Stochastic Description Research Articles

Dynamic Monte Carlo reactor modeling of calcium looping with sorbent purge and utilization decay

This work presents a new dynamic modelling approach for calcium looping systems that allows explicit sorbent deactivation and purge/makeup. These are common in plant operations, but often neglected in modelling. This model adopts a Monte Carlo approach, tracking merely 100 particles between the carbonator and the calciner for 60 reaction cycles with the particle residence time in each reactor determined stochastically. The simulated results agree well with the experimental data in both the transient and steady-state stages. This model provides a promising approach to predicting the dynamic behaviour of calcium looping systems under relatively realistic conditions at low computational cost. The stochastic description of a multiphase reactor via the Monte Carlo process can be widely adapted in various scenarios.

Stochastic approach to gravitational waves from inflation

We propose a coarse-graining procedure for describing the superhorizon dynamics of inflationary tensor modes. Our aim is to formulate a stochastic description for the statistics of spin-2 modes which seed the background of gravitational waves from inflation. Using basic principles of quantum mechanics, we determine a probability density for coarse-grained tensor fields, which satisfies a stochastic Fokker-Planck equation at superhorizon scales. The corresponding noise and drift are computable, and depend on the cosmological system under consideration. Our general formulas are applied to a variety of cosmological scenarios, also considering cases seldom considered in the context of stochastic inflation, and which are important for their observational consequences. We start obtaining the expected expressions for noise and drift in pure de Sitter and power-law inflation, also including a discussion of effects of non-attractor phases. We then apply our methods to describe scenarios with a transition from inflation to standard cosmological eras of radiation and matter domination. We show how the interference between modes flowing through the cosmological horizon, and modes spontaneously produced at superhorizon scales, can affect the stochastic evolution of coarse-grained tensor quantities. In appropriate limits, we find that the corresponding spectrum of tensor modes at horizon crossing matches with the results of quantum field theory calculations, but we also highlight where differences can arise.

Nonlinear atomic force microscopy: Squeezing and skewness of micromechanical oscillators interacting with a surface

We propose a two-frequency driving scheme in dynamic atomic force microscopy that maximizes the interaction time between tip and sample. Using a stochastic description of the cantilever dynamics, we predict large classical squeezing and a small amount of skewness of the tip's phase-space probability distribution. Strong position squeezing will require close contact between tip and surface, while momentum squeezing would also be possible in the van der Waals region of the tip-surface force. Employing a generalized Caldeira-Leggett model, we predict that surface-dependent dissipative forces may be the dominant source of quantum effects and propose a procedure to isolate quantum effects from thermal fluctuations.

Instantaneous turbulent kinetic energy modelling based on Lagrangian stochastic approach in CFD and application to wind energy

We present the construction of an original stochastic model for the instantaneous turbulent kinetic energy at a given point of a flow, and we validate estimator methods on this model with observational data examples. Motivated by the need for wind energy industry of acquiring relevant statistical information of air motion at a local place, we adopt the Lagrangian description of fluid flows to derive, from the 3D+time equations of the physics, a 0D+time-stochastic model for the time series of the instantaneous turbulent kinetic energy at a given position. Specifically, based on the Lagrangian stochastic description of a generic fluid-particles, we derive a family of mean-field dynamics featuring the square norm of the turbulent velocity. By approximating at equilibrium the characteristic nonlinear terms of the dynamics, we recover the so called Cox-Ingersoll-Ross process, which was previously suggested in the literature for modelling wind speed. We then propose a calibration procedure for the parameters employing both direct methods and Bayesian inference. In particular, we show the consistency of the estimators and validate the model through the quantification of uncertainty, with respect to the range of values given in the literature for some physical constants of turbulence modelling.

