Linear Growth Time Research Articles

Characterization of nano formation and growth in a LiNbO3 glass

The growth mechanism of nanosized crystal grains of LiNbO3 with changing temperature and time has been studied. The nanograin LiNbO3 crystal is initially obtained by crystallizing the pure LiNbO3 glass that is formed using a polymeric complex method. X-ray diffraction and scanning electron microscopy are used to measure the grain sizes. Our study shows that, when the temperature linearly changes from 650 to 1000 °C, the nanograins linearly grow from about 55 nm to 100 nm with increasing temperature. At a fixed temperature of 700 °C, the nanograins also grow linearly within a delay time of 180 min. Meanwhile, at higher fixed temperatures of 800 and 900 °C, the linear dependence of grain growth occurs in the early stage; at a later time, the growth stops and the size saturates. Because nanograins with various sizes are required in many fields, the growth mechanism of nanograins with a linear growth in time and temperature and the size saturation effect in this study are expected to provide valuable information for applications.

Dynamics of gravitational collapse and holographic entropy production

We consider the time evolution of two entropy-like quantities, the holographic entanglement entropy and causal holographic information, in a model of holographic thermalization dual to the gravitational collapse of a thin planar shell. Unlike earlier calculations valid in different limits, we perform a full treatment of the dynamics of the system, varying both the shell's equation of state and initial position. In all cases considered, we find that between an early period related to the acceleration of the shell and a late epoch of saturation towards the thermal limit, the entanglement entropy exhibits universal linear growth in time in accordance with the prediction of Liu and Suh. As intermediate steps of our analysis, we explicitly construct a coordinate system continuous at the location of an infinitely thin shell and derive matching conditions for geodesics and extremal surfaces traversing this region.

Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards

We perform numerical measurements of the moments of the position of a tracer particle in a two-dimensional periodic billiard model (Lorentz gas) with infinite corridors. This model is known to exhibit a weak form of superdiffusion, in the sense that there is a logarithmic correction to the linear growth in time of the mean-squared displacement. We show numerically that this expected asymptotic behavior is easily overwhelmed by the subleading linear growth throughout the time range accessible to numerical simulations. We compare our simulations to analytical results for the variance of the anomalously rescaled limiting normal distributions.

Aspects of the non-zonal transition

The non-zonal transition, a process which can bring about very large heat fluxes in gyrokinetic simulations, occurs once a certain threshold plasma β is reached. This threshold is parameterized via a simulation database, yielding an expression estimating at what β a given system may approach the transition. Furthermore, the diffusive outward transport of a heat blob in a temperature profile marginally stable with respect to the non-zonal transition is discussed: the resulting transport timescale combines the underlying turbulent transport timescale and the linear instability growth time, thus demonstrating that the non-zonal transition provides a mechanism for very fast heat dissipation.

An approach to generate deterministic Brownian motion

We propose an approach for generation of deterministic Brownian motion. By adding an additional degree of freedom to the Langevin equation and transforming it into a system of three linear differential equations, we determine the position of switching surfaces, which act as a multi-well potential with a short fluctuation escape time. Although the model is based on the Langevin equation, the final system does not contain a stochastic term, and therefore the obtained motion is deterministic. Nevertheless, the system behavior exhibits important characteristic properties of Brownian motion, namely, a linear growth in time of the mean square displacement, a Gaussian distribution, and a −2 power law of the frequency spectrum. Furthermore, we use the detrended fluctuation analysis to prove the Brownian character of this motion.

Dynamical bar-mode instability in rotating and magnetized relativistic stars

We present three-dimensional simulations of the dynamical bar-mode instability in magnetized and differentially rotating stars in full general relativity. Our focus is on the effects that magnetic fields have on the dynamics and the onset of the instability. In particular, we perform ideal-magnetohydrodynamics simulations of neutron stars that are known to be either stable or unstable against the purely hydrodynamical instability, but to which a poloidal magnetic field in the range of $10^{14}$--$10^{16}$ G is superimposed initially. As expected, the differential rotation is responsible for the shearing of the poloidal field and the consequent linear growth in time of the toroidal magnetic field. The latter rapidly exceeds in strength the original poloidal one, leading to a magnetic-field amplification in the the stars. Weak initial magnetic fields, i.e. $ \lesssim 10^{15}$ G, have negligible effects on the development of the dynamical bar-mode instability, simply braking the stellar configuration via magnetic-field shearing, and over a timescale for which we derived a simple algebraic expression. On the other hand, strong magnetic fields, i.e. $\gtrsim 10^{16}$ G, can suppress the instability completely, with the precise threshold being dependent also on the amount of rotation. As a result, it is unlikely that very highly magnetized neutron stars can be considered as sources of gravitational waves via the dynamical bar-mode instability.

