Metastable Behavior Research Articles

Metastability for small random perturbations of a PDE with blow-up

We study random perturbations of a reaction–diffusion equation with a unique stable equilibrium and solutions that blow-up in finite time. If the strength of the perturbation ε>0 is small and the initial data is in the domain of attraction of the stable equilibrium, the system exhibits metastable behavior: its time averages remain stable around this equilibrium until an abrupt and unpredictable transition occurs which leads to explosion in a finite time (but exponentially large in ε−2). Moreover, for initial data in the domain of explosion we show that the explosion times converge to the one of the deterministic solution.

Metastability for Glauber dynamics on random graphs

In this paper, we study metastable behaviour at low temperature of Glauber spin-flip dynamics on random graphs. We fix a large number of vertices and randomly allocate edges according to the configuration model with a prescribed degree distribution. Each vertex carries a spin that can point either up or down. Each spin interacts with a positive magnetic field, while spins at vertices that are connected by edges also interact with each other via a ferromagnetic pair potential. We start from the configuration where all spins point down, and allow spins to flip up or down according to a Metropolis dynamics at positive temperature. We are interested in the time it takes the system to reach the configuration where all spins point up. In order to achieve this transition, the system needs to create a sufficiently large droplet of up-spins, called critical droplet, which triggers the crossover. In the limit as the temperature tends to zero, and subject to a certain key hypothesis implying metastable behaviour, the average crossover time follows the classical Arrhenius law, with an exponent and a prefactor that are controlled by the energy and the entropy of the critical droplet. The crossover time divided by its average is exponentially distributed. We study the scaling behaviour of the exponent as the number of vertices tends to infinity, deriving upper and lower bounds. We also identify a regime for the magnetic field and the pair potential in which the key hypothesis is satisfied. The critical droplets, representing the saddle points for the crossover, have a size that is of the order of the number of vertices. This is because the random graphs generated by the configuration model are expander graphs.

Metastable phase selection from undercooled Zr77Rh23 liquid alloys

From measurements of X-ray and neutron scattering of electrostatically levitated Zr77Rh23 liquids, a variety of metastable crystallization behavior was observed. The metastable phase selection in deeply undercooled liquid droplets is characterized and their crystallization pathways discussed. A metastable phase previously identified as a primary devitrification product from the metallic glass formed when undercooling was maximized to near the hypercooling limit. The direct formation of α–Zr and the equilibrium C16 phase as well as a newly discovered Zr5Rh3 (Mg5Si3-type) phase are also reported.

Performance stability of photovoltaic modules in different climates

AbstractThe current–voltage (I‐V) characteristics of 15 different photovoltaic modules are monitored during more than 2 years of operation at four locations (Germany, Italy, India and Arizona) corresponding to four different climate zones. The electrical stability of the photovoltaic modules during the time of outdoor exposure is investigated in terms of measured I‐V curve translated to standard test conditions. This translation compensates the influence of module temperature, irradiance, spectral effects and soiling on the I‐V curves. The changes of output power after these corrections are attributed to initial consolidation phases, to long‐term degradation of the electrical properties and to seasonal cycles associated with metastabilities. Modules made from crystalline Si turn out to show no or only minor effects. Thin‐Film modules (CdTe, Cu(In,Ga)Se2 and thin‐film Si) exhibit a wide spread of metastable behaviour with consistent patterns for identical modules in different climates but with significant differences amongst different manufacturers of the same thin‐film technology. We show further that this metastable behaviour influences the energy yield of the modules. Copyright © 2017 John Wiley & Sons, Ltd.

Metastability of the Nonlinear Wave Equation: Insights from Transition State Theory