Protein pattern formation induced by the joint effect of noise and delay in a multi-cellular system

We explore the combined effect of the intrinsic noise and time delay on the spatial pattern formation within the framework of a multi-scale mobile lattice model mimicking two-dimensional epithelium tissues. Every cell is represented by an elastic polygon changing its form and size under pressure from the surrounding cells. The model includes the procedure of minimization of the potential energy of tissue. The protein fluctuations in the tissue are driven by transcription/translation processes in epithelial cells exchanging chemical and mechanical signals. Network architecture includes a simple autorepressor model with time-delayed negative feedback, in which the only gene defines the oscillatory activity. Simultaneously, the expressed protein of the autorepressor acts as a positive regulator of the signaling protein by activating its transcription. The signaling species is assumed to spread from one cell to the other by the diffusion mechanism. We provide both deterministic and stochastic descriptions. The numerical simulation of spatially-extended stochastic oscillations is performed using a generalized Gillespie algorithm. We developed this method earlier to account for the non-Markovian properties of random biochemical events with delay. Finally, we demonstrate that time delay, intrinsic noise, and spatial signaling can cause a system to develop the protein pattern even when its deterministic counterpart exhibits no pattern formation.

Can we understand the mechanisms of tumor formation by analyzing dynamics of cancer initiation?

Cancer is a collection of related genetic diseases exhibiting uncontrolled cell growth that interferes with normal functioning of human organisms. It results from accumulation of unfavorable mutations in tissues. While the biochemical picture of how cancer appears is known, the molecular mechanisms of tumor formation remain not fully understood despite tremendous efforts of researchers in multiple fields. New approaches for investigating cancer are constantly sought. In this paper, we discuss a powerful method of clarifying better a more microscopic picture of cancer by analyzing the dynamics of tumor formation. Using physics- and chemistry-inspired discrete-state stochastic description of cancer initiation, it is shown how the mechanisms of tumor formation can be uncovered. This approach is suggested as a powerful new physical-chemical tool for a better understanding of complex processes associated with cancer.

Modeling Neurons in 3D at the Nanoscale.

For decades, neurons have been modeled by methods developed by early pioneers in the field such as Rall, Hodgkin and Huxley, as cable-like morphological structures with voltage changes that are governed by a series of ordinary differential equations describing the conductances of ion channels embedded in the membrane. In recent years, advances in experimental techniques have improved our knowledge of the morphological and molecular makeup of neurons, and this has come alongside ever-increasing computational power and the wider availability of computer hardware to researchers. This has opened up the possibility of more detailed 3D modeling of neuronal morphologies and their molecular makeup, a new, emerging component of the field of computational neuroscience that is expected to play an important role in building our understanding of neurons and their behavior into the future.Many readers may be familiar with 1D models yet unfamiliar with the more detailed 3D description of neurons. As such, this chapter introduces some of the techniques used in detailed 3D, molecular modeling, and shows the steps required for building such models from a foundation of the more familiar 1D description. This broadly falls into two categories; morphology and how to build a 3D computational mesh based on a cable-like description of the neuronal geometry or directly from imaging studies, and biochemically how to define a discrete, stochastic description of the molecular neuronal makeup. We demonstrate this with a full Purkinje cell model, implemented in 3D simulation in software STEPS.

Pore Size Dependence of Permeability in Bicontinuous Nanoporous Media.

In this work, we employ many-body dissipative particle dynamics (mDPD) simulations to investigate the fluid flow process through bicontinuous nanoporous media, which are representative models for a broad class of nanoporous materials. The mDPD formulation includes attractive and repulsive interactions describing accurately fluid-fluid and fluid-solid interactions. As a mesoscale simulation method, mDPD can bridge the length and time scale gap between continuum and atomistic simulations. The bicontinuous nanoporous models are constructed considering a defined morphology, the porosity level, and varying pore sizes in the range from 3.41 to 13.63 nm. All models have a 0.65 porosity level and the same topology. The models provide a stochastic description of the morphology and pore size distribution and allow for a direct investigation of the dependence of permeability on the average pore size. The stationary nanoporous models are filled with fluid particles, and flow is induced by the action of confining pistons. Simulation results, obtained by imposing different pressure differences on the surfaces of the nanoporous media, indicate a linear pressure drop within the nanoporous model. Regardless of the complexities and different scales of the porous media considered, the steady-state fluid flow through the nanoporous models is proportional to the pressure gradient applied, in agreement with Darcy's law. The calculated pore size dependence of permeability is well described by the Hagen-Poiseuille law, considering a single shape correction factor that accounts for the flow resistance due to the complex nanoporous morphology. This work highlights the effect of the average pore size of a complex stochastic bicontinuous nanoporous medium on fluid properties. The results indicate rather a relatively simple dependence of permeability on the average pore size. The novel method we employ to generate the stochastic bicontinuous nanoporous structure allows the control of different geometric features that can be explored in future studies.