Small for Gestational Age and Age at Puberty: Evidence From Hong Kong's "Children of 1997" Birth Cohort

The predictive-adaptive response paradigm postulates that slow fetal growth advances puberty as a life-history strategy for reproductive success, when constraints on postnatal growth are minimal. The authors examined the association of birth weight for gestational age and small for gestational age (SGA) status (birth weight for gestational age <10th percentile, 6.9%) with clinically assessed age at onset of Tanner stage II in a non-Western developed population using interval-censored regression in 7,366 children (89% follow-up) from a population-representative Chinese birth cohort, "Children of 1997" in Hong Kong. Neither SGA status nor birth weight z score for gestational age was associated with age at onset of puberty, adjusted for sex, mother's place of birth, parental height, income, and parental education. Greater childhood height and linear growth were associated with younger age at onset of puberty. SGA status was associated with earlier puberty after adjustment for childhood height (time ratio = 0.984, 95% confidence interval: 0.972, 0.995) but later puberty after adjustment for linear growth (time ratio = 1.017, 95% confidence interval: 1.005, 1.030). In this developed city of China, SGA status was not associated with timing of puberty. However, the observation may be contextually specific depending on how other attributes, such as childhood growth, differ between SGA and other children.

Optical measurement of cycle-dependent cell growth

Determining the growth patterns of single cells offers answers to some of the most elusive questions in contemporary cell biology: how cell growth is regulated and how cell size distributions are maintained. For example, a linear growth in time implies that there is no regulation required to maintain homeostasis; an exponential pattern indicates the opposite. Recently, there has been great effort to measure single cells using microelectromechanical systems technology, and several important questions have been explored. However, a unified, easy-to-use methodology to measure the growth rate of individual adherent cells of various sizes has been lacking. Here we demonstrate that a newly developed optical interferometric technique, known as spatial light interference microscopy, can measure the cell dry mass of many individual adherent cells in various conditions, over spatial scales from micrometers to millimeters, temporal scales ranging from seconds to days, and cell types ranging from bacteria to mammalian cells. We found evidence of exponential growth in Escherichia coli, which agrees very well with other recent reports. Perhaps most importantly, combining spatial light interference microscopy with fluorescence imaging provides a unique method for studying cell cycle-dependent growth. Thus, by using a fluorescent reporter for the S phase, we measured single cell growth over each phase of the cell cycle in human osteosarcoma U2OS cells and found that the G2 phase exhibits the highest growth rate, which is mass-dependent and can be approximated by an exponential.

A study of planar Richtmyer-Meshkov instability in fluids with Mie-Grüneisen equations of state