This paper is concerned with the long-time dynamics of the nonlinear wave equation in one-space dimension, $$ u_{tt} - \delta^2 u_{xx} +V'(u) =0 \qquad x\in [0,1] $$ where $\delta>0$ is a parameter and $V(u)$ is a potential bounded from below and growing at least like $u^2$ as $|u|\to\infty$. Infinite energy solutions of this equation preserve a natural Gibbsian invariant measure and when the potential is double-welled, for example when $V(u) = \tfrac14(1-u^2)^2$, there is a regime such that two small disjoint sets in the system's phase-space concentrate most of the mass of this measure. This suggests that the solutions to the nonlinear wave equation can be metastable over these sets, in the sense that they spend long periods of time in these sets and only rarely transition between them. Here we quantify this phenomenon by calculating exactly via Transition State Theory (TST) the mean frequency at which the solutions of the nonlinear wave equation with initial conditions drawn from its invariant measure cross a dividing surface lying in between the metastable sets. Numerical results suggest that the dynamics of the nonlinear wave equation is ergodic and rapidly mixing with respect to the Gibbs invariant measure when the parameter $\delta$ in small enough. This is a regime in which the dynamics of the nonlinear wave equation displays a metastable behavior that is not fundamentally different from that observed in its stochastic counterpart in which random noise and damping terms are added to the equation. For larger $\delta$, however, the dynamics either stops being ergodic, or its mixing time becomes larger than the inverse of the TST frequency, indicating that successive transitions between the metastable sets are correlated and the coarse-graining to a Markov chain fails.

Metastable state-induced consecutive step-like negative differential resistance behaviors in single crystalline VO2 nanobeams.

We demonstrate the current-dependent consecutive appearance of two different negative differential resistance (NDR) transitions in a single crystalline VO2 nanobeam epitaxially grown on a c-cut sapphire substrate. It is revealed that the first NDR occurs at an approximately constant current level as a result of the carrier injection-induced transition, independent of a thermally induced phase transition. In contrast, it is observed that the second NDR exhibits a temperature-dependent behavior and current values triggering the metal-insulator transition (MIT) are strongly mediated by Joule heating effects in a phase coexisting temperature range. Moreover, we find that the electrically and thermally triggered MIT behavior can be closely related with the alternate occurrence of current-induced multiple insulating and metallic phase coexistence in the nanobeam. These findings indicate that the current density passing through VO2 plays a critical role in both the electrical and structural phase transitions.

Spectral Theory for Random Poincaré Maps

We consider stochastic differential equations, obtained by adding weak Gaussian white noise to ordinary differential equations admitting $N$ asymptotically stable periodic orbits. We construct a discrete-time, continuous-space Markov chain, called a random Poincaré map, which encodes the metastable behavior of the system. We show that this process admits exactly $N$ eigenvalues which are exponentially close to $1$, and we provide expressions for these eigenvalues and their left and right eigenfunctions in terms of committor functions of neighborhoods of periodic orbits. The eigenvalues and eigenfunctions are well-approximated by principal eigenvalues and quasi-stationary distributions of processes killed upon hitting some of these neighborhoods. The proofs rely on Feynman--Kac-type representation formulas for eigenfunctions, Doob's $h$-transform, spectral theory of compact operators, and a recently discovered detailed-balance property satisfied by committor functions.

Full Metastable Asymptotic of the Fisher Information

We establish an expansion by Gamma-convergence of the Fisher information relative to the reference measure exp(-beta V), where V is a generic multiwell potential and beta goes to infinity. The expansion reveals a hierarchy of multiple scales reflecting the metastable behavior of the underlying overdamped Langevin dynamics: distinct scales emerge and become relevant depending on whether one considers probability measures concentrated on local minima of V, probability measures concentrated on critical points of V, or generic probability measures on R^d. We thus fully describe the asymptotic behavior of minima of the Fisher information over regular sets of probabilities. The analysis mostly relies on spectral properties of diffusion operators and the related semiclassical Witten Laplacian and covers also the case of a compact smooth manifold as underlying space.

Suppressing escape events in maps of the unit interval with demographic noise.

We explore the properties of discrete-time stochastic processes with a bounded state space, whose deterministic limit is given by a map of the unit interval. We find that, in the mesoscopic description of the system, the large jumps between successive iterates of the process can result in probability leaking out of the unit interval, despite the fact that the noise is multiplicative and vanishes at the boundaries. By including higher-order terms in the mesoscopic expansion, we are able to capture the non-Gaussian nature of the noise distribution near the boundaries, but this does not preclude the possibility of a trajectory leaving the interval. We propose a number of prescriptions for treating these escape events, and we compare the results with those obtained for the metastable behavior of the microscopic model, where escape events are not possible. We find that, rather than truncating the noise distribution, censoring this distribution to prevent escape events leads to results which are more consistent with the microscopic model. The addition of higher moments to the noise distribution does not increase the accuracy of the final results, and it can be replaced by the simpler Gaussian noise.

