Critical Time Scale Research Articles

Analysis of large unreliable stochastic networks

In this paper, a stochastic model of a large distributed system where users’ files are duplicated on unreliable data servers is investigated. Due to a server breakdown, a copy of a file can be lost, it can be retrieved if another copy of the same file is stored on other servers. In the case where no other copy of a given file is present in the network, it is definitively lost. In order to have multiple copies of a given file, it is assumed that each server can devote a fraction of its processing capacity to duplicate files on other servers to enhance the durability of the system. A simplified stochastic model of this network is analyzed. It is assumed that a copy of a given file is lost at some fixed rate and that the initial state is optimal: each file has the maximum number $d$ of copies located on the servers of the network. The capacity of duplication policy is used by the files with the lowest number of copies. Due to random losses, the state of the network is transient and all files will be eventually lost. As a consequence, a transient $d$-dimensional Markov process $(X(t))$ with a unique absorbing state describes the evolution this network. By taking a scaling parameter $N$ related to the number of nodes of the network, a scaling analysis of this process is developed. The asymptotic behavior of $(X(t))$ is analyzed on time scales of the type $t\mapsto N^{p}t$ for $0\leq p\leq d-1$. The paper derives asymptotic results on the decay of the network: Under a stability assumption, the main results state that the critical time scale for the decay of the system is given by $t\mapsto N^{d-1}t$. In particular, the duration of time after which a fixed fraction of files are lost is of the order of $N^{d-1}$. When the stability condition is not satisfied, that is, when it is initially overloaded, it is shown that the state of the network converges to an interesting local equilibrium which is investigated. As a consequence, it sheds some light on the role of the key parameters $\lambda$, the duplication rate and $d$, the maximal number of copies, in the design of these systems. The techniques used involve careful stochastic calculus for Poisson processes, technical estimates and the proof of a stochastic averaging principle.

Enhanced spin–orbit coupling and charge carrier density suppression in LaAl1−xCrxO3/SrTiO3 hetero-interfaces

We report a gradual suppression of the two-dimensional electron gas (2DEG) at the LaAlO3/SrTiO3 interface on substitution of chromium at the Al sites. The sheet carrier density at the interface (n□) drops monotonically from ∼2.2 × 1014 cm−2 to ∼2.5 × 1013 cm−2 on replacing ≈60% of the Al sites by Cr and the sheet resistance (R□) exceeds the quantum limit for localization (h/2e2) in the concentrating range 40–60% of Cr. The samples with Cr ⩽40% show a distinct minimum (Tm) in metallic R□(T) whose position shifts to higher temperatures on increasing the substitution. Distinct signatures of Rashba spin–orbit interaction (SOI) induced magnetoresistance (MR) are seen in R□ measured in out of plane field (H⊥) geometry at T ⩽ 8 K. Analysis of these data in the framework of Maekawa–Fukuyama theory allows extraction of the SOI critical field (HSO) and time scale (τSO) whose evolution with Cr concentration is similar as with the increasing negative gate voltage in LAO/STO interface. The MR in the temperature range 8 K ⩽ T ⩽ Tm is quadratic in the field with a +ve sign for H⊥ and −ve sign for H∥. The behaviour of H∥ MR is consistent with Kondo theory which in the present case is renormalized by the strong Rashba SOI at T < 8 K.

Testing for Wilson's quantum field theory in less than 4 dimensions

Abstract Wilson's quantum field theory (QFT) in less than 4 dimensions has achieved a great success in the study of critical phenomenon but is still not tested within the scope of particle physics. To guarantee the validity of Wilson's QFT in less than 4 dimensions, Newton–Leibniz's differential-integral formulas must be extended to the noninteger dimensional situation. We show that this leads to a new prediction that Planck's constant will be expressed in terms of three fundamental constants: critical time scale, dimension of time axis, and total energy of universe. We propose the corresponding methods to measure these three constants. It will be thus interesting to compare the well-known value of Planck's constant with the potential theoretical value consisting of three fundamental constants.

Interlayer structure and self-healing in suspensions of brush-stabilized nanoplatelets with smectic order.

