scholarly journals On Differences between Deterministic and Stochastic Models of Chemical Reactions: Schlögl Solved with ZI-Closure

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 678 ◽  
Author(s):  
Michail Vlysidis ◽  
Yiannis Kaznessis
Keyword(s):  
Stochastic Models ◽  
Deterministic Model ◽  
Kinetic Constant ◽  
Stable Solution ◽  
System Size ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
Deterministic Models ◽  
Chemical Reaction Kinetics ◽  
Schlögl Model

Deterministic and stochastic models of chemical reaction kinetics can give starkly different results when the deterministic model exhibits more than one stable solution. For example, in the stochastic Schlögl model, the bimodal stationary probability distribution collapses to a unimodal distribution when the system size increases, even for kinetic constant values that result in two distinct stable solutions in the deterministic Schlögl model. Using zero-information (ZI) closure scheme, an algorithm for solving chemical master equations, we compute stationary probability distributions for varying system sizes of the Schlögl model. With ZI-closure, system sizes can be studied that have been previously unattainable by stochastic simulation algorithms. We observe and quantify paradoxical discrepancies between stochastic and deterministic models and explain this behavior by postulating that the entropy of non-equilibrium steady states (NESS) is maximum.

Model-Based Analysis of the Surface Generation in Microendmilling—Part II: Experimental Validation and Analysis

2006 ◽  
Vol 129 (3) ◽  
pp. 461-469 ◽  
Author(s):  
Xinyu Liu ◽  
Richard E. DeVor ◽  
Shiv G. Kapoor
Keyword(s):  
Surface Roughness ◽  
Stochastic Models ◽  
Feed Rate ◽  
Deterministic Model ◽  
Surface Generation ◽  
Tool Geometry ◽  
Process Conditions ◽  
Deterministic Models ◽  
Floor Surface ◽  
Wide Range

The surface-generation models for the microendmilling process developed in Part I (Liu, DeVor, and Kapoor, 2007, J. Manuf. Sci. Eng., 129(3), pp. 453–460) are experimentally calibrated and validated. Partial immersion peripheral downmilling and full-immersion slotting tests are performed over a wide range of feed rates (0.25–12μm∕flute) using two tools with different edge radii (3μm and 2μm) and runout levels (2μm and 3μm) for the investigation of sidewall and floor surface generation, respectively. The deterministic models are validated using large feed-rate tests with errors within 18% for both sidewall and floor surfaces. For low feed-rate tests, the stochastic portion of the surface roughness data are determined from the observed roughness data and the validated deterministic model. The stochastic models are then calibrated and validated using independent data sets. The combination of the deterministic and stochastic models predicts the total surface roughness within 15% for both the sidewall and floor surface over a range of feed rates. The models are then used to simulate micromachined surfaces under a variety of conditions to gain a deeper understanding of the effects of tool geometry (edge radius and edge serration), process conditions, tool tip runout, process kinematics and dynamics on the machined surface roughness.

STATIONARY PROBABILITY DISTRIBUTIONS FOR FITZHUGH-NAGUMO SYSTEMS

2003 ◽  
Vol 03 (02) ◽  
pp. L155-L166 ◽  
Author(s):  
MARCIN KOSTUR ◽  
XAVER SAILER ◽  
LUTZ SCHIMANSKY-GEIER
Keyword(s):  
Probability Distribution ◽  
Noise Intensity ◽  
Additive Noise ◽  
Deterministic Model ◽  
Probability Distributions ◽  
Saddle Points ◽  
Planck Equation ◽  
Stationary Solutions ◽  
Stationary Probability ◽  
Stationary Probability Distribution

We study numerically the stationary solutions of the Fokker-Planck equation for the FitzHugh-Nagumo model with additive noise. In the parameter regimes where the deterministic model is excitable we find various sets of maxima, minima, and saddle points of the stationary probability distribution depending on the noise intensity and separation of the time scales between activator and inhibitor.

scholarly journals Determinism

2021 ◽  
Vol 20 (5) ◽  
pp. 1-34
Author(s):  
Edward A. Lee
Keyword(s):  
Quantum Physics ◽  
Deterministic Model ◽  
Physical World ◽  
Deterministic Models ◽  
Newtonian Physics ◽  
Physical Systems ◽  
Newton’S Laws ◽  
And Behavior