In Situ Studies and Magnetic Properties of the Cmcm Polymorph of LiCoPO4 with a Hierarchical Dumbbell-Like Morphology Synthesized by Easy Single-Step Polyol Synthesis

LiCoPO4 (LCP) exists in three different structural modifications: LCP-Pnma (olivine structure), LCP-Pn21a (KNiPO4 structure type), and LCP-Cmcm (Na2CrO4 structure type). The synthesis of the LCP-Cmcm polymorph has been reported via high pressure/temperature solid-state methods and by microwave-assisted solvothermal synthesis. Phase transitions from both LCP-Pn21a and LCP-Cmcm to LCP-Pnma upon heating indicates a metastable behavior. However, a precise study of the structural changes during the heating process and the magnetic properties of LCP-Cmcm are hitherto unknown. Herein, we present the synthesis and characterization of LCP-Cmcm via a rapid and facile soft-chemistry approach using two different kinetically controlled pathways, solvothermal and polyol syntheses, both of which only require relatively low temperatures (~200 °C). Additionally, by polyol, method a dumbbell-like morphology is obtained without the use of any additional surfactant or template. A temperature-dependent in situ powder XRD shows a transition from LCP-Cmcm at room temperature to LCP-Pnma and finally to LCP-Pn21a at 575 and 725 °C, respectively. In addition to that, the determination of the magnetic susceptibility as a function of temperature indicates a long-range antiferromagnetic order below TN = 11 K at 10 kOe and 9.1 K at 25 kOe. The magnetization curves suggests the presence of a metamagnetic transition.

When and how the melittin ion channel exhibits ohmic behavior

Melittin exhibits an ohmic behavior in a lipid bilayer having a DOPC distal leaflet and interposed between a 2.25nm tetraethyleneoxy chain tethered to a mercury drop and an aqueous solution. This behavior is induced in a pH6.8 buffer solution of 0.8μg/mL melittin by a pretreatment consisting of series of electrochemical impedance spectroscopy measurements at bias potentials varied by 50mV steps over a transmembrane potential range from about +250 to −250mV. This metastable ohmic behavior remains unchanged for hours, even after acidifying the solution to pH3. The midpoint potential E1/2 between the positive and negative peaks of the resulting cyclic voltammogram is almost coincident with that of the ohmic channels gramicidin and syringopeptin 25A and shifts by the same amount toward more positive potentials with a pH decrease from 6.8 to 3. This common cyclic voltammetry behavior is explained by a tilt of the DOPC polar heads around the channel mouth of these three peptides and is simulated by a modelistic approach. The ohmic behavior of melittin is explained by the persistence of the peptide orientation initially assumed at trans-negative potentials even after application of trans-positive ones, at sufficiently high peptide-to-lipid molar ratios.

Simulation of metastable changes in time resolved photoluminescence of Cu(In,Ga)Se2 thin film solar cells upon light soaking treatment

Cu(In,Ga)Se2 thin film solar cells are known to show a metastable behavior of their open circuit voltage when exposed to light. Time resolved measurements of the photoluminescence decay after pulsed excitation (TRPL) reveal that also the photoluminescence decay (PL) changes with conditioning, yielding shorter effective lifetimes after annealing under illumination (light soaking) than after annealing in the dark. By using time resolved device simulations we applied a model that explains the TRPL metastable behavior. For this we used a model of capture and reemission of minority charge carriers via two characteristic trap states near the conduction band, while also accounting for changes in trap density and doping concentration caused by light soaking. To further understand the mechanisms behind PL decay we analyse the influence that the charge carrier dynamics play in TRPL decay by a variation of select simulation parameters.

Influence of Martensitic Transformation on the Fatigue of Low Temperature Metastable Stainless Steel

The influence of martensitic transformation on the fatigue of AISI 316 type stainless steel, which presents a low temperature metastable behavior, was investigated at-196 and 25°C. Fatigue tests were conducted on both annealed and cold worked conditions. The results from stress versus number of cycles curves showed that the fatigue resistance is not affected by the “dynamically” induced martensite in annealed specimens at both temperatures. On the contrary, the previously induced “static” martensite in work hardened specimens contributes to decrease the fatigue resistance limit. The contribution of the martensite stability is discussed.