We have investigated the rheology of an uncured epoxy fluid containing high aspect ratio (length/thickness ≈ 160) α-zirconium phosphate (ZrP) nanoplatelets with smectic order. The nanoplatelets were exfoliated into monocrystalline sheets with uniform thickness using a monoamine-terminated oligomer. The oligomers were densely grafted to the plate surfaces and behave as a molecular brush. Suspensions containing ∼ 2 vol.% ZrP and above show liquid crystalline order with scattering peaks characteristic of a smectic (layered) mesophase. At much higher loading, ∼ 4 vol.% ZrP, there is a sharp transition in visual appearance, steady shear rheology, and linear and non-linear viscoelasticity that is attributed to the reversible interdigitation of oligomer chains between closely spaced layers. The oligomers are proposed to serve as inter-lamellar bridges that store elastic stresses for intermediate rates of deformation, but are able to relax on longer time scales. Under steady shearing conditions, the smectic suspensions with "overlapped" microstructure show a discontinuous flow curve characteristic of shear banding that is attributed to the dynamic pull-out of oligomer chains from the overlap region. At high shear rates, the limiting viscosity of the concentrated suspensions is on the same order of magnitude as the unfilled suspending fluid. When the rate of deformation is reduced below a critical time scale, the original network strength, and corresponding microstructure, is recovered through a passive self-healing process. The unique combination of concentration-dependent yield stress, low post-yield viscosity, and self-healing is potentially useful for various applications in the liquid state, and desirable for scalable processing of nanocomposite materials for structural applications.

Temporal Response of Bacterial Cells to High Pressures

Cells encounter physicochemical fluctuations in their environment on a routine basis. Bacterial cells have developed regulatory mechanisms to cope with many of these stress fluctuations. The response of the cells to different stresses may depend on the type, magnitude and the duration of the stress [1]. While the stress response of bacterial cells to temperature fluctuations is relatively known, the stress response to fluctuations in high pressures is not explored. We experimentally study the effect of pressure fluctuations on a mesophilic bacterium, Escherichia coli. Specifically, we study the cell morphology, cell division, and gene expression of the one of the major cytoskeletal proteins MreB, which is implicated in maintaining both the cell shape and cell division, as a function of temporal fluctuations in pressure. We find that the probability of high pressure induced elongation of bacterial cells or lack of cell division decreases as the cells are allowed to relax at lower hydrostatic pressure. Moreover, we find that for a given pressure, the rate of cell division depends on the ratio of doubling time and the time scale of relaxation to normal pressure. From our data we can quantify the critical time-scale over which cell morphology and cell division does not exhibit any apparent changes compared to bacterial cells at normal growth conditions. Our experimental results can be explained by a two state stochastic model of bacterial cell elongation at high pressure and relaxation to normal morphology at normal pressures [3]. Our results shed light on the cellular response of bacterial cells to novel stress fluctuations.

CONDITION FOR CAPTURE INTO FIRST-ORDER MEAN MOTION RESONANCES AND APPLICATION TO CONSTRAINTS ON THE ORIGIN OF RESONANT SYSTEMS

We investigate the condition for capture into first-order mean motion resonances using numerical simulations with a wide range of various parameters. In particular, we focus on deriving the critical migration timescale for capture into the 2:1 resonance; additional numerical experiments for closely spaced resonances (e.g., 3:2) are also performed. We find that the critical migration timescale is determined by the planet-to-stellar mass ratio, and its dependence exhibits power-law behavior with index -4/3. This dependence is also supported by simple analytic arguments. We also find that the critical migration timescale for systems with equal-mass bodies is shorter than that in the restricted problem; for instance, for the 2:1 resonance between two equal-mass bodies, the critical timescale decreases by a factor of 10. In addition, using the obtained formula, the origin of observed systems that include first-order commensurabilities is constrained. Assuming that pairs of planets originally form well separated from each other and then undergo convergent migration and are captured in resonances, it is possible that a number of exoplanets experienced rapid orbital migration. For systems in closely spaced resonances, the differential migration timescale between the resonant pair can be constrained well; it is further suggested that several exoplanets underwent migration that can equal or even exceed the type I migration rate predicted by the linear theory. This implies that some of them may have formed in situ. Future observations and the use of our model will allow us to statistically determine the typical migration speed in a protoplanetary disk.

