscous behaviors of the Oldroyd-B fluid. A finite difference method (MATLAB bvp4c) is employed to solve the strongly non-linear electrical potential and concentration equations. A time-periodic oscillation effect on the velocity profile, concentration distribution, and mass transport is analyzed under low or high zeta potentials at the wall. The results highlight that the elastic nature of fluid increases the oscillatory behavior of flow velocity compared to that of the viscous fluid. The zeta potential (ξ¯)in the flow field is responsible for increasing the flow velocity amplitude. Notably, the flow velocity for a high zeta potential (ξ¯=4) at the wall has increased approximately 142.75% at the peak point compared to the low zeta potential(ξ¯=0.1). The velocity amplitude for a higher elastic fluid (elasticity, E = 0.2) compared to a lower elastic fluid (E = 0.1) amplifies 21.01% at the peak point. Finally, it is noteworthy to demonstrate that for the Oldroyd-B fluid, a resonant behavior of the mass transfer rate is dependent on the combination of the parameters, namely angular Reynolds number and elasticity. In this context, it is evident that the interaction of the fluid elasticity and the oscillatory character of flow enhance the mass transfer rate. This innovative concept can be adapted to amplify the fluid mass transfer rate.

Artificial and natural silk materials have high mechanical property variability regardless of sample size

Silk fibres attract great interest in materials science for their biological and mechanical properties. Hitherto, the mechanical properties of the silk fibres have been explored mainly by tensile tests, which provide information on their strength, Young’s modulus, strain at break and toughness modulus. Several hypotheses have been based on these data, but the intrinsic and often overlooked variability of natural and artificial silk fibres makes it challenging to identify trends and correlations. In this work, we determined the mechanical properties of Bombyx mori cocoon and degummed silk, native spider silk, and artificial spider silk, and compared them with classical commercial carbon fibres using large sample sizes (from 10 to 100 fibres, in total 200 specimens per fibre type). The results confirm a substantial variability of the mechanical properties of silk fibres compared to commercial carbon fibres, as the relative standard deviation for strength and strain at break is 10–50%. Moreover, the variability does not decrease significantly when the number of tested fibres is increased, which was surprising considering the low variability frequently reported for silk fibres in the literature. Based on this, we prove that tensile testing of 10 fibres per type is representative of a silk fibre population. Finally, we show that the ideal shape of the stress–strain curve for spider silk, characterized by a pronounced exponential stiffening regime, occurs in only 25% of all tested spider silk fibres.

Relation between Charging Times and Storage Properties of Nanoporous Supercapacitors.

An optimal combination of power and energy characteristics is beneficial for the further progress of supercapacitors-based technologies. We develop a nanoscale dynamic electrolyte model, which describes both static capacitance and the time-dependent charging process, including the initial square-root dependency and two subsequent exponential trends. The observed charging time corresponds to one of the relaxation times of the exponential regimes and significantly depends on the pore size. Additionally, we find analytical expressions providing relations of the time scales to the electrode’s parameters, applied potential, and the final state of the confined electrolyte. Our numerical results for the charging regimes agree with published computer simulations, and estimations of the charging times coincide with the experimental values.

Universal Scaling for the Exit Dynamics of Block Copolymers from Micelles at Short and Long Time Scales.

The correlation function for the exit of poloxamer copolymers from equilibrated micelles is found to show up to four regimes depending on the chain flexibility: an initial fast reorganization, a logarithmic intermediate regime, followed by an exponential intermediate regime, and a final exponential decay. The logarithmic intermediate regime has been observed experimentally and attributed to the polydispersity of the polymer samples. However, we present dynamic single-chain mean-field theory simulations with chains of variable flexibility which show the same logarithmic relaxation but with strictly monodisperse systems. In agreement with our previous studies, we propose that this logarithmic response arises from a degeneracy of energy states of the hydrophobic block in the micelle core. For this to occur, a sufficiently large number of degenerate conformational states are required, which depend on the polymer flexibility and therefore should not be present for rigid polymers. Experimental results for monodisperse polymeric samples claiming the absence of such a logarithmic response may also lack a sufficient number of hydrophobic blocks for the required number of configurational states for this type of response to be seen. The insight gained from analyzing the simulation results allows us to propose a modified Eyring equation capable of reproducing the observed dynamic behavior. On scaling experimental results from different sources and systems according to this equation, we find a unique master curve showing a universal nature of the intermediate regimes: the logarithmic regime together with the secondary exponential decay. The terminal exponential regime at long times proposed by the standard Halperin and Alexander model is beyond the range of the data analyzed in this article. The universality observed suggests an entropic origin of the short-time dynamic response of this class of systems rather than the polydispersity.

Inhomogeneous phase separation kinetics in liquid binary mixtures: Sensitivity to initial local composition †

The kinetics of phase separation subsequent to a finite temperature quench is assumed to be driven by diffusion on the altered free energy surface and is generally assumed to be slow. The situation can be different in phase separating liquid binary mixtures, especially for systems characterized by the large difference in mutual interactions between solute and solvent molecules. In such cases, the phase separation kinetics could be fast and may get completed within a short time (ns) scale. As a result, in these systems, one may observe diverse dynamical features arising out of local heterogeneity leading to the onset of phase separation through pattern formation, spinodal decomposition, nucleation, and growth. By using a coarse-grained analysis, we examine phase separation kinetics in each spatial grid and indeed observe important effects of initial heterogeneity on the subsequent evolution. Interestingly, we observe slower separation kinetics for those regions that correspond to the composition at the minimum of the high-temperature surface. The heterogeneous dynamics has been captured here through the non-linear susceptibility function, which shows a pattern similar to what is observed in the supercooled liquid. Each grid shows somewhat different dynamics in the three-stage (exponential, power-law, and logarithmic regime) phase separation dynamics. The late stage of phase separation kinetics is usually attributed to the coarsening of the phase-separated domains. However, in a liquid binary mixture, the late-stage power-law decay undergoes a further change. A new dynamical regime arises characterized by a logarithmic time dependence, which is due to the “smoothening” of the rough interface of already well-separated phases. This can also be described as opposite to the roughening transition described by Chui and Weeks [Phys. Rev. Lett. 40, 733 (1978)]. This reverse roughening transition can explain the logarithmic time dependence observed in the simulation.