Stochastic Motion Research Articles

Regionally adjusted stochastic earthquake ground motion models, associated variabilities and epistemic uncertainties

An optimisation-based calibration technique, using the area metric, is applied to determine the input parameters of a stochastic earthquake-waveform simulation method. The calibration algorithm updates a model prior, specifically an estimate of a region’s seismological (source, path and site) parameters, typically developed using waveform data, or existing models, from a wide range of sources. In the absence of calibration, this can result in overestimates of a target region’s ground motion variability, and in some cases, introduce biases. The proposed method simultaneously attains optimum estimates of median, range and distribution (uncertainty) of these seismological parameters, and resultant ground motions, for a specific target region, through calibration of physically constrained parametric models to local ground motion data. We apply the method to Italy, a region of moderate seismicity, using response spectra recorded in the European Strong Motion (ESM) dataset. As a prior, we utilise independent seismological models developed using strong motion data across a wider European context. The calibration obtains values of each seismological parameter considered (such as, but not limited to, quality factor, geometrical spreading and stress drop), to develop a suite of optimal parameters for locally adjusted stochastic ground motion simulation. We consider both the epistemic and aleatory variability associated with the calibration process. We were able to reduce the area metric (misfit) value by up to 88% for the simulations using updated parameters, compared to the initial prior. This framework for the calibration and updating of seismological models can help achieve robust and transparent regionally adjusted estimates of ground motion, and to consider epistemic uncertainty through correlated parameters.

Confinement of transitioning particles in bi-helical Wendelstein-type configurations

Recently, stochastic motion of 3.5-MeV alpha particles with orbits that vary between locally trapped and locally passing states (transitioning particles) in a Helias reactor was observed numerically. This validated the theoretical predictions that (i) the stochastic diffusion represents a mechanism of considerable delayed loss of fast ions in quasi-isodynamic stellarators and (ii) it is possible to prevent the escape of particles to the wall by closing the separatrix between the locally trapped and passing states. It was concluded that, in principle, the separatrix could be made closed, resulting in a reduction of fast ion losses, by compensating for the effect of the helical component of the magnetic field $(1, 1)$ with an enhanced ‘anti-helical’ harmonic $(1, -1)$ ; the enlargement of this harmonic was proposed previously for other reasons. This possibility is explored in this work. Equations of previous relevant works were generalised to include the $(1, -1)$ harmonic. Calculations were carried out for several magnitudes of the ratio of anti-helical to helical magnetic field harmonics. Positive effects were found already at the smallest anti-helical harmonic considered: when the latter ratio is 0.25, transitioning particles with the smallest and intermediate pitch parameters are confined and, moreover, their fraction decreased. When the ratio is 0.85, almost all transitioning particles are confined and their fraction is minimal; well-confined localised orbits dominate at the smallest pitch parameters.

Micro‐mechanism of stress‐dilatancy anisotropy in granular materials: Affine and nonaffine deformation

AbstractThe stress‐induced dilatancy anisotropy is an important kinematic response in granular materials and is very complex due to the stochastic motion of particles. In this study, the local stress‐dilatancy relationship is investigated referring to the local contact force and relative displacement in a certain contact direction. The micromechanism of this anisotropy is investigated from the view of affine and nonaffine approaches based on DEM simulation. Numerical results indicate that the local nonaffine dilatancy/contraction tendency is opposite with that of local affine deformation in macroscopic compression and extension direction, which mainly stems from the counteraction effect between the normal components of affine and nonaffine displacements. The local stress‐dilatancy of affine and nonaffine components is linear and orientation‐dependent, which results in the nonlinear character of macroscopic stress‐dilatancy. By establishing an analytical relation of stress‐dilatancy, the local affine/nonaffine dilatancy is decomposed into the component of stress‐induced dilatancy synchronized with the local stress anisotropy; and the component of stress‐induced dilatancy asynchronized with the local stress anisotropy. The former is related to the unchanged isotropic microstructure, and the latter attributes to the anisotropic microstructure. Besides, the contribution of nonsliding contacts to nonaffine dilatancy is larger than that of sliding contacts even for the case of relatively higher sliding ratio, which indicates that the heterogeneous deformation attributes not only to the friction dissipation, but to the blocked energy at micro contacts in granular materials.

