Simulations discrete-stochastic

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. 

All other discrete stochastic models, obtained from polystochastic chains, attached to an investigated process, present the capacity to be transformed into an asymptotic model. When the original and its asymptotic model are calculated numerically, we can rapidly observe if they converge by direct simulation. In this case, the comparison between the behaviour of the original model and the generator function of the asymptotic stochastic model is not necessary. 

Pardoux, E., and Talay, D., Discretization and Simulation of Stochastic Differential-Equations, Acta Applicandae Mathematicae 3 (1) 23 7 (1985). 

Instead of solving the evolution equation in terms of the orientation tensor, one can simulate the stochastic equation such as Eq. 5.7 for the orientation vector p without the need of closure approximations, using the numerical technique for the simulation of stochastic processes (Ottinger 1996) known as the Brownian dynamics simulation. Once trajectories for aU fibers are obtained, the orientation tensor can be calculated in terms of the ensemble average of the discrete form ... 

Two important challenges exist for multiscale systems. The first is multiple time scales, a problem that is familiar in chemical engineering where it is called stiffness, and we have good solutions to it. In the stochastic world there doesn t seem to be much knowledge of this phenomenon, but I believe that we recently have found a solution to this problem. The second challenge—one that is even more difficult—arises when an exceedingly large number of molecules must be accounted for in stochastic simulation. I think the solution will be multiscale simulation. We will need to treat some reactions at a deterministic scale, maybe even with differential equations, and treat other reactions by a discrete stochastic method. This is not an easy task in a simulation. 

Fig. 9. Fluxes across six set of reactions in LPG biosynthetic pathway of L. rmjor predicted through discrete stochastic simulation event.

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. 

Figure 18.1 Regimes of the problem space for multiscale stochastic simulations of chemical reaction kinetics. The r-axis represents the number of molecules of reacting species, x, and the ) -axis measures the frequency of reaction events, A. The threshold variables demarcate the partitions of modeling formalisms. In area I, the number of molecules is so small and the reaction events are so infrequent that a discrete-stochastic simulation algorithm, like the SSA, is needed. In contrast, in area V, which extends to infinity, the thermodynamic limit assumption becomes vahd and a continuous-deterministic modehng formalism becomes valid. Other areas admit different modehng formalisms, such as ones based on chemical Langevin equations, or probabilistic steady-state assumptions.

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). 

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. 

V, ip, x, and t) in the PDF transport equation makes it intractable to solve using standard discretization methods. Instead, Lagrangian PDF methods (Pope 1994a) can be used to express the problem in terms of stochastic differential equations for so-called notional particles. In Chapter 7, we will discuss grid-based Eulerian PDF codes which also use notional particles. However, in the Eulerian context, a notional particle serves only as a discrete representation of the Eulerian PDF and not as a model for a Lagrangian fluid particle. The Lagrangian Monte-Carlo simulation methods discussed in Chapter 7 are based on Lagrangian PDF methods. 

In this section, we consider the description of Brownian motion by Markov diffusion processes that are the solutions of corresponding stochastic differential equations (SDEs). This section contains self-contained discussions of each of several possible interpretations of a system of nonlinear SDEs, and the relationships between different interpretations. Because most of the subtleties of this subject are generic to models with coordinate-dependent diffusivities, with or without constraints, this analysis may be more broadly useful as a review of the use of nonlinear SDEs to describe Brownian motion. Because each of the various possible interpretations of an SDE may be defined as the limit of a discrete jump process, this subject also provides a useful starting point for the discussion of numerical simulation algorithms, which are considered in the following section. 

Brownian Dynamics (BD) methods treat the short-term behavior of particles influenced by Brownian motion stochastically. The requirement must be met that time scales in these simulations are sufficiently long so that the random walk approximation is valid. Simultaneously, time steps must be sufficiently small such that external force fields can be considered constant (e.g., hydrodynamic forces and interfacial forces). Due to the inclusion of random elements, BD methods are not reversible as are the MD methods (i.e., a reverse trajectory will not, in general, be the same as the forward using BD methods). BD methods typically proceed by discretization and integration of the equation for motion in the Langevin form... 

Discrete models treat individual atoms, molecules, or particles and can be deterministic or stochastic. Examples of the former include MD simulations. Examples of the latter are various MC methods, BD, DPD, DSMC, and LB simulations. There are different ensembles in which these simulations can be performed, depending on the quantities that one is interested in computing. 

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. 

Furthermore, it has recently been found that the discrete nature of a molecule population leads to qualitatively different behavior than in the continuum case in a simple autocatalytic reaction network [29]. In a simple autocatalytic reaction system with a small number of molecules, a novel steady state is found when the number of molecules is small, which is not described by a continuum rate equation of chemical concentrations. This novel state is first found by stochastic particle simulations. The mechanism is now understood in terms of fluctuation and discreteness in molecular numbers. Indeed, some state with extinction of specific molecule species shows a qualitatively different behavior from that with very low concentration of the molecule. This difference leads to a transition to a novel state, referred to as discreteness-induced transition. This phase transition appears by decreasing the system size or flow to the system, and it is analyzed from the stochastic process, where a single-molecule switch changes the distributions of molecules drastically. 

A discrete state stochastic Markov process simulates the movement of the evacuees. Transition from node-to-node is simulated as a random process where the prohahility of transition depends on the dynamically changed states of the destination and origin nodes and on the link between them. Solution of the Markov process provides the expected distribution of the evacuees in the nodes of the area as a function of time. 

The simulation of lignin liquefaction combined a stochastic interpretation of depolymerization kinetics with models for catalyst deactivation and polymer diffusion. The stochastic model was based on discrete mathematics, which allowed the transformations of a system between its discrete states to be chronicled by comparing random numbers to transition probabilities. The transition probability was dependent on both the time interval of reaction and a global reaction rate constant. McDermott s ( analysis of the random reaction trajectory of the linear polymer shown in Figure 6 permits illustration. 

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