Stochastic discrete modeling

Figure 11.3 State transition diagram for stochastic discrete model for the single unimolecular reaction of Equation (11.5). It is assumed that there is a total of N molecules in the system k is the number of molecules in state A. The intrinsic rate constants are Ai>2.

Linear growth In a simple stochastic (discrete) model for linear growth, represented by... 

On the other hand, if the numbers of reacting molecules are very small, for example in the order of O(10 iV ), then integer numbers of molecules must be modeled along with discrete changes upon reaction. Importantly, the reaction occurrences can no longer be considered deterministic, but probabilistic. In this chapter we present the theory to treat reacting systems away from the thermodynamic limit. We first present a brief overview of continuous-deterministic chemical kinetics models and then discuss the development of stochastic-discrete models. 

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. 

In the CME framework of discrete molecular numbers, X changes one by one stochastically. The stochastic CME model follows the master equation graph of Figure 11.5 ... 

The stochastic models can present discrete or continuous forms. The former discussion was centred on discrete models. The continuous models are developed according to the same base as the discrete ones. Example 4.3.1 has already shown this method, which leads to a continuous stochastic model. This case can be gen-... 

Chapter 4 is devoted to the description of stochastic mathematical modelling and the methods used to solve these models such as analytical, asymptotic or numerical methods. The evolution of processes is then analyzed by using different concepts, theories and methods. The concept of Markov chains or of complete connected chains, probability balance, the similarity between the Fokker-Plank-Kolmogorov equation and the property transport equation, and the stochastic differential equation systems are presented as the basic elements of stochastic process modelling. Mathematical models of the application of continuous and discrete polystochastic processes to chemical engineering processes are discussed. They include liquid and gas flow in a column with a mobile packed bed, mechanical stirring of a liquid in a tank, solid motion in a liquid fluidized bed, species movement and transfer in a porous media. Deep bed filtration and heat exchanger dynamics are also analyzed. 

Another possibility to quantify the response of a stochastic system to periodic signals is to generalize the notion of synchronization, which is known from deterministic nonlinear oscillators. We will pursue this idea in what follows. To this end we review in section 2.2 the notion of effective synchronization in stochastic systems. The mean number of synchronized system cycles turns out to be an appropriate quantity to characterize the synchronization properties of the system to the periodic signal. However the task remains to calculate this quantity. This calculation will be based on discrete renewal models for bistable and excitable dynamics. These discrete models are introduced in section 2.3. We first recapitulate the well known two state model for the stochastic dynamics of an overdamped particle in a doublewell system [10] and afterwards introduce a phenomenological discrete model for excitable dynamics. In section 2.4 a theory to calculate the mean frequency and effective diffusion coefficient in periodically driven renewal processes is presented. These two quantities allow to calculate the mean number of synchronized cycles. Finally in section 2.5 we apply this theory to investigate synchronization in bistable and excitable systems. 

One possibility to simplify a continuous stochastic system is the reduction to a description in terms of a few discrete states. The system s behavior is then specified by the transition times between these discrete states. For example when investigating a neurons behavior, the important aspect are often only the times when a spike is emitted and not the complex evolution of the membrane potential [16]. In a doublewell potential system, depending on the questions asked, it may be sufficient to know in which of the two wells the system is located, neglecting the fluctuations in the wells as well as the actual dynamics when crossing from one well to the other. In these cases a reduction to a discrete description can be considered as an appropriate simplification. We first review the two state description of bistable systems [10] and then introduce a phenomenological discrete model for excitable dynamics. 

We have presented a method to calculate the mean frequency and effective diffusion coefficient of the numbers of cycles(events) in periodically driven renewal processes. Based on these two quantities one can evaluate the number of locked cycles in order to quantify stochastic synchronization. Applied to a discrete model of bistable dynamics the theory can be evaluated analytically. The system shows only 1 1 synchronization, however in contrast to spectral based stochastic resonance measures the mean number of locked cycles has a maximum at an optimal driving frequency, i.e. the system shows bona fide resonance [6]. For the discrete model of... 

AD-equation a Peclet number of Pe=10 is used. This gives a spread in the RTD (dispersion) that is within the range of observed values in the field. The standard deviation of the transmissivities of the channels in the network model was chosen such that the RTD of a nonsorbing tracer also has a Pe=10. In the time scale presented there seem not be dramatic differences between the model results for the Network and the AD-models. It should be noted, however, that the early arrival times are of special interest for radionuclides that decay. There the differences are considerable. Similar results were obtained in comparisons of different models using very similar data bases (Selroos et a. 2002). In that comparison a discrete fracture network model, a channel network model and a stochastic continuum model were used. 

Cacas, M. C., Ledoux, E., de Marsily, G., Tillie, B., Barbreau, B., Durand, A., Feuga B. and Peaudecerf, P. 1990. Modeling fracture flow with a stochastic discrete fracture network Calibration and validation, 1, The flow model. Water Resour. Res., 26(3), pp. 479-489. 

To estimate the Vasicek continuous stochastic time model, the model must be discretized. We discretized and estimated the continuous time model as follows ... 

One of the most widely used approaches for the simulation of sprays is the stochastic discrete droplet model introduced by Williams [30]. In this approach, the droplets are described by a probability density fxmction (PDF),/(t,X), which represents the probable number of droplets per unit volume at time t and in state X. The state of a droplet is described by its parameters that are the coordinates in the particle state space. Typically, the state parameters include the location x, the velocity v, the radius r, the temperature Td, the deformation parameter y, and the rate of deformation y. As discussed in more detail in Chapter 16, this spray PDF is the solution of a spray transport equation, which in component form is given by... 

Keywords Atomization Chemical reactions Craiservation equations Constitutive equations Drop breakup Drop deformation Drop collisions Evaporation LES Newtonian fluids RANS Spray modeling Spray PDF Stochastic discrete particle method Source terms Turbulence... 

Drop size distributions are typically described using raie of four methods empirical, maximum entropy formalism (MEF), discrete probability function (DPF) method, or stochastic. The empirical method was most popular before about the year 2000, when drop size distributions were usually determined by fitting spray data to predetermined mathematical functions. Problems arose when extrapolating to regimes outside the range of experimental data. Two analytical approaches were proposed to surmount this, MEF and DPF, as well as one numerical approach, the stochastic breakup model. 

A great amount of stochastic physics investigates the approximation of jump processes by diffusion processes, i.e. of the master equation by a Fokker-Planck equation, since the latter is easier to solve. The rationale behind this procedure is the fact that the usual deterministic (CCD) and stochastic (CDS) models differ from each other in two aspects. The CDS model offers a stochastic description with a discrete state space. In most applications, where the number of particles is large and may approach Avogadro s number, the discreteness should be of minor importance. Since the CCD model adopts a continuous state-space, it is quite natural to adopt CCS model as an approximation for fluctuations. 

An example fn>m the literature (Alperovits Shamir, 1977) is presented. The results show the improved behaviour, particularly in terms of robustness and consistency, achieved through the combination of the stochastic optimization of a discrete model, user interaction, and rigorous minlp solution. 

What is the relationship between the molecular dynamics simulations of a continuous model and an isothermal Monte Carlo trajectory of an otherwise similar discretized (or lattice) model When only local (and small distance) moves are applied in a properly controlled random (or rather pseudorandom) scheme, the discrete models mimic the coarse-grained Brownian dynamics of the chain. The Monte Carlo trajectory could be then interpreted as the numerical solution to a stochastic equation of motion. Of course, the short-time dynamics... 

The theory of probabilistic languages was developed by Garg, Kumar Marcus (1999) in order to model the stochastic Discrete Event Systems (DES) behavior. 

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