Employing Cantor sets for earthquake time series analysis in two zones of western Turkey

In this study, a fractal approach is employed to analyze the seismic history of Fethiye and Simav zones in the West of Turkey. In this scope, a database, including the set of earthquakes in two seismogenic zones, is compiled. Applying fractal dimension and probability concepts, it is aimed to find out the occurrence probability of earthquakes having magnitudes equal to or greater than a threshold level. The results are analyzed in detail and relationships among the fractal dimension, threshold magnitude, probability intercept, and critical time scale are presented. The analyses revealed that activities in the Fethiye Zone are more frequent and continuous than those in the Simav Zone. As mentioned in previous studies, the critical time scales at certain magnitude thresholds can be referred to as the lower limit of the recurrence period of earthquakes in these zones.

Development of the trailing shear layer in a starting jet during pinch-off

AbstractExperiments on a circular starting jet generated by a piston–cylinder arrangement, over a range of Reynolds number from $2600$ to $5600$, are conducted so as to investigate the development of the trailing shear layer during the leading vortex ring formation, as well as the corresponding effects on the pinch-off process. Results obtained by digital particle image velocimetry (DPIV) show that secondary vortices start to develop in the trailing jet only after the critical time scale, the ‘formation number’, is achieved. The subsequent growth of the secondary vortices reduces the vorticity flux being fed into the leading vortex ring and, as a consequence, constrains the growth of leading vortex ring with larger circulation. Evolution of perturbation waves into secondary vortices is found to associate with growth and translation of the leading vortex ring during the formation process. A dimensionless parameter ‘$A$’, defined as ${\Gamma }_{\mathit{ring}} / ({x}_{\mathit{core}} \mrm{\Delta} U$), is therefore proposed to characterize the effect of the leading vortex ring on suppressing the nonlinear development of instability in the trailing shear layer, i.e. the initial roll-up of the secondary vortices. In a starting jet, $A$ follows a decreasing trend with the formation time ${t}^{\ensuremath{\ast} } $. A critical value ${A}_{c} = 1. 1\pm 0. 1$ is identified experimentally, which physically coincides with the initiation of the first secondary vortex roll-up and, therefore, indicates the onset of pinch-off process.

Solute transport in divergent radial flow with multistep pumping

An efficient approach is developed to analytically evaluate solute transport in a horizontal, divergent radial flow field with a multistep injection flow rate and an arbitrary input concentration history. By assuming a piecewise steady state flow and transforming the time domain to the cumulative injected flow domain, the concentration distribution is found to be completely determined by the total volume of injected flow and independent of specific flow rates. Thus, on the cumulative flow domain, the transport problem with a temporally varying velocity field can be transformed into a steady state flow problem. Linear convolution can then be applied on the cumulative injected flow domain to evaluate the solution for an arbitrarily time‐dependent input concentration. Solutions on the regular time domain can be conveniently obtained by mapping the solution on the cumulative injected flow domain to the time domain. Furthermore, we theoretically examine the conditions for the assumption of piecewise steady state flow to be valid. On the basis of the critical time scale of the “pseudosteady state condition,” defined as when velocity changes accomplish 99% of their steady state differences, and the relative error in the mean travel time of plume front, we obtain conditions for neglecting the transitional period between two pumping steps. Such conditions include the following: (1) the duration of a pumping step, tp, must be longer than the critical time scale, tc, i.e., tp ≥ tc = 25r2S/T, where r is the radial distance, S is the storage coefficient, and T is the transmissivity, or similarly, a maximum problem domain needs to be defined for a given pumping strategy. (2) the maximum well pumping rate, qmax, should satisfy qmax ≤ πθT/25S, where θ is the effective porosity. When both conditions are satisfied, transitional periods may be neglected.