Understanding the Reaction Dynamics on Heterogeneous Catalysts Using a Simple Stochastic Approach.

Recent experimental advances on investigating nanoparticle catalysts with multiple active sites provided a large amount of quantitative information on catalytic processes. These observations stimulated significant theoretical efforts, but the underlying molecular mechanisms are still not well-understood. We introduce a simple theoretical method to analyze the reaction dynamics on catalysts with multiple active sites based on a discrete-state stochastic description and obtain a comprehensive description of the dynamics of chemical reactions on such catalysts. We explicitly determine how the dynamics of catalyzed chemical reactions depend on the number of active sites, on the number of intermediate chemical transitions, and on the topology of underlying chemical reactions. It is argued that the theory provides quantitative bounds for realistic dynamic properties of catalytic processes that can be directly applied to analyze the experimental observations. In addition, this theoretical approach clarifies several important aspects of the molecular mechanisms of chemical reactions on catalysts.

Physical light-matter interaction in hermite-gauss space

Our purpose in this paper is two-fold: introduce a computationally-tractable decomposition of the coherence properties of light; and, present a general-purpose light-matter interaction framework for partially-coherent light. In a recent publication, Steinberg and Yan [2021] introduced a framework that generalises the classical radiometry-based light transport to physical optics. This facilitates a qualitative increase in the scope of optical phenomena that can be rendered, however with the additional expressibility comes greater analytic difficulty: This coherence of light, which is the core quantity of physical light transport, depends initially on the characteristics of the light source, and mutates on interaction with matter and propagation. Furthermore, current tools that aim to quantify the interaction of partially-coherent light with matter remain limited to specific materials and are computationally intensive. To practically represent a wide class of coherence functions, we decompose their modal content in Hermite-Gauss space and derive a set of light-matter interaction formulae, which quantify how matter scatters light and affects its coherence properties. Then, we model matter as a locally-stationary random process, generalizing the prevalent deterministic and stationary stochastic descriptions. This gives rise to a framework that is able to formulate the interaction of arbitrary partially-coherent light with a wide class of matter. Indeed, we will show that our presented formalism unifies a few of the state-of-the-art scatter and diffraction formulae into one cohesive theory. This formulae include the sourcing of partially-coherent light, scatter by rough surfaces and microgeometry, diffraction grating and interference by a layered structure.

Modeling epidemics by means of the stochastic description of complex systems

The aim of this article is to show a way in which the problem of predicting the evolution of an epidemic may be tackled by describing it in the framework of Boltzmann's kinetic theory, as it has been developed and applied in the last years to complex systems by a suitable modification of the Boltzmann equation, via a suitable reinterpretation of state variables and the introduction of the notion of «functional subsystems». Accordingly, in this article we model an arbitrary (national) population S as a complex system, split in two functional subsystems, the first containing all single individuals of S and the second containing the «care tools», that are to be meant as available places in hospitals with a sufficient number of physicians and of equipments for intensive cares. The state variable on the first subsystem will be the «health state», and the state variable on the other will be the «effectiveness». We shall then write a system of nonlinear ordinary differential equations which gives the evolution of the probability distribution on the set of possible values of the health states. By assigning data partly on the basis of plausibility assumptions and partly as estimated from those furnished by institutions of Campania region, the system takes a form allowing the numerical simulation of such evolution, which will be performed and presented in a forthcoming paper.