Power reserve control for utility-scale PV power plants under cloud conditions

The photovoltaic generation growth has posed challenges for the control and operation of contemporary power systems, degrading the system frequency response and stability margins. The decrease of the system equivalent inertia and the performance deterioration of the primary frequency control are the main causes of these problems. Such challenges can be overcome by the provision of ancillary services from photovoltaic power plants. However, the provision of ancillary services based on active power, such as inertial response, frequency regulation, and curtailment, requires the de-loaded operation of the photovoltaic power plants. In such context, this work proposes a de-loaded control approach (or, equivalently, power reserve control) for large photovoltaic power plants under shading conditions caused by cloud movements. This kind of shading condition poses a significant challenge for the de-loaded control, as it causes large and fast variations in the maximum power point of the multiple photovoltaic generation units that compose the photovoltaic power plant. A test system corresponding to a 100 MWac photovoltaic power plant with 25 photovoltaic generation units of 4 MWac is employed to evaluate and validate the proposed control approach. The stochastic distribution and movement of clouds, as well as the solar irradiance over the photovoltaic power plant are generated by an approach based on the fractal geometry. The proposed control was effective and presented a good performance under different cloud coverage levels.

Two fluid dynamics in solar prominences

Aims. Solar prominences contain a significant number of neutral species. The dynamics of the ionised and neutral fluids composing the prominence plasma can be slightly different if the collisional coupling is not strong enough. The differential dynamics can be discerned by tracing line-of-sight velocities using observational techniques. Large-scale velocities can be quantified by measuring the global local and instantaneous displacement of spectral lines by the Doppler effect. Small-scale velocities leave their imprint on the width of spectral lines. In addition, these small-scale velocities can have a thermal (pure stochastic motion) nature or a non-thermal (small-scale unresolved instabilities, high-frequency waves, etc.) origin. For this work, we used one spectral line of ionised and two spectral lines of neutral elements to measure the resolved and unresolved velocities in a prominence with the aim to investigate the possible decoupling of the observed charged and neutral species. Methods. A faint prominence was observed with the German Vacuum Tower Telescope (VTT) on June 17, 2017. Time series consisting of repeated ten-position scans over the prominence were performed while simultaneously recording the intensity spectra of the Ca II IR 854.2 nm, Hα 656.28 nm, and He I D3 587.56 nm lines. The line-of-sight velocities and the Doppler width of the three spectral lines were determined at every spatial position and temporal moment. To make sure all spectral lines were sampling the same plasma volume, we applied selection criteria to identify locations with optically thin plasma. In addition, asymmetric or double-peaked profiles were also excluded for the analysis, since (even in an optically thin regime) they are indicative of the presence of strong velocity gradients or multiple components in the line of sight. Thus, only optically thin, symmetric, single-lobed profiles were retained for this study. As an additional reliability test of the selection criteria, we have also compared our results with optical thickness calculations. Results. After the application of all the selection criteria, only a region close to the prominence border met all requirements. The velocities of the three spectral lines turned out to be very similar over this region, with the ionised Ca II IR showing velocity excursions systematically larger compared to those of the neutral lines of Hα and He I at some moments. The latter was found to be much closer to each other. Most of the velocity differences were below 1 km s−1. The analysis of the Doppler widths indicated that the Ca II IR line shows an excess of unresolved motions. We cannot establish whether these velocities are related to a different temperature of the ions or to unresolved small-scale motions due to any non-thermal mechanism. Conclusions. The dynamics of the ionised and neutral plasma components in the observed prominence were very close to one another. The differences found may indicate that a localised decoupling between ions and neutrals may appear at particular spatial locations or instants of time. Indications of different unresolved motions between those species have also been obtained.

Nonlinear squeezing of stochastic motion

Linearized stochastic nanomechanical systems operating at nonzero temperatures and constant frequency and damping are restricted in their capacity to reduce noise in nonlinear combinations of the canonical variables. Nonlinear dynamics are then required in order to overcome these limits. Here we demonstrate how to make these limits explicit in the form of a threshold for nonlinear squeezing of the motional variables. Noise suppression below the threshold cannot be explained by linearized dynamics and is helpful in low-noise nonlinear devices at an ambient temperature. We predict that a state of the art levitating particle, exposed to cubic or quartic trapping potentials for a short interval will display nonlinear squeezing of stochastic motion that cannot be replicated by linear motion.

