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Simulations continuous-stochastic

Given the simplicity of the current system, it was possible to analytically solve the resulting chemical master equation. However, this is not always the case and one is limited to simulating individual realizations of the stochastic process in order to reconstruct the probability distributions out from several simulations. Below, we introduce the celebrated Gillespie algorithm (Gillespie 1977) to simulate the stochastic evolution of continuous-time discrete-state stochastic processes, like the one analyzed in the present chapter. [Pg.21]

For certain types of stochastic or random-variable problems, the sequence of events may be of particular importance. Statistical information about expected values or moments obtained from plant experimental data alone may not be sufficient to describe the process completely. In these cases, computet simulations with known statistical iaputs may be the only satisfactory way of providing the necessary information. These problems ate more likely to arise with discrete manufactuting systems or solids-handling systems rather than the continuous fluid-flow systems usually encountered ia chemical engineering studies. However, there ate numerous situations for such stochastic events or data ia process iadustries (7—10). [Pg.73]

A final comment on the interpretation of stochastic simulations We are so accustomed to writing continuous functions—differential and integrated rate equations, commonly called deterministic rate equations—that our first impulse on viewing these stochastic calculations is to interpret them as approximations to the familiar continuous functions. However, we have got this the wrong way around. On a molecular level, events are discrete, not continuous. The continuous functions work so well for us only because we do experiments on veiy large numbers of molecules (typically 10 -10 ). If we could experiment with very much smaller numbers of molecules, we would find that it is the continuous functions that are approximations to the stochastic results. Gillespie has developed the stochastic theory of chemical kinetics without dependence on the deterministic rate equations. [Pg.114]

After years of experiments and the nonetheless effective use of these tools making it possible to refine and systematically optimise control of facilities, the tools due to increased performance and possibilities made available by software and other computer equipment, has been improved by increasing the calculation horizon, beyond two gas years by using adapted models (closer to classic reservoir models by taking into account a simplified physical representation). At the present stage with R D works still continuing, they now include more and more complexity (risk management, stochastic simulations, etc), and works are focused on ... [Pg.190]

A computational method was developed by Gillespie in the 1970s [381, 388] from premises that take explicit account of the fact that the time evolution of a spatially homogeneous process is a discrete, stochastic process instead of a continuous, deterministic process. This computational method, which is referred to as the stochastic simulation algorithm, offers an alternative to the Kolmogorov differential equations that is free of the difficulties mentioned above. The simulation algorithm is based on the reaction probability density function defined below. [Pg.267]

In the light of the previous discussion it is quite apparent that a detailed mathematical simulation of the combined chemical reaction and transport processes, which occur in microporous catalysts, would be highly desirable to support the exploration of the crucial parameters determining conversion and selectivity. Moreover, from the treatment of the basic types of catalyst selectivity in multiple reactions given in Section 6.2.6, it is clear that an analytical solution to this problem, if at all possible, will presumably not favor a convenient and efficient treatment of real world problems. This is because of the various assumptions and restrictions which usually have to be introduced in order to achive a complete or even an approximate solution. Hence, numerical methods are required. Concerning these, one basically has to distinguish between three fundamentally different types, namely molecular-dynamic models, stochastic models, and continuous models. [Pg.360]

After completing a step (a jump in state) the corresponding outward rate constants for the new state are now all different. To continue the simulation, we draw another two random numbers, make another move, and so on. A stochastic trajectory is thus obtained. One notes that the trajectory has randomly variable time steps, a feature indicative of the Gillespie algorithm. [Pg.278]