We present a numerical comparison study of planar Richtmyer-Meshkov instability with the intention of exposing the role of the equation of state. Results for Richtmyer-Meshkov instability in fluids with Mie-Grüneisen equations of state derived from a linear shock-particle speed Hugoniot relationship (Jeanloz, J. Geophys. Res. 94, 5873, 1989; McQueen et al., High Velocity Impact Phenomena (1970), pp. 294–417; Menikoff and Plohr, Rev. Mod. Phys. 61(1), 75 1989) are compared to those from perfect gases under nondimensionally matched initial conditions at room temperature and pressure. The study was performed using Caltech’s Adaptive Mesh Refinement, Object-oriented C++ (AMROC) (Deiterding, Adaptive Mesh Refinement: Theory and Applications (2005), Vol. 41, pp. 361–372; Deiterding, “Parallel adaptive simulation of multi-dimensional detonation structures,” Ph.D. thesis (Brandenburgische Technische Universität Cottbus, September 2003)) framework with a low-dissipation, hybrid, center-difference, limiter patch solver (Ward and Pullin, J. Comput. Phys. 229, 2999 (2010)). Results for single and triple mode planar Richtmyer-Meshkov instability when a reflected shock wave occurs are first examined for mid-ocean ridge basalt (MORB) and molybdenum modeled by Mie-Grüneisen equations of state. The single mode case is examined for incident shock Mach numbers of 1.5 and 2.5. The planar triple mode case is studied using a single incident Mach number of 2.5 with initial corrugation wavenumbers related by k1=k2+k3. Comparison is then drawn to Richtmyer-Meshkov instability in perfect gases with matched nondimensional pressure jump across the incident shock, post-shock Atwood ratio, post-shock amplitude-to-wavelength ratio, and time nondimensionalized by Richtmyer’s linear growth time constant prediction. Differences in start-up time and growth rate oscillations are observed across equations of state. Growth rate oscillation frequency is seen to correlate directly to the oscillation frequency for the transmitted and reflected shocks. For the single mode cases, further comparison is given for vorticity distribution and corrugation centerline shortly after shock interaction. Additionally, we examine single mode Richtmyer-Meshkov instability when a reflected expansion wave is present for incident Mach numbers of 1.5 and 2.5. Comparison to perfect gas solutions in such cases yields a higher degree of similarity in start-up time and growth rate oscillations. The formation of incipient weak waves in the heavy fluid driven by waves emanating from the perturbed transmitted shock is observed when an expansion wave is reflected.

Coarsening in the electrohydrodynamic patterning of thin polymer films

Periodic pillarlike microstructures can be created from initially flat polymer films via the electrohydrodynamic instabilities. Those patterns, however, are metastable. Our experimental observations show that the average pillar size increases slowly after linear growth. Major coarsening events then take place over times several orders of magnitude longer than the linear growth time. For all fill ratios, a logarithmic time dependence of the average pillar size can be identified, i.e., <S> proportional to ln t. Thicker films, however, have faster coarsening rates than thinner films. Linear stability analysis of the pseudosteady states reveals two major coarsening mechanisms, collision and Ostwald ripening, which can also be identified from experimental images. We then reduce the original partial differential equation (PDE) into a pair of ODEs, which govern the interaction between pillars due to the above two coarsening mechanisms. From this, a logarithm scaling law is obtained for both low and high fill ratios and the coarsening rate is slower for lower fill ratios, consistent with experimental observations. We also find that arrays with more uniform sizes tend to start coarsening later, but they coarsen faster than more "disperse" arrays, which could be possibly utilized in experiments for controlling the onset and speed of coarsening. The logarithm scaling in the electrohydrodynamic coarsening phenomenon, which differs from coarsening in spinodal decomposition and dewetting of thin liquid films, is due to the significant nonlinear effect of Maxwell stresses and geometric confinement on the disjoining pressure at both top and bottom electrodes.

Startup process in the Richtmyer–Meshkov instability

An analytical model for the initial growth period of the planar Richtmyer–Meshkov instability is presented for the case of a reflected shock, which corresponds in general to light-to-heavy interactions. The model captures the main features of the interfacial perturbation growth before the regime with linear growth in time is attained. The analysis provides a characteristic time scale τ for the startup phase of the instability, expressed explicitly as a function of the perturbation wavenumber k, the algebraic transmitted and reflected shock speeds US1&amp;lt;0 and US2&amp;gt;0 (defined in the frame of the accelerated interface), and the postshock Atwood number A+: τ=[(1−A+)∕US2+(1+A+)∕(−US1)]∕(2k). Results are compared with computations obtained from two-dimensional highly resolved numerical simulations over a wide range of incident shock strengths S and preshock Atwood ratios A. An interesting observation shows that, within this model, the amplitude of small perturbations across a light-to-heavy interface evolves quadratically in time (and not linearly) in the limit A→1−.

Evolution of entanglement entropy following a quantum quench: Analytic results for theXYchain in a transverse magnetic field

The nonequilibrium evolution of the block entanglement entropy is investigated in the $XY$ chain in a transverse magnetic field after the Hamiltonian parameters are suddenly changed from and to arbitrary values. Using Toeplitz matrix representation and multidimensional phase methods, we provide analytic results for large blocks and for all times, showing explicitly the linear growth in time followed by saturation. The consequences of these analytic results are discussed and the effects of a finite block length is taken into account numerically.

