Time-dependent Probability Density Functions Research Articles

Time-dependent probability density functions and information geometry in a stochastic prey–predator model of fusion plasmas

Time-dependent probability density functions and information geometry in a stochastic prey–predator model of fusion plasmas

Spectral incremental dynamic methodology for nonlinear structural systems endowed with fractional derivative elements subjected to fully non-stationary stochastic excitation

A novel spectral incremental dynamic analysis methodology for analysing structural response in nonlinear systems with fractional derivative elements is presented, aligning with modern seismic design codes, like Eurocode 8. Drawing inspiration from the concept of fully non-stationary stochastic processes, the vector of the imposed seismic excitations is characterised by time and frequency evolving power spectra stochastically compatible with elastic response spectra of specified damping ratio and ground acceleration. The proposed method efficiently determines the nonlinear system time-dependent probability density functions for the non-stationary system response amplitude by employing potent nonlinear stochastic dynamics concepts, such as stochastic averaging and statistical linearization. Unlike traditional incremental dynamic analysis curves found in the literature, the herein proposed method introduces a three-dimensional alternative counterpart, that of stochastic engineering demand parameter surfaces, providing with higher-order statistics of the system response. An additional noteworthy aspect involves the derivation of response evolutionary power spectra as function of spectral acceleration, offering a deeper insight into the underlying system dynamics. Besides its capabilities, the method maintains the coveted element of a particularly low associated computational cost, increasing its attractiveness and practicality among diverse applications of engineering interest. Numerical examples comprising the bilinear hysteretic model endowed with fractional derivative elements subject to an Eurocode 8 elastic design spectrum demonstrate the capabilities and reliability of the proposed methodology. Its accuracy is assessed by juxtaposing the derived results with germane Monte Carlo Simulation data.

Uncertainty quantification for the random HIV dynamical model driven by drug adherence

In HIV drug therapy, the high variability of CD4+ T cells and viral loads brings uncertainty to the determination of treatment options and the ultimate treatment efficacy, which may be the result of poor drug adherence. We develop a dynamical HIV model coupled with pharmacokinetics, driven by drug adherence as a random variable, and systematically study the uncertainty quantification, aiming to construct the relationship between drug adherence and therapeutic effect. Using adaptive generalized polynomial chaos, stochastic solutions are approximated as polynomials of input random parameters. Numerical simulations show that results obtained by this method are in good agreement, compared with results obtained through Monte Carlo sampling, which helps to verify the accuracy of approximation. Based on these expansions, we calculate the time-dependent probability density functions of this system theoretically and numerically. To verify the applicability of this model, we fit clinical data of four HIV patients, and the goodness of fit results demonstrate that the proposed random model depicts the dynamics of HIV well. Sensitivity analyses based on the Sobol index indicate that the randomness of drug effect has the greatest impact on both CD4+ T cells and viral loads, compared to random initial values, which further highlights the significance of drug adherence. The proposed models and qualitative analysis results, along with monitoring CD4+ T cells counts and viral loads, evaluate the influence of drug adherence on HIV treatment, which helps to better interpret clinical data with fluctuations and makes several contributions to the design of individual-based optimal antiretroviral strategies.

Stochastic dynamics of mechanical systems with impacts via the Step Matrix multiplication based Path Integration method

AbstractIn this work we propose the Step Matrix Multiplication based Path Integration method (SMM-PI) for nonlinear vibro-impact oscillator systems. This method allows the efficient and accurate deterministic computation of the time-dependent response probability density function by transforming the corresponding Chapman–Kolmogorov equation to a matrix–vector multiplication using high-order numerical time-stepping and interpolation methods. Additionally, the SMM-PI approach yields the computation of the joint probability distribution for response and impact velocity, as well as the time between impacts and other important characteristics. The method is applied to a nonlinear oscillator with a pair of impact barriers, and to a linear oscillator with a single barrier, providing relevant densities and analysing energy accumulation and absorption properties. We validate the results with the help of stochastic Monte-Carlo simulations and show the superior ability of the introduced formulation to compute accurate response statistics.

