No abstract

The stochastic nature of pitting corrosion of metallic structures has been widely recognized. It is assumed that this kind of deterioration retains no memory of the past, so only the current state of the damage influences its future development. This characteristic allows pitting corrosion to be categorized as a Markov process. In this paper, two different models of pitting corrosion, developed using Markov chains, are presented. Firstly, a continuous-time, nonhomogeneous linear growth (pure birth) Markov process is used to model external pitting corrosion in underground pipelines. A closed-form solution of the system of Kolmogorov's forward equations is used to describe the transition probability function in a discrete pit depth space. The transition probability function is identified by correlating the stochastic pit depth mean with the empirical deterministic mean. In the second model, the distribution of maximum pit depths in a pitting experiment is successfully modeled after the combination of two stochastic processes: pit initiation and pit growth. Pit generation is modeled as a nonhomogeneous Poisson process, in which induction time is simulated as the realization of a Weibull process. Pit growth is simulated using a nonhomogeneous Markov process. An analytical solution of Kolmogorov's system of equations is also found for the transition probabilities from the first Markov state. Extreme value statistics is employed to find the distribution of maximum pit depths.

Abstract.In cloud modeling studies, the time evolution of droplet size distributions due to collision-coalescence events is usually modeled with the Smoluchowski coagulation equation, also known as the kinetic collection equation (KCE). However, the KCE is a deterministic equation with no stochastic fluctuations or correlations. Therefore, the full stochastic description of cloud droplet growth in a coalescing system must be obtained from the solution of the multivariate master equation, which models the evolution of the state vector for the number of droplets of a given mass. Unfortunately, due to its complexity, only limited results were obtained for certain types of kernels and monodisperse initial conditions. In this work, a novel numerical algorithm for the solution of the multivariate master equation for stochastic coalescence that works for any type of kernels, multivariate initial conditions and small system sizes is introduced. The performance of the method was seen by comparing the numerically calculated particle mass spectrum with analytical solutions of the master equation obtained for the constant and sum kernels. Correlation coefficients were calculated for the turbulent hydrodynamic kernel, and true stochastic averages were compared with numerical solutions of the kinetic collection equation for that case. The results for collection kernels depending on droplet mass demonstrates that the magnitudes of correlations are significant and must be taken into account when modeling the evolution of a finite volume coalescing system.

In this work, the statistical methods for the reliability of repairable systems have been used to produce a methodology capable to estimate the annualized failure rate of a pipeline population from the historical failure data of multiple pipelines systems. The proposed methodology provides point and interval estimators of the parameters of the failure intensity function for two of the most commonly applied stochastic models; the homogeneous Poisson process and the power law process. It also provides statistical tests to assess the adequacy of the stochastic model assumed for each system and to test whether all systems have the same model parameters. In this way, the failure data of multiple pipeline systems are only pooled to produce a generic failure intensity function when all systems follow the same stochastic model. This allows addressing both statistical and tolerance uncertainty adequately. The proposed methodology is outlined and illustrated using real life failure data of multiple oil and gas pipeline systems.

Abstract. The kinetic collection equation (KCE) describes the evolution of the average droplet spectrum due to successive events of collision and coalescence. Fluctuations and non-zero correlations present in the stochastic coalescence process would imply that the size distributions may not be correctly modeled by the KCE.In this study we expand the known analytical studies of the coalescence equation with some numerical tools such as Monte Carlo simulations of the coalescence process. The validity time of the KCE was estimated by calculating the maximum of the ratio of the standard deviation for the largest droplet mass over all the realizations to the averaged value. A good correspondence between the analytical and the numerical approaches was found for all the kernels. The expected values from analytical solutions of the KCE, were compared with true expected values of the stochastic collection equation (SCE) estimated with Gillespie's Monte Carlo algorithm and analytical solutions of the SCE, after and before the breakdown time.The possible implications for cloud physics are discussed, in particular the possibility of application of these results to kernels modified by turbulence and electrical processes.

In this paper, a statistical analysis of log-return fluctuations of the IPC, the Mexican Stock Market Index is presented. A sample of daily data covering the period from 04/09/2000 − 04/09/2010 was analyzed, and fitted to different distributions. Tests of the goodness of fit were performed in order to quantitatively asses the quality of the estimation. Special attention was paid to the impact of the size of the sample on the estimated decay of the distributions tail. In this study a forceful rejection of normality was obtained. On the other hand, the null hypothesis that the log-fluctuations are fitted to a α-stable Lévy distribution cannot be rejected at 5% significance level.

Abstract. The impact of stochastic fluctuations in cloud droplet growth is a matter of broad interest, since stochastic effects are one of the possible explanations of how cloud droplets cross the size gap and form the raindrop embryos that trigger warm rain development in cumulus clouds. Most theoretical studies on this topic rely on the use of the kinetic collection equation, or the Gillespie stochastic simulation algorithm. However, the kinetic collection equation is a deterministic equation with no stochastic fluctuations. Moreover, the traditional calculations using the kinetic collection equation are not valid when the system undergoes a transition from a continuous distribution to a distribution plus a runaway raindrop embryo (known as the sol-gel transition). On the other hand, the stochastic simulation algorithm, although intrinsically stochastic, fails to adequately reproduce the large end of the droplet size distribution due to the huge number of realizations required. Therefore, the full stochastic description of cloud droplet growth must be obtained from the solution of the master equation for stochastic coalescence.In this study the master equation is used to calculate the evolution of the droplet size distribution after the sol-gel transition. These calculations show that after the formation of the raindrop embryo, the expected droplet mass distribution strongly differs from the results obtained with the kinetic collection equation. Furthermore, the low-mass bins and bins from the gel fraction are strongly anticorrelated in the vicinity of the critical time, this being one of the possible explanations for the differences between the kinetic and stochastic approaches after the sol-gel transition. Calculations performed within the stochastic framework provide insight into the inability of explicit microphysics cloud models to explain the droplet spectral broadening observed in small, warm clouds.

Abstract. The kinetic collection equation (KCE) has been widely used to describe the evolution of the average droplet spectrum due to the collection process that leads to the development of precipitation in warm clouds. This deterministic, integro-differential equation only has analytic solution for very simple kernels. For more realistic kernels, the KCE needs to be integrated numerically. In this study, the validity time of the KCE for the hydrodynamic kernel is estimated by a direct comparison of Monte Carlo simulations with numerical solutions of the KCE. The simulation results show that when the largest droplet becomes separated from the smooth spectrum, the total mass calculated from the numerical solution of the KCE is not conserved and, thus, the KCE is no longer valid. This result confirms the fact that for kernels appropriate for precipitation development within warm clouds, the KCE can only be applied to the continuous portion of the mass distribution.