Degenerate Modes of Symmetric Instability

A basic ow, which is symmetric in one of the horizontal direction (say, y direction), is unstable for y-symmetric disturbances when the basic potential vorticity is negative (so called symmetric instability). The temporal dependence of the growing modes is exponential, that is, eσt. In the case of vanishing growth rate, that is, σ → 0, there emerge degenerate modes as shown by Xu (2007). Under the thermal wind-balanced initial condition, Xu (2007) showed that the temporal dependence of the degenerate modes is linear, that is, they grow as a linear function of time. In this note, we show that in general the degener ate modes grow quadratically in time, and that in particular they are reduced to the result of Xu (2007), (i.e., linear growth in time) under the thermal-wind-balanced initial conditions.

An analysis of the fluctuations of the geomagnetic dipole

The time evolution of the strength of the Earth’s virtual axial dipole moment (VADM) is analyzed by relating it to the Fokker–Planck equation, which describes a random walk with VADM-dependent drift and diffusion coefficients. We demonstrate first that our method is able to retrieve the correct shape of the drift and diffusion coefficients from a time series generated by a test model. Analysis of the Sint-2000 data shows that the geomagnetic dipole mode has a linear growth time of 2 0 − 7 + 13 kyear, and that the nonlinear quenching of the growth rate follows a quadratic function of the type [ 1 − ( x / x 0 ) 2 ] . On theoretical grounds, the diffusive motion of the VADM is expected to be driven by multiplicative noise, and the corresponding diffusion coefficient to scale quadratically with dipole strength. However, analysis of the Sint-2000 VADM data reveals a diffusion which depends only very weakly on the dipole strength. This may indicate that the magnetic field quenches the amplitude of the turbulent velocity in the Earth’s outer core.

Temporal extensivity of Tsallis’ entropy and the bound on entropy production rate

The Tsallis entropy, which is a generalization of the Boltzmann-Gibbs entropy, plays a central role in nonextensive statistical mechanics of complex systems. A lot of efforts have recently been made on establishing a dynamical foundation for the Tsallis entropy. They are primarily concerned with nonlinear dynamical systems at the edge of chaos. Here, it is shown by generalizing a formulation of thermostatistics based on time averages recently proposed by Carati [A. Carati, Physica A 348, 110 (2005)] that, whenever relevant, the Tsallis entropy indexed by q is temporally extensive: linear growth in time, i.e., finite entropy production rate. Then, the universal bound on the entropy production rate is shown to be 1/ absolute value (1-q). The property of the associated probabilistic process, i.e., the sojourn time distribution, determining randomness of motion in phase space is also analyzed.

On the role of endothelial progenitor cells in tumor neovascularization

The exact role that bone marrow (BM)-derived endothelial progenitor cells (EPCs) play in tumor neovascularization is heavily debated. We develop a quantitative three-compartment model with predictive power regarding the dynamics of tumorigenesis. There are two distinct processes by which tumor neovasculature can be built: angiogenesis is the formation of new blood vessels from preexisting vessels; vasculogenesis is the formation of new vessels by recruiting circulating EPCs. We show that vasculogenesis-driven and angiogenesis-driven tumors grow in different ways. (i) If angiogenesis is the prevailing process, then the tumor mass (and volume) will grow as a cubic power of time, and BM-derived EPCs will stay at a constant level. (ii) If vasculogenesis is the dominant process, then the tumor mass will be characterized by a linear growth in time, and the number of circulating EPCs (after possibly increasing to a maximum) will decrease to low levels. With this information, one can identify the “signature” of each of the processes in the observations of tumor growth and the dynamics of the relevant characteristics, such as the level of BM-derived EPCs. We show how our results can help explain some apparently contradictory experimental data. We also propose ways to couple this study with directed experiments to identify the exact role of vasculogenesis in tumor progression.

Relativistic Euler equations for isentropic fluids: Stability of Riemann solutions with large oscillation

We analyze global entropy solutions of the 2 × 2 relativistic Euler equations for isentropic fluids in special relativity and establish the uniqueness of Riemann solutions in the class of entropy solutions in L∞ ∩ BVloc with arbitrarily large oscillation. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions implies their inviscid time-asymptotic stability under arbitrarily large L1 ∩ L∞ ∩ BVloc perturbation of the Riemann initial data, provided that the corresponding solutions are in L∞ and have local bounded total variation that allows the linear growth in time. This approach is also extended to deal with the stability of Riemann solutions containing vacuum in the class of entropy solutions in L∞ with arbitrarily large oscillation.