Upscaling of fractured rock mass properties – An example comparing Discrete Fracture Network (DFN) modeling and empirical relations based on engineering rock mass classifications

In this study a comparison of Discrete Fracture Network (DFN) modeling and empirical relationships based on conventional engineering rock mass classifications is carried out in order to determine the elastic properties of fractured rock masses. Uncertainties are considered by a stochastic description of the fracture network geometry and a probabilistic approach, utilizing Monte Carlo simulation techniques, for the various other input parameters. The proposed workflow is applied to two sandstone outcrops. Geometric information on the fracture systems is obtained from terrestrial laser scanning providing the input parameters (orientation, size and intensity) for each fracture set including probability density functions required for DFN modeling. After adding the mechanical properties of intact rock and fractures obtained by laboratory tests, the DFN approach allows to calculate values for rock mass deformation modulus and Poisson's ratio assuming either isotropy or vertical-transverse isotropy. These estimates are compared to values derived from empirical relationships based on engineering rock mass classifications. The comparison of the two approaches and, in particular, of the different empirical relationships reveals a significant scatter in the estimated rock mass properties. Only those empirical equations which incorporate intact rock properties in addition to the classification score yield plausible results, i.e., a pronounced weakening due to the fractures, and are in agreement with the estimates of the DFN approach. The additional effort of generating a DFN model shows its benefits if an anisotropic rock mass and/or spatially varying properties have to be considered, thus, providing a comprehensive hydromechanical characterization of anisotropic fractured rock masses.

First passage time properties of miRNA-mediated protein translation

An important function of microRNAs in gene regulation is to repress the protein synthesis in a multi-step process with implications in timing efficiency of the regulatory network. We propose a stochastic description of translation-initiation mechanism and solve for the steady state distribution of protein number in the linear regime. The time-dependent moments have been approximately calculated and the role of miRNAs in determining the First Passage Time (FPT) properties of protein dynamics has been established. We analytically show that the modulation of slow rates of the translation process will result in efficient and robust timing mechanism. For a general nonlinear model our numerical simulation results are in qualitative agreement with the linear model.

An enhanced stochastic operating cycle description including weather and traffic models

• The new stochastic operating cycle includes models for the weather and traffic. • A primary model for the weather category is introduced to account for seasonality. • A regression analysis is made to investigate spread in CO 2 emissions. • It is found that both seasonal and traffic settings influence CO 2 emissions. The present paper extends the concept of a stochastic operating cycle (sOC) by introducing additional models for weather and traffic. In regard to the weather parameters, dynamic models for air temperature, atmospheric pressure, relative humidity, precipitation, wind speed and direction are included. The traffic models is instead based on a macroscopic approach which describes the density dynamically by means of a simple autoregressive process. The enhanced format is structured in a hierarchical fashion, allowing for ease of implementation and modularity. The novel models are parametrised starting from data available from external databases. The possibility of generating synthetic data using the statistical descriptors introduced in the paper is also discussed. To investigate the impact of the novel parameters over energy efficiency, a sensitivity analysis is conducted with a combinatorial test design. Simulation results show that both seasonality and traffic conditions are responsible for introducing major variations in the CO 2 emissions.

Three-dimensional stochastic simulation of chemoattractant-mediated excitability in cells.