Stochastic ground motion simulation of the 26 September 2019 Mw 5.8 Silivri (Istanbul) earthquake

Stochastic ground motion simulation of the 26 September 2019 Mw 5.8 Silivri (Istanbul) earthquake

Introducing a novel method for simulating stochastic movement and occupancy in residential spaces using time-use survey data

In the context of growing concerns over energy consumption and sustainability, accurate modelling of occupancy patterns within residential buildings is critical. In this study, a novel stochastic occupancy model is introduced for simulating human behaviour within residential buildings by employing Time Use Survey (TUS) data and utilising Markov chains and probabilistic sampling algorithms. The novelty of this research lies in its approach to represent the dynamic nature of occupancy across different functional spaces and age groups, a gap not yet adequately addressed in existing studies. The model's accuracy is ascertained through ten-fold cross-validation, achieving an average R2 value of 0.91 across key functional rooms (bedroom, bathroom, kitchen, living room), indicating a high degree of precision. Applied to a case study of a two-story detached house in the UK, the model effectively reflects varied behaviour patterns and room occupancy among different age groups. For instance, the average daily appliance energy consumption for occupants aged 8–14 ranged from 0 to 3.77 kWh (median 1.71 kWh), for ages 15–64 from 0 to 4.93 kWh (median 2.61 kWh), and for over 65 from 0.87 to 5.65 kWh (median 3.60 kWh). This model, with its scalability and accuracy in capturing the inherent randomness of human behaviour, is a valuable tool for improving energy consumption simulations and contributing to sustainable residential building design and management.

Kinematically excited beam vibrations under random perturbations

There is numerous information in the technical and scientific literature that often the sources of forced kinematically excited oscillations have a clearly expressed random nature. Kinematically excited vibrations of beams are considered. Their sources are random disturbances. Steady (in a probabilistic sense) vibrations of beams are studied. As a consequence, the transverse vibrations of the beam will also be random, and it becomes necessary and expedient to switch to stochastic motion models.
 Introduction. For a random perturbation process, a spectral matrix is specified. Initial conditions for the equations are not required. The boundary conditions are similar to those used for free and kinematically excited deterministic oscillations. The task is to find the spectral matrix of the random field of beam deviations and dispersion for a given spectral matrix. The question of determining the mathematical expectation is not raised. It is argued that it can be easily reduced to known deterministic problems.
 Methods. To carry out specific calculations, a model of kinematic disturbances is used in the form of a vector N-dimensional stationary random process with stationary connected components that have hidden periodicities (characteristic frequencies). A test example was completed to determine the dispersion and standard deviation of displacements. The numerical integration step was adopted after numerical experiments. When performing calculations, the parity of the integrand was taken into account. The lower limit of integration is taken to be zero, followed by doubling of the final result.
 Results. For small values of the broadband parameter, the results of the stochastic and deterministic problems differ little. Examples are presented that are stochastic analogs of harmonic oscillations. For beams under kinematic harmonic disturbances, test examples were performed to determine dispersions and standard deviations.
 The discussion of the results. The results of calculations in the article are presented by curves having numbers that coincide with the numbers of the matrices. The first matrix corresponds to the absolute correlation of disturbances. The second matrix corresponds to the absolute correlation between the first and third, second and fourth disturbances, while these pairs are absolutely uncorrelated. The third matrix describes random perturbations when the indicated pairs, being perfectly positively correlated within, are absolutely perfectly negatively correlated between pairs. The fourth curve corresponds to the case of ideal correlation of the first three disturbances with each other, while the movements of the fourth support are in antiphase with them. In the latter case, the movements of the outer supports are in antiphase with the movements of the middle ones, which favors the greatest bending of the beam. Therefore, the elastic line has a curvature that is significantly greater than the others.

Conformational Dynamics in Proteins: Entangled Slow Fluctuations and Nonequilibrium Reaction Events.