Statistical analysis of magnetic divertor configuration influence on H-mode transitions

DIII-D plasmas are compared for two upper divertor configurations: with the outer strike point on the small angle slot (SAS) divertor target and with the outer strike point on the horizontal divertor target (HT). Scanning the vertical distance between the magnetic null point and the divertor target over a range 0.10–0.16 m is shown to increase the threshold power, Pth , and edge plasma power, PLoss , for the low-to-high confinement (L–H) and H–L transitions respectively, by up to a factor of 1.4. The X-point height scans were performed at three L-mode core plasma line average electron densities, n¯e= 1.2, 2.2 and 3.6 ×1019m−3 , to investigate the density dependence of divertor magnetic configuration influence on Pth . The X-point height, Zx-pt , was further extended across the range 0.16–0.22 m with the more open HT divertor configuration, for which a clear decrease in Pth with increasing Zx-pt is observed. The dependence of Pth on divertor magnetic geometry is further investigated using a time-dependent probability density function (PDF) model and information geometry to elucidate the roles played by pedestal plasma turbulence and perpendicular velocity flows. The degree of stochasticity of the plasma turbulence is observed to be sensitive to the plasma heating rate. The calculated square of the information rate shows changes in the relative density fluctuations and perpendicular velocity PDFs begin 2–5 ms prior to the L–H transition for three plasmas; providing a crucial measurement of the dynamic timescale of external transport barrier formation. Additionally, both information length and rate provide potential predictors of the L–H transition for these plasmas.

Multi-hazard stochastic response analysis and reliability evaluation of a 5 MW monopile offshore wind turbine based on extended platform FAST-S

This study investigates the random dynamic response of a 5 MW monopile offshore wind turbine (MOWT) subjected to wind, wave and earthquake loads. Based on the aero-hydro-servo-elastic code FAST, a multi-field coupled simulation module considering earthquake, FAST-S, is developed for the fully coupled multi-hazard analysis of OWTs. Compared to the original FAST code, a seismic analysis module (SeismoDyn) and a soil-structure interaction (SSI) module (SoilDyn) have been supplemented in FAST-S. SeismoDyn is developed by modifying the Kane’s kinetic equation in the FAST, and the SoilDyn is developed through Winkler foundation model in order to consider the nonlinear SSI. Moreover, FAST-S has been validated by other recognized codes. Based on FAST-S, a fully coupled numerical analysis model of the NREL 5 MW MOWT is established, the random responses and reliability evaluation of the offshore wind turbine under wind, wave and earthquake loads are analyzed by probability density evolution method. The results show that the aerodynamic and hydrodynamic actions increase the damping of the wind turbine system, which has a certain effect on inhibiting the tower’s vibration caused by the earthquake. Therefore, compared with the coupled wind-wave-earthquake case, the sole earthquake case may be more unfavorable, and should be paid more attention to in high risk earthquake regions. The results show that the probability density functions of offshore wind turbine response under strong earthquakes obviously deviates from the typical regular distribution models. Based on the theory of equivalent extreme value event, the dynamic reliability of the offshore wind turbine is calculated. Seismic actions maybe the major factor in the design of the support structure. Nevertheless, owing to the aerodynamic and hydrodynamic damping effects of wind and wave loadings, the structural responses under wind-wave-earthquake are more moderate than ones subjected to solely earthquake. The strong seismic loads would cause significant complex variations of time-dependent probability density function of the OWT response.

Time-dependent probability density functions, information geometry and entropy production in a stochastic prey–predator model of fusion plasmas

A stochastic, prey–predator model of the L–H transition in fusion plasma is investigated. The model concerns the regulation of turbulence by zonal and mean flow shear. Independent delta-correlated Gaussian stochastic noises are used to construct Langevin equations for the amplitudes of turbulence and zonal flow shear. We then find numerical solutions of the equivalent Fokker–Planck equation for the time-dependent joint probability distribution of these quantities. We extend the earlier studies [Kim and Hollerbach, Phys. Rev. Res. 2, 023077 (2020) and Hollerbach et al., Phys. Plasmas 27, 102301 (2020)] by applying different functional forms of the time-dependent external heating (input power), which is increased and then decreased in a symmetric fashion to study hysteresis. The hysteresis is examined through the probability distribution and statistical measures, which include information geometry and entropy. We find strongly non-Gaussian probability distributions with bi-modality being a persistent feature across the input powers; the information length to be a better indicator of distance to equilibrium than the total entropy. Both dithering transitions and direct L-–H transitions are (also) seen when the input power is stepped in time. By increasing the number of steps, we see less hysteresis (in the statistical measures) and a reduced probability of H-mode access; intermittent zonal flow shear is seen to have a role in the initial suppression of turbulence by zonal flow shear and stronger excitation of intermittent zonal flow shear for a faster changing input power.