Metastable structural transformations and pressure-induced amorphization in natural (Mg,Fe)2SiO4olivine under static compression: A Raman spectroscopic study

Raman spectroscopic data were obtained for (Mg,Fe)2SiO4 samples during compression to 57 GPa. Single crystals of San Carlos olivine compressed hydrostatically above 41 GPa showed appearance of a new “defect” peak in the 820–840 cm−1 region associated with SiOSi linkages appearing between adjacent Embedded Image tetrahedra to result in five- or sixfold-coordinated silicate species. Appearance of this feature is accompanied by a broad amorphous background. The changes occur at lower pressure than metastable crystalline transitions of end-member Mg2SiO4 forsterite (Fo-I) into Fo-II and Fo-III phases described recently. We complemented our experimental study using density functional theory (DFT) calculations and anisotropic ion molecular dynamics (AIMD) simulations to study the Raman spectra and vibrational density of states (VDOS) of metastably compressed Mg2SiO4 olivine, Fo-II and Fo-III, and quenched melts at high and low pressures. By 54 GPa all sharp crystalline peaks disappeared from observed Raman spectra indicating complete pressure-induced amorphization (PIA). The amorphous (Mg,Fe)2SiO4 spectrum contains Si-O stretching bands at lower wavenumber than expected for Embedded Image indicating high coordination of the silicate units. The amorphous spectrum persisted on decompression to ambient conditions but with evidence for reappearance of tetrahedrally coordinated units. Non-hydrostatic compression of polycrystalline olivine samples showed similar appearance of the defect feature and broad amorphous features between 43–44 GPa. Both increased in intensity as the sample was left at pressure overnight but they disappeared during decompression below 17 GPa with recovery of the starting olivine Raman signature. A hydrated San Carlos olivine sample containing 75–150 ppm OH was also studied. Significant broadening of the Embedded Image stretching peaks was observed above 43 GPa but without immediate appearance of the defect or broad amorphous features. However, both of these characteristics emerged after leaving the sample at 47 GPa overnight followed by complete amorphization that occurred upon subsequent pressurization to 54 GPa. During decompression the high-density amorphous spectrum was retained to 3 GPa but on final pressure release a spectrum similar to thermally quenched low-pressure olivine glass containing isolated Embedded Image groups was obtained. Leaving this sample overnight resulted in recrystallization of olivine. Our experimental data provide new insights into the metastable structural transformations and relaxation behavior of olivine samples including material recovered from meteorites and laboratory shock experiments.

Metastability of the Two-Dimensional Blume–Capel Model with Zero Chemical Potential and Small Magnetic Field

We consider the two-dimensional Blume-Capel model with zero chemical potential and small magnetic field evolving on a large but finite torus. We obtain sharp estimates for the transition time, we characterize the set of critical configurations, and we prove the metastable behavior of the dynamics as the temperature vanishes.

Multi-scale metastable dynamics and the asymptotic stationary distribution of perturbed Markov chains

We assume that the transition matrix of a Markov chain depends on a parameter ε, and converges as ε→0. The chain is irreducible for ε>0 but may have several essential communicating classes when ε=0. This leads to metastable behavior, possibly on multiple time scales. For each of the relevant time scales, we derive two effective chains. The first one describes the (possibly irreversible) metastable dynamics, while the second one is reversible and describes metastable escape probabilities. Closed probabilistic expressions are given for the asymptotic transition probabilities of these chains. As a consequence, we obtain efficient algorithms for computing the committor function and the limiting stationary distribution.

Bulk Properties of the Oxygen Reduction Catalyst SrCo0.9Nb0.1O3−δ

The perovskite SrCo0.9Nb0.1O3−δ (SCN) has excellent electrochemical activity toward oxygen reduction, and it is also valuable as a possible model material for other state-of-the-art perovskite catalysts based on strontium and cobalt, such as Ba0.5Sr0.5Co0.8Fe0.2O3−δ (BSCF). Here we report thermogravimetric, conductivity, and diffraction measurements from SCN. We find that the thermodynamic stability limits of SCN are slightly more favorable than those reported for BSCF, although both materials exhibit a slow oxidative partial decomposition under likely operating conditions. In SCN, this decomposition is thermodynamically preferred when the average formal oxidation state of cobalt is greater than ∼3.0+, but due to sluggish kinetics, metastable SCN with higher cobalt valence can be observed. The oxygen stoichiometry 3−δ varies from 2.45 to 2.70 under the conditions studied, 500–1000 °C and 10–4–1 bar O2, which encompass both stable and metastable behavior. The electronic conductivity is p-type and thermally...