ANALYTICAL STAR FORMATION RATE FROM GRAVOTURBULENT FRAGMENTATION

We present an analytical determination of the star formation rate (SFR) in molecular clouds, based on a time-dependent extension of our analytical theory of the stellar initial mass function (IMF). The theory yields SFR's in good agreement with observations, suggesting that turbulence {\it is} the dominant, initial process responsible for star formation. In contrast to previous SFR theories, the present one does not invoke an ad-hoc density threshold for star formation; instead, the SFR {\it continuously} increases with gas density, naturally yielding two different characteristic regimes, thus two different slopes in the SFR vs gas density relationship, in agreement with observational determinations. Besides the complete SFR derivation, we also provide a simplified expression, which reproduces reasonably well the complete calculations and can easily be used for quick determinations of SFR's in cloud environments. A key property at the heart of both our complete and simplified theory is that the SFR involves a {\it density-dependent dynamical time}, characteristic of each collapsing (prestellar) overdense region in the cloud, instead of one single mean or critical freefall timescale. Unfortunately, the SFR also depends on some ill determined parameters, such as the core-to-star mass conversion efficiency and the crossing timescale. Although we provide estimates for these parameters, their uncertainty hampers a precise quantitative determination of the SFR, within less than a factor of a few.

Time-scale for critical growth of partial and supercavitation development over impulsively translating projectiles

A physical mechanism is proposed to explain an experimentally observed critical time scale that governs the partial cavity development over blunt free flying cylindrical projectiles. The projectiles were ejected using a modified gas-gun mechanism consisting of a barrel and explosive charge. Upon ignition, high-pressure gases forced a projectile down the launch barrel and into quiescent water. Results indicate that initial small cavities created at the projectile forebody are convected downstream where they subsequently grow towards the forebody, partially enveloping the projectile in a vapor cavity. The time at which the initially stable bubbles rapidly expand signifies that the partial cavity development process has begun. When this time is quantified and is non-dimensionalized appropriately, a time-scale for the critical growth (CGTS) for the cavitation is revealed. A plausible explanation of the partial cavity development process observed in these experiments is that the process is due to the interaction between small cavitation bubbles shed from the projectile forebody and the vortex ring generated by the impulsively started projectile. This interaction mediates the destabilization and spontaneous growth of small unstable bubbles resulting in the formation of partial cavitation over the projectile. An additional supercavitation formation mechanism was observed and is attributed to the launch mechanism. This process is not due to pure hydrodynamic cavitation, but rather an effect we term “gas-leakage” whereby the driving gases contaminated the aft flow field near the projectile and thus facilitated supercavitation to occur on a reduced time scale. This mechanism is equivalent to off-body ventilated supercavitation.

Recovery efficiency of aquifer storage and recovery (ASR) with mass transfer limitation

Aquifer storage and recovery (ASR) is an effective strategy for water resources management and has been widely used in many contaminated and saline aquifers. However, its recovery efficiency (RE) may be significantly affected by mass transfer limitations. A numerical model is developed to simulate ASR performance by combining the convergent and divergent dispersion models with a first‐order mass transfer model. By analyzing the concentration history at the pumping well, we obtain simple and effective relationships for investigating ASR efficiency under various mass transfer parameters, including capacity ratio and mass transfer timescale, and operational parameters, including injection durations and well‐pumping rates. On the basis of such relationships, one can conveniently determine whether a site with mass transfer limitations is appropriate or not for ASR and how many ASR cycles are required for achieving a positive RE. Results indicate that the immobile domain may function as a contaminant source or sink or both during the recovery phase and RE usually improves with well‐flow rate, the decrease of capacity ratio, and the ASR cycles. However, RE is a nonmonotonic function of the mass transfer timescale and the injection duration. A critical timescale is given for quantifying this nonmonotonic behavior. When the injection period is greater than such a critical value, increasing injection period results in a higher RE. Contrarily, when the injection period is less than the critical value, increasing the injection period may even yield a lower RE.