Life-Cycle Seismic Reliability Analysis of a Railway Cable-Stayed Bridge Considering Material Corrosion and Degradation

To study the life-cycle seismic reliability analysis (SRA) of cable-stayed bridges (CSBs) taking into account chloride-induced corrosion and degradation of components, an actual railway CSB with uncertainties in structural geometry and material corrosion coefficients was employed in this investigation, and time-dependent models of CSB components at different service times were studied. Based on the OpenSees batch program, we adapted a mass numerical computation to obtain time-dependent non-linear seismic response and probability density function (PDF) of response via the multiplier dimensional-reduction method (MDRM) and the maximum entropy method with fractional moments (FM-MEM). Next, the time-dependent failure possibility of every component and the association coefficient between the failure modes of different parts were acquired. In the end, the product of the conditional marginal (PCM) approach was employed to obtain the life-cycle failure possibility of the CSB system. The results showed that the system failure possibility of the CSB in a corrosive environment increases significantly with increasing servicing time.

Time-dependent probability density function for partial resetting dynamics

Stochastic resetting is a rapidly developing topic in the field of stochastic processes and their applications. It denotes the occasional reset of a diffusing particle to its starting point and effects, inter alia, optimal first-passage times to a target. Recently the concept of partial resetting, in which the particle is reset to a given fraction of the current value of the process, has been established and the associated search behaviour analysed. Here we go one step further and we develop a general technique to determine the time-dependent probability density function (PDF) for Markov processes with partial resetting. We obtain an exact representation of the PDF in the case of general symmetric Lévy flights with stable index . For Cauchy and Brownian motions (i.e. ), this PDF can be expressed in terms of elementary functions in position space. We also determine the stationary PDF. Our numerical analysis of the PDF demonstrates intricate crossover behaviours as function of time.

Time-dependent probability density function analysis of H-mode transitions

The first application of time-dependent probability density function (PDF) analysis to the L-H transition in fusion plasmas is presented. PDFs are constructed using Doppler Backscattering data of perpendicular fluctuation velocity, , and turbulence from the edge region of the DIII-D tokamak. These raw time-series data are sliced into millisecond-long sliding time-windows to create PDFs. During the transition, the PDFs develop strong right tails, indicative of turbulence-suppressing localised flows in the plasma edge; such features and other subtle behaviours are explored using novel information geometry techniques. This letter examines the applicability of these techniques to predict L-H transitions and investigate predator-prey self-regulation theories between turbulence and perpendicular velocity.

A stochastic model of edge-localized modes in magnetically confined plasmas.

Magnetically confined plasmas are far from equilibrium and pose considerable challenges in statistical analysis. We discuss a non-perturbative statistical method, namely a time-dependent probability density function (PDF) approach that is potentially useful for analysing time-varying, large, or non-Gaussian fluctuations and bursty events associated with instabilities in the low-to-high confinement transition and the H-mode. Specifically, we present a stochastic Langevin model of edge-localized modes (ELMs) by including stochastic noise terms in a previous ODE ELM model. We calculate exact time-dependent PDFs by numerically solving the Fokker-Planck equation and characterize time-varying statistical properties of ELMs for different energy fluxes and noise amplitudes. The stochastic noise is shown to introduce phase-mixing and plays a significant role in mitigating extreme bursts of large ELMs. Furthermore, based on time-dependent PDFs, we provide a path-dependent information geometric theory of the ELM dynamics and demonstrate its utility in capturing self-regulatory relaxation oscillations, bursts and a sudden change in the system. This article is part of a discussion meeting issue 'H-mode transition and pedestal studies in fusion plasmas'.

Non-equilibrium statistical properties, path-dependent information geometry, and entropy relations in edge-localized modes in fusion plasmas

We investigate time-varying turbulence statistical properties of edge-localized modes (ELMs) in fusion plasmas. By utilizing a simplified stochastic model, we calculate a time-dependent probability density function and various entropy-related quantities such as entropy, entropy production, entropy flux, mutual information, and information flow and path-dependent information geometry. A thorough analysis is performed to elucidate the effects on ELM dynamics (evolution, suppression, mitigation, etc.) of different values of stochastic noise and different forms of a time-varying input power. Furthermore, the time-irreversibility and hysteresis are investigated through the employment of forward and back processes where a time-varying input power varies mirror-symmetrically in time. Among all the statistical quantities, the path-dependent information geometry is shown to be a robust diagnostic for quantifying hysteresis and self-regulation as well as for an early detection of subtle changes in ELM dynamics, for example, caused by a sudden change in the input power.