Proteins exhibit conformational fluctuations and changes over various time scales, ranging from rapid picosecond-scale local atomic motions to slower microsecond-scale global conformational transformations. In the presence of these intricate fluctuations, chemical reactions occur and functions emerge. These conformational fluctuations of proteins are not merely stochastic random motions but possess distinct spatiotemporal characteristics. Moreover, chemical reactions do not always proceed along a single reaction coordinate in a quasi-equilibrium manner. Therefore, it is essential to understand spatiotemporal conformational fluctuations of proteins and the conformational change processes associated with reactions. In this Perspective, we shed light on the complex dynamics of proteins and their role in enzyme catalysis by presenting recent results regarding dynamic couplings and disorder in the conformational dynamics of proteins and rare but rapid enzymatic reaction events obtained from molecular dynamics simulations.

Equity Price Dynamics under Shocks: In Distress or Short Squeeze

This paper proposes a simple bounded stochastic motion to model equity price dynamics under shocks. The stochastic process has a quasi-bounded boundary which can be breached if the probability leakage condition is met. The quasi-boundedness of the process at the boundary can thus provide an indicator of the possible risk of equities under price shocks or in distress. Empirical calibration of the model parameters of the proposed process for equities can be performed easily due to the availability of an analytically tractable probability density function which generates fat-tailed distributions consistent with empirical observations. The volatility and mean-reversion of the S&P500 dynamics calibrated by the process are positively and negatively co-integrated, respectively, with the VIX index representing the level of market distress. The process captures the high likelihood of Hertz’s default about two months earlier, using only information until that point, and before the firm filed for Chapter 11 bankruptcy in May 2020 as a result of the COVID-19 pandemic. Empirical calibration of the process for GameStop’s stock price shows that the short squeeze in the stock occurred when the condition for breaching the upper boundary was met on 14 January 2021, i.e., about two weeks before major short-sellers closed out their positions with significant losses. The trading volume of the stock was positively co-integrated with the probability leakage ratio.

Stochastic motion under nonlinear friction representing shear thinning.

We study stochastic motion under a nonlinear frictional force that levels off with increasing velocity. Specifically, our frictional force is of the so-called Coulomb-tanh type. At small speed, it increases approximately linearly with velocity, while at large speed, it approaches a constant magnitude, similarly to solid (dry, Coulomb) friction. In one spatial dimension, a formal analogy between the associated Fokker-Planck equationand the Schrödinger equationfor a quantum mechanical oscillator in a nonharmonic Pöschl-Teller potential is revealed. Then, the stationary velocity statistics can be treated analytically. From such analytical considerations, we determine associated diffusion coefficients, which we confirm by agent-based simulations. Moreover, from such simulations and from numerically solving the associated Fokker-Planck equation, we find that the spatial distribution function, starting from an initial Gaussian shape, develops tails that appear exponential at intermediate timescales. At small magnitudes of stochastic driving, the velocity distribution resembles the case of linear friction, while at large magnitudes, it rather approaches the case of solid (dry, Coulomb) friction. The same is true for diffusion coefficients. In a certain sense thus interpolating between the two extreme cases of linear friction and solid (dry, Coulomb) friction, our approach should be useful to describe several cases of practical relevance. For instance, a reduced increase in friction with increasing relative speed is typical of shear-thinning behavior. Therefore, driven motion in shear-thinning environments is one specific example to which our description may be applied.

Deciphering Solar Coronal Heating: Energizing Small-scale Loops through Surface Convection

The solar atmosphere is filled with clusters of hot small-scale loops commonly known as coronal bright points (CBPs). These ubiquitous structures stand out in the Sun by their strong X-ray and/or extreme-ultraviolet (EUV) emission for hours to days, which makes them a crucial piece when solving the solar coronal heating puzzle. In addition, they can be the source of coronal jets and small-scale filament eruptions. Here we present a novel 3D numerical model using the Bifrost code that explains the sustained CBP heating for several hours. We find that stochastic photospheric convective motions alone significantly stress the CBP magnetic field topology, leading to important Joule and viscous heating concentrated around the CBP’s inner spine at a few megameters above the solar surface. We also detect continuous upflows with faint EUV signals resembling observational dark coronal jets and small-scale eruptions when Hα fibrils interact with the reconnection site. We validate our model by comparing simultaneous CBP observations from the Solar Dynamics Observatory (SDO) and the Swedish 1‐m Solar Telescope (SST) with observable diagnostics calculated from the numerical results for EUV wavelengths as well as for the Hα line using the Multi3D synthesis code. Additionally, we provide synthetic observables to be compared with Hinode, Solar Orbiter, and the Interface Region Imaging Spectrograph (IRIS). Our results constitute a step forward in the understanding of the many different facets of the solar coronal heating problem.