For consequence analysis, we have developed a dynamic simulation model of the refinery SC, called Integrated Refinery In-Silico (IRIS) (Pitty et al., 2007). It is implemented in Matlab/Simulink (MathWorks, 1996). Four types of entities are incorporated in the model external SC entities (e.g. suppliers), refinery functional departments (e.g. procurement), refinery units (e.g. crude distillation), and refinery economics. Some of these entities, such as the refinery units, operate continuously while others embody discrete events such as arrival of a VLCC, delivery of products, etc. Both are considered here using a unified discrete-time model. The model explicitly considers the various SC activities such as crude oil supply and transportation, along with intra-refinery SC activities such as procurement planning, scheduling, and operations management. Stochastic variations in transportation, yields, prices, and operational problems are considered. The economics of the refinery SC includes consideration of different crude slates, product prices, operation costs, transportation, etc. The impact of any disruptions or risks such as demand uncertainties on the profit and customer satisfaction level of the refinery can be simulated through IRIS. [Pg.41]

Juxtaposition seal can be complicated by a number of factors. These include unsystematic lateral lithologic variations, for instance those of channel belts. While it is common to model such lithologic variations stochastically for input to reservoir simulations, this is not to our knowledge used in fault seal analysis. For small fault throws any continuous (i.e., not wedged out) sand horizons may provide across-fault connectivity which could be misinterpreted as a seal if interpreted by juxtaposition analysis alone. Other complications relate to so-called damage and relay zones. [Pg.151]

It is often stated that MC methods lack real time and results are usually reported in MC events or steps. While this is immaterial as far as equilibrium is concerned, following real dynamics is essential for comparison to solutions of partial differential equations and/or experimental data. It turns out that MC simulations follow the stochastic dynamics of a master equation, and with appropriate parameterization of the transition probabilities per unit time, they provide continuous time information as well. For example, Gillespie has laid down the time foundations of MC for chemical reactions in a spatially homogeneous system.f His approach is easily extendable to arbitrarily complex computational systems when individual events have a prescribed transition probability per unit time, and is often referred to as the kinetic Monte Carlo or dynamic Monte Carlo (DMC) method. The microscopic processes along with their corresponding transition probabilities per unit time can be obtained via either experiments such as field emission or fast scanning tunneling microscopy or shorter time scale DFT/MD simulations discussed earlier. The creation of a database/lookup table of transition... [Pg.1718]

Complex pharmacokinetic/pharmacodynamic (PK/PD) simulations are usually developed in a modular manner. Each component or subsystem of the overall simulation is developed one-by-one and then each component is linked to run in a continuous manner (see Figure 33.2). Simulation of clinical trials consists of a covariate model and input-output model coupled to a trial execution model (10). The covariate model defines patient-specific characteristics (e.g., age, weight, clearance, volume of distribution). The input-output model consists of all those elements that link the known inputs into the system (e.g., dose, dosing regimen, PK model, PK/PD model, covariate-PK/PD relationships, disease progression) to the outputs of the system (e.g., exposure, PD response, outcome, or survival). In a stochastic simulation, random error is introduced into the appropriate subsystems. For example, between-subject variability may be introduced among the PK parameters, like clearance. The outputs of the system are driven by the inputs... [Pg.854]

For a stochastic simulation replicated many times, it would be inefficient to analyze every single trial one by one. Besides, in a stochastic simulation, the analyst is not interested in what happens with a single trial but what happens in the long run. Recall that the probability of observing a sample drawn from a continuous probability distribution is zero that is, the probability of observing the number... [Pg.857]

The correlation function, <-P2[am(0) ( )]>. provides a measure of the internal motions of particular residues in the protein.324 333 Figure 46 shows the results obtained for Trp-62 and Trp-63 from the stochastic boundary molecular dynamics simulations of lysozyme used to analyze the displacement and velocity autocorrelation functions. The net influence of the solvent for both Trp-62 and Trp-63 is to cause a slower decay in the anisotropy than occurs in vacuum. In vacuum, the anisotropy decays to a plateau value of 0.36 to 0.37 (relative to the initial value of 0.4) for both residues within a picosecond. In solution there is an initial rapid decay, corresponding to that found in vacuum, followed by a slower decay (without reaching a plateau value) that continues beyond the period (10 ps) over which the correlation function is ex-... [Pg.151]


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Stochastic simulation

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