Monte Carlo Simulation of Stochastic Differential Equation to Study Information Geometry.

Information Geometry is a useful tool to study and compare the solutions of a Stochastic Differential Equations (SDEs) for non-equilibrium systems. As an alternative method to solving the Fokker–Planck equation, we propose a new method to calculate time-dependent probability density functions (PDFs) and to study Information Geometry using Monte Carlo (MC) simulation of SDEs. Specifically, we develop a new MC SDE method to overcome the challenges in calculating a time-dependent PDF and information geometric diagnostics and to speed up simulations by utilizing GPU computing. Using MC SDE simulations, we reproduce Information Geometric scaling relations found from the Fokker–Planck method for the case of a stochastic process with linear and cubic damping terms. We showcase the advantage of MC SDE simulation over FPE solvers by calculating unequal time joint PDFs. For the linear process with a linear damping force, joint PDF is found to be a Gaussian. In contrast, for the cubic process with a cubic damping force, joint PDF exhibits a bimodal structure, even in a stationary state. This suggests a finite memory time induced by a nonlinear force. Furthermore, several power-law scalings in the characteristics of bimodal PDFs are identified and investigated.

Singularity Behaviour of the Density, Information, and Entropy Functions Defining a Uniform Non-stationary Stochastic Process

Precise definitions and derivatives of the time-dependent continuous and discrete uniform probability density functions and related information and entropy functions are investigated. A stochastic system is formed that can represent a uniform noise source having a time-dependent variance and forming a uniform non-stationary stochastic process. The information and entropy function of the system are defined, and their properties are investigated in the time domain, including the limit cases defined for infinite and zero values of the time-dependent variance. In particular, the singularity properties of the entropy function will be investigated when the time-dependent variance reaches infinity. Like in thermodynamics, where the physical entropy of a system increases all the time, the information entropy of the stochastic system in information theory is also expected to increase towards infinity when the variance increases. All investigations are conducted for both the continuous and discrete random variables and their density functions. The presented theory is of particular interest in analyzing the Gaussian density function having infinite variance and tending to a uniform density function.

Autocorrelation functions and ergodicity in diffusion with stochastic resetting

Diffusion with stochastic resetting is a paradigm of resetting processes. Standard renewal or master equation approach are typically used to study steady state and other transport properties such as average, mean squared displacement etc. What remains less explored is the two time point correlation functions whose evaluation is often daunting since it requires the implementation of the exact time dependent probability density functions of the resetting processes which are unknown for most of the problems. We adopt a different approach that allows us to write a stochastic solution for a single trajectory undergoing resetting. Moments and the autocorrelation functions between any two times along the trajectory can then be computed directly using the laws of total expectation. Estimation of autocorrelation functions turns out to be pivotal for investigating the ergodic properties of various observables for this canonical model. In particular, we investigate two observables (i) sample mean which is widely used in economics and (ii) time-averaged-mean-squared-displacement (TAMSD) which is of acute interest in physics. We find that both diffusion and drift–diffusion processes with resetting are ergodic at the mean level unlike their reset-free counterparts. In contrast, resetting renders ergodicity breaking in the TAMSD while both the stochastic processes are ergodic when resetting is absent. We quantify these behaviors with detailed analytical study and corroborate with extensive numerical simulations. Our results can be verified in experimental set-ups that can track single particle trajectories and thus have strong implications in understanding the physics of resetting.

Exact Time-Dependent Solutions and Information Geometry of a Rocking Ratchet

The noise-induced transport due to spatial symmetry-breaking is a key mechanism for the generation of a uni-directional motion by a Brownian motor. By utilising an asymmetric sawtooth periodic potential and three different types of periodic forcing G(t) (sinusoidal, square and sawtooth waves) with period T and amplitude A, we investigate the performance (energetics, mean current, Stokes efficiency) of a rocking ratchet in light of thermodynamic quantities (entropy production) and the path-dependent information geometric measures. For each G(t), we calculate exact time-dependent probability density functions under different conditions by varying T, A and the strength of the stochastic noise D in an unprecedentedly wide range. Overall similar behaviours are found for different cases of G(t). In particular, in all cases, the current, Stokes efficiency and the information rate normalised by A and D exhibit one or multiple local maxima and minima as A increases. However, the dependence of the current and Stokes efficiency on A can be quite different, while the behaviour of the information rate normalised by A and D tends to resemble that of the Stokes efficiency. In comparison, the irreversibility measured by a normalised entropy production is independent of A. The results indicate the utility of the information geometry as a proxy of a motor efficiency.