During the last decade, a consensus has emerged that the stochastic triggering of an excitable system drives pseudopod formation and subsequent migration of amoeboid cells. The presence of chemoattractant stimuli alters the threshold for triggering this activity and can bias the direction of migration. Though noise plays an important role in these behaviors, mathematical models have typically ignored its origin and merely introduced it as an external signal into a series of reaction-diffusion equations. Here we consider a more realistic description based on a reaction-diffusion master equation formalism to implement these networks. In this scheme, noise arises naturally from a stochastic description of the various reaction and diffusion terms. Working on a three-dimensional geometry in which separate compartments are divided into a tetrahedral mesh, we implement a modular description of the system, consisting of G-protein coupled receptor signaling (GPCR), a local excitation-global inhibition mechanism (LEGI), and signal transduction excitable network (STEN). Our models implement detailed biochemical descriptions whenever this information is available, such as in the GPCR and G-protein interactions. In contrast, where the biochemical entities are less certain, such as the LEGI mechanism, we consider various possible schemes and highlight the differences between them. Our simulations show that even when the LEGI mechanism displays perfect adaptation in terms of the mean level of proteins, the variance shows a dose-dependence. This differs between the various models considered, suggesting a possible means for determining experimentally among the various potential networks. Overall, our simulations recreate temporal and spatial patterns observed experimentally in both wild-type and perturbed cells, providing further evidence for the excitable system paradigm. Moreover, because of the overall importance and ubiquity of the modules we consider, including GPCR signaling and adaptation, our results will be of interest beyond the field of directed migration.

Structural reliability of ultra high-performance fibre reinforced concrete beams in flexure

A study of the model uncertainty and structural reliability for ultra high-performance fibre reinforced concrete (UHPFRC) beams under flexure is presented. Model factors are calibrated by comparison to finite element analyses for nine UHPFRC beams with volume percentage of steel fibres in the range of 1–4%, a feasible percentage range to produce UHPFRC. Moment capacity values obtained by the finite element models were systematically higher than values predicted by the analytical models described in ACI-318, ACI-544, and FHWA-HIF-13. The model factors were found to indicate strong positive correlation with the yield stress of steel (fy) and strong negative correlation with beam’s section geometry (d/h). When combined with stochastic descriptions of the remaining parameters that describe these analytical models, reliability values for beams designed to the ACI-318 and ACI-544 specifications are found to be in good agreement with the target values for these standards, while values for FHWA-HIF-13 fall about 15% above this value.

Deciphering the Chemistry of Cultural Heritage: Targeting Material Properties by Coupling Spectral Imaging with Image Analysis.

The chemical study of materials from natural history and cultural heritage, which provide information for art history, archeology, or paleontology, presents a series of specific challenges. The complexity of these ancient and historical materials, which are chemically heterogeneous, the product of alteration processes, and inherently not reproducible, is a major obstacle to a thorough understanding of their making and long-term behavior (e.g., fossilization). These challenges required the development of methodologies and instruments coupling imaging and data processing approaches that are optimized for the specific properties of the materials. This Account discusses how these characteristics not only constrain their study but also open up specific innovative avenues for providing key historical information. Synchrotron methods have extensively been used since the late 1990s to study heritage objects, in particular for their potential to provide speciation information from excitation spectroscopies and to image complex heritage objects and samples in two and three dimensions at high resolution. We examine in practice how the identification of key intrinsic chemical specificities has offered fertile ground for the development of novel synchrotron approaches allowing a better stochastic description of the properties of ancient and historical materials. These developments encompass three main aspects: (1) The multiscale heterogeneity of these materials can provide an essential source of information in the development of probes targeting their multiple scales of homogeneity. (2) Chemical alteration can be described in many ways, e.g., by segmenting datasets in a semiquantitative way to jointly inform morphological and chemical transformation pathways. (3) The intrinsic individuality of chemical signatures in artifacts triggers the development of specific strategies, such as those focusing on weak signal detection. We propose a rereading of the advent of these new methodologies for analysis and characterization and examine how they have led to innovative strategies combining materials science, instrument development, history, and data science. In particular, we show that spectral imaging and the search for correlations in image datasets have provided a powerful way to address what archeologists have called the uncertainty and ambiguity of the material record. This approach has implications beyond synchrotron techniques and extends in particular to a series of rapidly developing approaches that couple spectral and spatial information, as in hyperspectral imaging and spatially resolved mass spectrometry. The preeminence of correlations holds promise for the future development of machine learning methods for processing data on historical objects. Beyond heritage, these developments are an original source of inspiration for the study of materials in many related fields, such as environmental, geochemical, or life sciences, which deal with systems whose alteration and heterogeneity cannot be neglected.