Freezing and melting behaviors of H2O‐NaCl‐CaCl2 solutions in fused silica capillaries and glass‐sandwiched films: implications for fluid inclusion studies

AbstractFluid inclusions of the H2O‐NaCl‐CaCl2 system are notorious for their metastable behavior during cooling and heating processes, which can render microthermometric measurement impossible or difficult and interpretation of the results ambiguous. This study addresses these problems through detailed microscopic examination of synthetic solutions during cooling and warming runs, development of methods to enhance nucleation of hydrates, and comparison of microthermometric results with different degrees of metastability with values predicted for stable conditions. Synthetic H2O‐NaCl‐CaCl2 solutions with different NaCl/(NaCl + CaCl2) ratios were prepared and loaded in fused silica capillaries and glass‐sandwiched films for microthermometric studies; pure solutions were used with the capillaries to simulate fluid inclusions, whereas alumina powder was added in the solutions to facilitate ice and hydrate crystallization in the sandwiched samples. The phase changes observed and the microthermometric data obtained in this study have led to the following conclusions that have important implications for fluid inclusion studies: (i) most H2O‐NaCl‐CaCl2 inclusions that appear to be completely frozen in the first cooling run to −185°C actually contain large amounts of residual solution, as also reported in some previous studies; (ii) inability of H2O‐NaCl‐CaCl2 inclusions to freeze completely may be related to their composition (low NaCl/(NaCl + CaCl2) ratios) and lack of solid particles; (iii) crystallization of hydrates, which is important for cryogenic Raman spectroscopic studies of fluid inclusion composition, can be greatly enhanced by finding an optimum combination of cooling and warming rates and temperatures; and (iv) even if an inclusion is not completely frozen, the melting temperatures of hydrohalite and ice are still valid for estimating the fluid composition.

An Explanation of Metastability in the Viscous Burgers Equation with Periodic Boundary Conditions via a Spectral Analysis

A “metastable solution to a differential equation typically refers to a family of solutions for which solutions with initial data near the family converge to the family much faster than evolution along the family. Metastable families have been observed both experimentally and numerically in various contexts; they are believed to be particularly relevant for organizing the dynamics of fluid flows. In this work, we propose a candidate metastable family for the Burgers equation with periodic boundary conditions. Our choice of family is motivated by our numerical experiments. We furthermore explain the metastable behavior of the family without reference to the Cole--Hopf transformation, but rather by linearizing the Burgers equation about the family and analyzing the spectrum of the resulting operator. We hope this may make the analysis more readily transferable to more realistic systems like the Navier--Stokes equations. Our analysis is motivated by ideas from singular perturbation theory and Melnikov theory.

Numerical Simulation of Copper Migration in Single Crystal CdTe

With better CdTe material properties and more effective contacts, the compensated Cu doping is crucial for enhancing the performance of CdTe solar cells. Moreover, the metastable behavior of CdTe devices is associated with Cu migration. To understand the formation of Cu acceptors in CdTe and Cu-related metastabilities, we developed a one-dimensional diffusion-reaction simulator to study the kinetics of Cu-related defects in CdTe material. A detailed kinetic model that explicitly describes the evolutions of point defects was developed and applied for the first time. Migration of intrinsic and Cu-related extrinsic defects in single crystal CdTe (sx-CdTe) bulk has been investigated in time-space domain self-consistently with free carrier transport. Cu profiles achieved from simulations match with the experimental profiles quantitatively. The simulation results provide explanations of limited Cu incorporation (also known as the Cu solubility limits) and the development of Cu compensation in course of annealing and cooling processes. Although the one-dimensional simulation has intrinsic limitations when applied to poly-crystalline CdTe solar cells, the results suggest that the approach still has strong potential in better understanding the performance and the metastabilities of poly-crystalline CdTe photovoltaic devices.