Phase transition theory of sprite halo

We present the phase transition theory for sprite halo using measurable lightning parameters (charge moment and discharge time) on the basis of steady state thermodynamics. A halo is located at the upper part of the tree‐like structure of a sprite and is produced through electron impact excitation of neutral species under the lightning‐induced electric field. We proposed in our previous studies that the occurrence criteria for halos and sprites are characterized by the above lightning parameters, and additionally, the intensity of a halo weakens rapidly with an increase in the discharge time T. We assume that this phenomenon is quite similar to the phase transition between the vapor and the liquid states of water; here the analogy is between the accelerated electrons and the water molecules. We demonstrate analytically a phase transition for a simply modeled halo based on the quasistatic theory of lightning‐induced electric field. Choosing the luminosity of a halo as an order parameter, we show that it has a dependence of T−0.25 − Tc−0.25 near the critical point Tc, which is characteristic of the phase transition. Furthermore, the critical time scale Tc ≈ 5.5 ms is provided naturally from our modeling and is somewhat larger than the typical time scale of the halo luminosity in observations. We consider that this kind of formalism is useful in understanding the detailed relationship between lightning activity and occurrence of halos. We discuss this point for future observations along with the possibilities of the transition model of column and carrot structures.

Duration, magnitude, and frequency of subaerial volcano deformation events: New results from Latin America using InSAR and a global synthesis

We combine new observations of volcano deformation in Latin America with more than 100 previous deformation studies in other areas of the world to constrain the frequency, magnitude, and duration of subaerial volcano deformation events. We discuss implications for eruptive hazards from a given deformation event and the optimum repeat interval for proposed InSAR satellite missions. We use L band (23.6 cm wavelength) satellite‐based interferometric synthetic aperture radar (InSAR) to make the first systematic search for deformation in all volcanic arcs of Latin America (including Mexico, Central America, the Caribbean, and the northern and southern Andes), spanning 2006–2008. We combine L and C band (5.6 cm wavelength) InSAR observations over the southern Andes volcanoes to extend the time series from 2002 to 2008 and assess the capabilities of the different radars: L band gives superior results in highly vegetated areas. Our observations reveal 11 areas of volcano deformation, some of them in areas that were thought to be dormant. There is a lack of observed deformation at several erupting volcanoes, probably due to temporal aliasing. The total number of deforming volcanoes in the central and southern Andes now totals 15 (from observations between 1992 and 2008), comparable to the Alaska/Aleutian arc. Globally, volcanoes deform across a variety of time scales (from seconds to centuries) often without eruption and with no apparent critical observation time scale, although observations made every minute are sometimes necessary to see precursors to eruption.

Experimental and analytical investigation of the turbulent burning velocity of two-component fuel mixtures of hydrogen, methane and propane

In this paper, we present some experimental and analytical model results of two-component fuel mixtures of methane, propane and hydrogen. Experimentally obtained turbulent burning velocity S T for outwardly propagating spherical lean turbulent premixed flames is examined with an algebraic flame surface wrinkling reaction model using 1) mean local burning velocity, and 2) the critical chemical time scale from the leading edge model by Zel'dovich and Frank-Kamenetskii. Based on the latter approach, the time scale that characterizes the effects of preferential diffusion phenomenon in critically curved spherical flames is incorporated into the reaction model. For this, a proposed simple linear model is used for estimating the effective Lewis number of the two-component fuel (CH 4–H 2 and C 3H 8–H 2)/Air mixtures. In general, both approaches are effective ways in achieving qualitatively consistent S T trends for both mixtures. However, in the second approach, model predictions show large S T deviation especially at high turbulence. This may be attributed to the use of approximate values of activation temperature and for the use of the effective Lewis number of both mixtures based on the simple linear model.

Unstable combustion induced by oblique shock waves at the non-attaching condition of the oblique detonation wave

The instability of oblique shock wave (OSW) induced combustion is examined for a wedge with a flow turning angle greater than the maximum attach angle of the oblique detonation wave (ODW), where archival results rarely exist for this case in previous literatures. Numerical simulations were carried out for wedges of different length scales to account for the ratio of the chemical and fluid dynamic time scales. The results reveal three different regimes of combustion. (1) No ignition or decoupled combustion was observed if a fluid dynamic time is shorter than a chemical time behind an OSW. (2) Oscillatory combustion was observed behind an OSW if a fluid dynamic time is longer than a chemical time behind an OSW and the fluid dynamic time is shorter than the chemical time behind a normal shock wave (NSW) at the same Mach number. (3) Detached bow shock-induced combustion (or detached overdriven detonation wave) was observed if a fluid dynamic time is longer than a chemical time behind a NSW. Since no ignition or decoupled combustion occurs as a very slow reaction and the detached wave occurs as an infinitely fast reaction, the finite rate chemistry is considered to be the key for the oscillating combustion induced by an OSW over a wedge of a finite length with a flow turning angle greater than the maximum attach angle for an ODW. Since this case has not been previously reported, grid independency was tested intensively to account for the interaction between the shock and reaction waves and to determine the critical time scale where the oscillating combustion can be observed.