Stability of Riemann solutions with large oscillation for the relativistic Euler equations

We are concerned with entropy solutions of the 2×2 relativistic Euler equations for perfect fluids in special relativity. We establish the uniqueness of Riemann solutions in the class of entropy solutions in L ∞∩ BV loc with arbitrarily large oscillation. Our proof for solutions with large oscillation is based on a detailed analysis of global behavior of shock curves in the phase space and on special features of centered rarefaction waves in the physical plane for this system. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily large L 1∩ L ∞∩ BV loc perturbation of the Riemann initial data, as long as the corresponding solutions are in L ∞ and have local bounded total variation that allows the linear growth in time. We also extend our approach to deal with the uniqueness and stability of Riemann solutions containing vacuum in the class of entropy solutions in L ∞ with arbitrarily large oscillation.

On the linear and nonlinear resistive tearing‐mode instabilities

The tearing‐mode instabilities of Harris sheet magnetic field configuration are studied based on the numerical simulations of two‐dimensional, resistive, compressible MHD models with isotropic pressure. The linear growth rate and the eigenmode structure are calculated from the linear numerical model for various parameters: magnetic Reynolds number, Rm; the ratio of wavelength to the layer thickness; magnetic By component; and plasma beta, as well as for different equations of state. The linear solutions are then used as initial perturbations of the nonlinear numerical model to allow the full evolution of resistive tearing‐mode instability. The linear calculations show that for the range of Rm = 10–105 the fastest growth rate increases with decreasing Rm and only slightly increases with increasing plasma beta and By but that it is not sensitive to the equation of state. While the nonlinear calculations snow that only for large Rm, where the diffusion time is much larger than the linear growth time, the magnetic island may possibly grow substantially and become saturated. For small Rm the plasmoids either diminish in the late stage or do not have apparent growing; that is, the linear analysis is not meaningful for large resistivity cases. The calculations are compared to the magnetic island structure at Earth's magnetopause reconstructed from the single‐spacecraft data by Hau and Sonnerup [1999].

Gravity Modes in ZZ Ceti Stars. I. Quasi‐adiabatic Analysis of Overstability

We analyze the stability of g-modes in white dwarfs with hydrogen envelopes. All relevant physical processes take place in the outer layer of hydrogen-rich material, which consists of a radiative layer overlaid by a convective envelope. The radiative layer contributes to mode damping, because its opacity decreases upon compression and the amplitude of the Lagrangian pressure perturbation increases outward. The convective envelope is the seat of mode excitation, because it acts as an insulating blanket with respect to the perturbed flux that enters it from below. A crucial point is that the convective motions respond to the instantaneous pulsational state. Driving exceeds damping by as much as a factor of 2 provided ωτ_c≥1, where ω is the radian frequency of the mode and τ_c≈4τ_(th), with τ_(th) being the thermal time constant evaluated at the base of the convective envelope. As a white dwarf cools, its convection zone deepens, and lower frequency modes become overstable. However, the deeper convection zone impedes the passage of flux perturbations from the base of the convection zone to the photosphere. Thus the photometric variation of a mode with constant velocity amplitude decreases. These factors account for the observed trend that longer period modes are found in cooler DA variables. Overstable modes have growth rates of order γ~1/(nτ_ω), where n is the mode's radial order and τ_ω is the thermal timescale evaluated at the top of the mode's cavity. The growth time, γ^(−1), ranges from hours for the longest period observed modes (P≈20 minutes) to thousands of years for those of shortest period (P≈2 minutes). The linear growth time probably sets the timescale for variations of mode amplitude and phase. This is consistent with observations showing that longer period modes are more variable than shorter period ones. Our investigation confirms many results obtained by Brickhill in his pioneering studies of ZZ Cetis. However, it suffers from two serious shortcomings. It is based on the quasiadiabatic approximation that strictly applies only in the limit ωτ_c » 1, and it ignores damping associated with turbulent viscosity in the convection zone. We will remove these shortcomings in future papers.