This article is about deterministic models, what they are, why they are useful, and what their limitations are. First, the article emphasizes that determinism is a property of models, not of physical systems. Whether a model is deterministic or not depends on how one defines the inputs and behavior of the model. To define behavior, one has to define an observer. The article compares and contrasts two classes of ways to define an observer, one based on the notion of “state” and another that more flexibly defines the observables. The notion of “state” is shown to be problematic and lead to nondeterminism that is avoided when the observables are defined differently. The article examines determinism in models of the physical world. In what may surprise many readers, it shows that Newtonian physics admits nondeterminism and that quantum physics may be interpreted as a deterministic model. Moreover, it shows that both relativity and quantum physics undermine the notion of “state” and therefore require more flexible ways of defining observables. Finally, the article reviews results showing that sufficiently rich sets of deterministic models are incomplete. Specifically, nondeterminism is inescapable in any system of models rich enough to encompass Newton’s laws.

scholarly journals On the analysis of the limited resources queuing system under MAP arrivals

2018 ◽  
Vol 16 ◽  
pp. 01008
Author(s):  
Eduard Sopin ◽  
Konstantin Samouylov
Keyword(s):  
Blocking Probability ◽  
Queuing System ◽  
Limited Resources ◽  
System Of Equations ◽  
Stationary Probability ◽  
Equilibrium System ◽  
Vector Form ◽  
Stationary Probability Distribution ◽  
Resource Requirements ◽  
Resource Allocation Schemes

In the paper, we analyse a multiserver queuing system with discrete limited resources and random resource requirements under MAP arrivals, which can adequately model resource allocation schemes in the contemporary wireless networks. The equilibrium system of equations is derived in the vector form and is solved numerically. With stationary probability distribution, we provide formulas for the average and the variance of the occupied resources, as well as for the blocking probability. The results are illustrated by a numerical example.

scholarly journals Estimating the distribution of time to extinction of infectious diseases in mean-field approaches

2020 ◽  
Author(s):  
Maryam Aliee ◽  
Kat S. Rock ◽  
Matt J. Keeling
Keyword(s):  
Infectious Diseases ◽  
Stochastic Models ◽  
Mean Field ◽  
Practical Method ◽  
Deterministic Models ◽  
Recovery Processes ◽  
Time To Extinction ◽  
And Control ◽  
Field Approaches ◽  
Birth Death

AbstractA key challenge for many infectious diseases is to predict the time to extinction under specific interventions. In general this question requires the use of stochastic models which recognise the inherent individual-based, chance-driven nature of the dynamics; yet stochastic models are inherently computationally expensive, especially when parameter uncertainty also needs to be incorporated. Deterministic models are often used for prediction as they are more tractable, however their inability to precisely reach zero infections makes forecasting extinction times problematic. Here, we study the extinction problem in deterministic models with the help of an effective “birth-death” description of infection and recovery processes. We present a practical method to estimate the distribution, and therefore robust means and prediction intervals, of extinction times by calculating their different moments within the birth-death framework. We show these predictions agree very well with the results of stochastic models by analysing the simplified SIS dynamics as well as studying an example of more complex and realistic dynamics accounting for the infection and control of African sleeping sickness (Trypanosoma brucei gambiense).

Optimization models for managing limited resources in logistics

2021 ◽  
Author(s):  
Aleksandr Mischenko ◽  
Anastasiya Ivanova
Keyword(s):  
Stochastic Models ◽  
Optimization Models ◽  
Limited Resources ◽  
Model Parameters ◽  
Systems Modeling ◽  
Deterministic Models ◽  
Logistics Systems ◽  
Industrial Enterprises ◽  
The Stability ◽  
Senior Students