Dynamics of shear-induced ATP release from red blood cells

Adenosine triphosphate (ATP) is a regulatory molecule for many cell functions, both for intracellular and, perhaps less well known, extracellular functions. An important example of the latter involves red blood cells (RBCs), which help regulate blood pressure by releasing ATP as a vasodilatory signaling molecule in response to the increased shear stress inside arterial constrictions. Although shear-induced ATP release has been observed widely and is believed to be triggered by deformation of the cell membrane, the underlying mechanosensing mechanism inside RBCs is still controversial. Here, we use an in vitro microfluidic approach to investigate the dynamics of shear-induced ATP release from human RBCs with millisecond resolution. We demonstrate that there is a sizable delay time between the onset of increased shear stress and the release of ATP. This response time decreases with shear stress, but surprisingly does not depend significantly on membrane rigidity. Furthermore, we show that even though the RBCs deform significantly in short constrictions (duration of increased stress <3 ms), no measurable ATP is released. This critical timescale is commensurate with a characteristic membrane relaxation time determined from observations of the cell deformation by using high-speed video. Taken together our results suggest a model wherein the retraction of the spectrin-actin cytoskeleton network triggers the mechanosensitive ATP release and a shear-dependent membrane viscosity controls the rate of release.

Multiscale Queueing Analysis

This paper introduces a new multiscale framework for estimating the tail probability of a queue fed by an arbitrary traffic process. Using traffic statistics at a small number of time scales, our analysis extends the theoretical concept of the critical time scale and provides practical approximations for the tail queue probability. These approximations are non-asymptotic; that is, they apply to any finite queue threshold. While our approach applies to any traffic process, it is particularly apt for long-range-dependent (LRD) traffic. For LRD fractional Brownian motion, we prove that a sparse exponential spacing of time scales yields optimal performance. Simulations with LRD traffic models and real Internet traces demonstrate the accuracy of the approach. Finally, simulations reveal that the marginals of traffic at multiple time scales have a strong influence on queueing that is not captured well by its global second-order correlation in non-Gaussian scenarios.

A rheological weak zone intensified by post-rift thermal relaxation as a possible origin of simple shear deformation associated with reactivation of rifting

Rheological heterogeneity brought about by the grain-size refinement associated with the spinel/plagioclase–lherzolite reaction is discussed in the context of post-rift thermal relaxation in continental rifts and passive margins. Above a critical temperature, the effect of grain-size refinement is not sufficient to obtain significant rheological heterogeneity. A critical time-scale for thermal relaxation, defined as a period when the temperature at the boundary of a reduced grain-size segment falls below the critical temperature, is evaluated in this study. It is found that a segmental rheological weak zone with a significant strength contrast can develop if the following conditions are satisfied: (1) a substantial grain-size reduction associated with spinel/plagioclase–lherzolite reaction occurs during an earlier extensional event, and (2) a successive extensional event occurs after the critical time-scale for thermal relaxation (a few tens of million years). The development of such rheologically weak zones may contribute to simple shear deformation in rift-related sedimentary basins developed by multi-rifting events.

Experiments on magma mixing

The arrival of a new hot batch of magma changes the thermal and the chemical state of a volcanic plumbing system. Formation of hybrid magma by chemical dilution (liquid miscibility model) of different magma batches is generally accepted. Observations of drop‐like domains of less evolved (newer) magma in more evolved (host) magma point to the contribution of a mechanical mixing process (liquid immiscibility). The interplay of both mechanisms seems to control the time scale from the dormant to the active state of a volcano. We carried out laboratory experiments using melts from natural volcanic rock compositions with the aim to validate model calculations for shear rate dependent domain sizes. The results confirmed the model and also the importance of hydro‐mechanically mixing for chemical dilution. The key process, however, that is able to change the state of a magma reservoir on a critical time scale is hydrodynamic magma mixing.