Ground motion simulations of the 1887 Pinal de Amoles (MW = 5.53) and the 1989 Landa de Matamoros (MW = 4.94) earthquakes, Querétaro, Mexico

Ground motions characteristics of the 1887 Pinal de Amoles (MW = 5.53) and the 1989 Landa de Matamoros (MW = 4.94) Querétaro earthquakes were studied. For this purpose, the specific barrier model (SBM) was implemented in the context of the stochastic ground motion representation of earthquakes. The SBM was calibrated by modeling the peak ground acceleration (PGA) and the Fourier acceleration spectra (FAS) of the 11 May 2016 (MW = 4.80) Guadalajara earthquake. The calibration results showed that the SBM and selected source and path parameters were able to reproduce synthetic PGA and spectra in good agreement with observed waveforms; in particular, a stress drop of 160 bars fits the results better. Several rupture scenarios for the 1887 and 1989 events were performed by obtaining synthetic seismograms considering different values of stress drop (10, 30, 50, 100, 150, 200, 300, and 500 bars). For a typical stress-drop value of 100 bars, the PGA at distances of 10 and 200 km are in the following intervals: 2.57–295 cm/s2 and 1.72–186 cm/s2 for the Pinal de Amoles and Landa de Matamoros earthquakes, respectively. The results obtained in this study contribute toward a better understanding of strong ground motions in the region.

An Analysis and Global Identification of Smoothless Variable Order of a Fractional Stochastic Differential Equation

We establish both the uniqueness and the existence of the solutions to a hidden-memory variable-order fractional stochastic partial differential equation, which models, e.g., the stochastic motion of a Brownian particle within a viscous liquid medium varied with fractal dimensions. We also investigate the inverse problem concerning the observations of the solutions, which eliminates the analytic assumptions on the variable orders in the literature of this topic and theoretically guarantees the reliability of the determination and experimental inference.

Reliability-based design optimization for seismic structures considering randomness associated with ground motions

When addressing the dynamic reliability analysis of structures, it becomes necessary to account for multiple limit state functions or their combinations. In scenarios where structures are subjected to random excitation, this can lead to intricate inter-dependencies among different limit states, and the computational workload can pose a substantial challenge in ensuring sufficient precision. Code-based design primarily ensures safety at the member level, while deterministic optimization fails to accommodate the inherent uncertainties associated with external excitation or the system as a whole. Therefore, in such cases, to address both the uncertainties in excitations and the presence of multiple limit states while mitigating computational challenges, equivalent extreme-value criteria are employed within the framework of the probability density evolution method to calculate the global reliability of the structure subjected to stochastic ground motions generated from the physically motivated stochastic ground motion model. Numerical optimization is subsequently conducted using genetic algorithms, aiming to minimize the cost of the superstructure while adhering to the design performance criteria related to the inter-story drift ratio and considering global reliability. Additionally, multi-objective optimization is carried out using NSGA-II, permitting the generation of multiple solutions, from which one can select the most suitable solution as needed. The numerical results illustrate the effectiveness of this technique in achieving an optimal balance between the cost of the structure and the consideration of global reliability, providing a comprehensive solution for dynamic reliability analysis and design optimization of structures under random excitations.