Stochastic Modeling of Protein Field with a Delayed Feedback

Protein fields synthesized by genes play a principal role in the functioning of living systems. The processes of gene regulation determine the properties of these fields. Since the number of nucleotides usually is not large, a deterministic description of the field dynamics is insufficient. In this work, we consider a special kind of protein field, the dynamic behavior of which is described by the non-Markov process. Generally, the dynamics of complex organic compounds is time-dependent and spatially extended, and it may depend on all the previous evolution of the system. We consider a time-delayed repressilator as a model system. We study this system numerically using a modified Gillespie algorithm. New dynamic phenomena, which are visible only within a stochastic description, are reported. We show that synchronization in a gene expression occurs much faster due to the non-linear interaction of noise and delay. It results in almost regular oscillations even below the neutral curve derived within the deterministic analysis. We apply a hybrid approach to study the spatial dynamics of the repressilator proteins. This approach includes a deterministic calculation of the diffusion fluxes between cells and the stochastic simulation of gene regulation processes. We found that the combined action of time-delay, noise, and spatial signaling can lead to pattern formation even when the deterministic description predicts the absolute stability of the system.

SiCaSMA: An Alternative Stochastic Description via Concatenation of Markov Processes for a Class of Catalytic Systems

The Chemical Master Equation is a standard approach to model biochemical reaction networks. It consists of a system of linear differential equations, in which each state corresponds to a possible configuration of the reaction system, and the solution describes a time-dependent probability distribution over all configurations. The Stochastic Simulation Algorithm (SSA) is a method to simulate sample paths from this stochastic process. Both approaches are only applicable for small systems, characterized by few reactions and small numbers of molecules. For larger systems, the CME is computationally intractable due to a large number of possible configurations, and the SSA suffers from large reaction propensities. In our study, we focus on catalytic reaction systems, in which substrates are converted by catalytic molecules. We present an alternative description of these systems, called SiCaSMA, in which the full system is subdivided into smaller subsystems with one catalyst molecule each. These single catalyst subsystems can be analyzed individually, and their solutions are concatenated to give the solution of the full system. We show the validity of our approach by applying it to two test-bed reaction systems, a reversible switch of a molecule and methyltransferase-mediated DNA methylation.

Rare extinction events in cyclic predator–prey games

In the May–Leonard model of three cyclically competing species, we analyze the statistics of rare events in which all three species go extinct due to strong but rare fluctuations. These fluctuations are from the tails of the probability distribution of species concentrations. They render a coexistence of three populations unstable even if the coexistence is stable in the deterministic limit. We determine the mean time to extinction (MTE) by using a WKB-ansatz in the master equation that represents the stochastic description of this model. This way, the calculation is reduced to a problem of classical mechanics and amounts to solving a Hamilton–Jacobi equation with zero-energy Hamiltonian. We solve the corresponding Hamilton’s equations of motion in six-dimensional phase space numerically by using the Iterative Action Minimization Method. This allows to project on the optimal path to extinction, starting from a parameter choice where the three-species coexistence-fixed point undergoes a Hopf bifurcation and becomes stable. Specifically for our system of three species, extinction events can be triggered along various paths to extinction, differing in their intermediate steps. We compare our analytical predictions with results from Gillespie simulations for two-species extinctions, complemented by an analytical calculation of the MTE in which the remaining third species goes extinct. From Gillespie simulations we also analyze how the distributions of times to extinction (TE) change upon varying the bifurcation parameter. Our results shed some light on the sensitivity of the TE to system parameters. Even within the same model and the same dynamical regime, which allows a stable coexistence of species in the deterministic limit, the MTE depends on the distance from the bifurcation point in a way that contains the system size dependence in the exponent. It is challenging and worthwhile to quantify how rare the rare events of extinction are.