In the proposed monograph, optimization models for managing limited resources in logical systems are considered. Such systems are primarily used by industrial enterprises, transport companies and trade organizations, including those that carry out wholesale activities. As a rule, the efficiency of these objects largely depends on how rational use of limited resources such as: consumer camera business, labor, vehicles, etc. In this paper, various approaches to managing such resources are considered both for deterministic models and for the situation when a number of model parameters are not specified exactly, that is, for stochastic models. In this case, it is proposed to evaluate the stability of models to the occurrence of various types of risk events, both by the structure of the solution and by the functionality. It is addressed to senior students, postgraduates and masters studying in the specialty "Management" and "Logistics", as well as specialists in the field of logistics systems modeling.

scholarly journals Translating environmental xenobiotic fate models across scales

1997 ◽  
Vol 1 (4) ◽  
pp. 895-904 ◽  
Author(s):  
O. Richter ◽  
B. Diekkrüger
Keyword(s):  
Stochastic Models ◽  
Process Models ◽  
Spatial Process ◽  
Soil Parameters ◽  
Deterministic Models ◽  
Deterministic Process ◽  
Statistical Variation ◽  
Scale Models ◽  
Classical Models ◽  
Environmental Xenobiotic

Abstract. The classical models developed for degradation and transport of xenobiotics have been derived with the assumption of homogeneous environments. Unfortunately, deterministic models function well in the laboratory under homogeneous conditions but such homogeneous conditions often do not prevail in the field. A possible solution is the incorporation of the statistical variation of soil parameters into deterministic process models. This demands the development of stochastic models of spatial variability. To this end, spatial soil parameter fields are conceived as the realisation of a random spatial process. Extrapolation of local fine scale models to large heterogeneous fields is achieved by coupling deterministic process models with random spatial field models.

The censored Markov chain and the best augmentation

1996 ◽  
Vol 33 (03) ◽  
pp. 623-629 ◽  
Author(s):  
Y. Quennel Zhao ◽  
Danielle Liu
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
The Best Approximation ◽  
Finite Matrix ◽  
Countable State ◽  
Stationary Probabilities

Computationally, when we solve for the stationary probabilities for a countable-state Markov chain, the transition probability matrix of the Markov chain has to be truncated, in some way, into a finite matrix. Different augmentation methods might be valid such that the stationary probability distribution for the truncated Markov chain approaches that for the countable Markov chain as the truncation size gets large. In this paper, we prove that the censored (watched) Markov chain provides the best approximation in the sense that, for a given truncation size, the sum of errors is the minimum and show, by examples, that the method of augmenting the last column only is not always the best.

STATISTICAL PROPERTIES OF INTENSITY FLUCTUATION OF SATURATION LASER MODEL DRIVEN BY CROSS-CORRELATED ADDITIVE AND MULTIPLICATIVE NOISES

2010 ◽  
Vol 24 (14) ◽  
pp. 2175-2188 ◽  
Author(s):  
PING ZHU ◽  
YI JIE ZHU
Keyword(s):  
Probability Distribution ◽  
Cross Correlation ◽  
Laser Intensity ◽  
Laser System ◽  
Stationary Probability ◽  
Stationary Probability Distribution ◽  
Model Driven ◽  
Multiplicative Noises ◽  
Laser Model ◽  
The Stability

Statistical properties of the intensity fluctuation of a saturation laser model driven by cross-correlation additive and multiplicative noises are investigated. Using the Novikov theorem and the projection operator method, we obtain the analytic expressions of the stationary probability distribution Pst(I), the relaxation time Tc, and the normalized variance λ2(0) of the system. By numerical computation, we discussed the effects of the cross-correlation strength λ, the cross-correlation time τ, the quantum noise intensity D, and the pump noise intensity Q for the fluctuation of the laser intensity. Above the threshold, λ weakens the stationary probability distribution, speeds up the startup velocity of the laser system from start status to steady work, and attenuates the stability of laser intensity output; however, τ strengthens the stationary probability distribution and strengths the stability of laser intensity output; when λ < 0, τ speeds up the startup; on the contrast, when λ > 0, τ slows down the startup. D and Q make the relaxation time exhibit extremum structure, that is, the startup time possesses the least values. At the threshold, τ cannot generate the effects for the saturation laser system, λ expedites the startup velocity and weakens the stability of laser intensity output. Below threshold, the effects of λ and τ not only relate to λ and τ, but also relate to other parameters of the system.

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