Log-growth rates of CO2: An empirical analysis

We study the parametric distribution of log-growth rates of CO2 and CO2 per capita emissions for 207 countries and territories taking data from 1994 to 2010. We define the log-growth rates for different duration periods, from one year apart to fifteen years apart. The considered probability distributions have been the following: the normal (N), the asymmetric double Laplace normal (adLN), the exponential tails normal (ETN) and a mixture of two normal (2N) or three normal (3N) distributions. The main result is that the best one is different depending on the period considered, in such a way that there is not a systematically dominant distribution. Thus, the behavior may change from one year to the next one, and possibly this is influenced by policy measures such as the Kyoto protocol or the Clean Development Mechanism. Moreover, a policy measure that can be derived from this paper is that some countries can still reduce their emissions of CO2 compared with others, as seen by the non-uniformity of the preferred probability distribution for each period. We also model a stochastic differential equation whose associated Fokker–Planck equation has as a solution the observed time-dependent probability density function.

Information Geometry, Fluctuations, Non-Equilibrium Thermodynamics, and Geodesics in Complex Systems.

Information theory provides an interdisciplinary method to understand important phenomena in many research fields ranging from astrophysical and laboratory fluids/plasmas to biological systems. In particular, information geometric theory enables us to envision the evolution of non-equilibrium processes in terms of a (dimensionless) distance by quantifying how information unfolds over time as a probability density function (PDF) evolves in time. Here, we discuss some recent developments in information geometric theory focusing on time-dependent dynamic aspects of non-equilibrium processes (e.g., time-varying mean value, time-varying variance, or temperature, etc.) and their thermodynamic and physical/biological implications. We compare different distances between two given PDFs and highlight the importance of a path-dependent distance for a time-dependent PDF. We then discuss the role of the information rate and relative entropy in non-equilibrium thermodynamic relations (entropy production rate, heat flux, dissipated work, non-equilibrium free energy, etc.), and various inequalities among them. Here, is the information length representing the total number of statistically distinguishable states a PDF evolves through over time. We explore the implications of a geodesic solution in information geometry for self-organization and control.

On Time-Dependent Trip Distance Distribution with For-Hire Vehicle Trips in Chicago

For transportation system analysis in a new space dimension with respect to individual trips’ remaining distances, vehicle trips demand has two main components: the departure time and the trip distance. In particular, the trip distance distribution (TDD) is a direct input to the bathtub model in the new space dimension, and is a very important variable to consider in many applications, such as the development of distance-based congestion pricing strategies or mileage tax. For a good understanding of the demand pattern, both the distribution of trip initiation and trip distance should be calibrated from real data. In this paper, it is assumed that the demand pattern can be described by the joint distribution of trip distance and departure time. In other words, TDD is assumed to be time-dependent, and a calibration and validation methodology of the joint probability is proposed, based on log-likelihood maximization and the Kolmogorov–Smirnov test. The calibration method is applied to empirical for-hire vehicle trips in Chicago, and it is concluded that TDD varies more within a day than across weekdays. The hypothesis that TDD follows a negative exponential, log-normal, or Gamma distribution is rejected. However, the best fit is systematically observed for the time-dependent log-normal probability density function. In the future, other trip distributions should be considered and also non-parametric probability density estimation should be explored for a better understanding of the demand pattern.

Stochastic inference of surface-induced effects using Brownian motion

Brownian motion in confinement and at interfaces is a canonical situation, encountered from fundamental biophysics to nanoscale engineering. Using the Lorenz-Mie framework, we optically record the thermally-induced tridimensional trajectories of individual microparticles, within salty aqueous solutions, in the vicinity of a rigid wall, and in the presence of surface charges. We construct the time-dependent position and displacement probability density functions, and study the non-Gaussian character of the latter which is a direct signature of the hindered mobility near the wall. Based on these distributions, we implement a novel, robust and self-calibrated multifitting method, allowing for the thermal-noise-limited inference of diffusion coefficients spatially-resolved at the nanoscale, equilibrium potentials, and forces at the femtoNewton resolution.