Reliability-based design optimization of tuned mass-damper-inerter for mitigating structural vibration

The tuned mass-damper-inerter (TMDI), comprising mass blocks, springs, dampers, and two-terminal inertial elements, has garnered significant attention as an advanced iteration of the tuned mass damper (TMD) in recent years. However, previous studies of TMDI's design are primarily conducted according to the theoretical formulas based on deterministic analysis assuming harmonic excitations or stochastic analysis assuming Gaussian white noise type excitations. This treatment falls short of capturing the non-stationary and non-Gaussian characteristics exhibited by engineering excitations and thus cannot ensure the safety of controlled structure. A few studies considering non-stationary excitations and based on reliability metrics, in conjunction with stochastic simulation methods, often confront excessive computational overhead. Aiming at such a challenge, the reliability-based design optimization (RBDO) of TMDI system, combining the probability density evolution method (PDEM) and genetic algorithm (GA), for mitigating structural vibration under non-stationary excitations is performed in this study. For the purposes of illustration, a single-degree-of-freedom system attached the TMDI and subjected to stochastic ground motions is studied. Considering the cases of different levels of inertance-to-mass ratios, the RBDO of the TMDI is carried out. The solutions by the theoretical formulas and by the variance-based design optimization are compared to verify the superiority of the proposed method. For assessing the seismic mitigation performance of the optimized TMDI system, probability density evolution analysis of structure-TMDI system with three levels of inertance-to-mass ratios is performed. Comparison between the RBDO of TMDI and that of TMD under consistent mass ratio is further addressed. Numerical results reveal the technical advantages of TMDI in a probabilistic sense, i.e., high vibration mitigation performance, considerable mass reduction, and less stroke demand.

Stochastic simulation of earthquake ground motions based on improved finite-fault model

In the seismic reliability analysis of engineering structures, it is often necessary to adopt stochastic ground motions as inputs. However, the current methods used to generate stochastic samples, such as the spectral representation method, the spectral transfer method, the dynamic source method, the physical source method, and the comprehensive method, etc., are not very convincing since they do not contain the complete information of seismic occurrence and propagation. In this study, an improved finite-fault model for stochastic simulation of ground motions is proposed. Compared to existing stochastic finite-fault model, the improved model considers the propagation law of phase spectrum, local site effect, and the impact of wave velocity on the arrival time of seismic waves. Besides, identification and statistics of model parameters are carried out based on ground motion records. Reasonable samples of stochastic ground motions are then obtained. In addition, parameter sensitivity analysis is conducted to determine critical random variables, which reduces the complexity of stochastic simulation. Simulation results are compared with recorded accelerograms from the 1999 Chi-Chi earthquake. It is shown that the improved stochastic finite-fault model is reliable, and the acceleration response spectra of simulated samples are consistent with those of station records collected at various site classes and the design spectra.

Site response analysis: Uncertain motions propagating through uncertain elastoplastic soil

Presented is a stochastic site response analysis of uncertain seismic motions propagating through uncertain elastoplastic soil. Time domain stochastic bedrock motions are simulated from GMPE of Fourier amplitude spectra and Fourier phase derivative. The uncertain motions are directly modeled as a non-stationary random process in the time domain, capturing all the important seismic characteristics. Uncertain heterogeneous parameters for the elastoplastic behavior of soils are also considered and modeled as random fields. Both uncertain motions and uncertain material parameters are discretized in the probabilistic space using orthogonal Hermite polynomial chaos basis. The probabilistic site response is efficiently solved using Galerkin stochastic elastoplastic finite element method, circumventing Monte Carlo simulations. The implication of the presented methodology for site-specific probabilistic seismic hazard analysis is discussed as well.

Diffusion limited aggregation, resetting and large deviations of Brownian motion

Models of fractal growth commonly consider particles diffusing in a medium and that stick irreversibly to the forming aggregate when making contact for the first time. As shown by the well-known diffusion limited aggregation (DLA) model and its generalisations, the fractal dimension is sensitive to the nature of the stochastic motion of the particles. Here, we study the structures formed by finite-lived Brownian particles, i.e. particles constrained to find the aggregate within a prescribed time, and which are removed otherwise. This motion can be modelled by diffusion with stochastic resetting, a class of processes which has been widely studied in recent years. In the short lifetime limit, a very small fraction of the particles manage to reach the aggregate. Hence, growth is controlled by atypical Brownian trajectories, that move nearly in straight line according to a large deviation principle. In d dimensions, the resulting fractal dimension of the aggregate decreases from the DLA value and tends to 1, instead of increasing to d as expected from ballistic aggregation. In the zero lifetime limit one recovers the non-trivial model of ‘aggregation by the tips’ proposed long ago by R. Jullien [J. Phys. A: Math. Gen. 19, 2129 (1986)].