Stochastic models definition

They point out that at the heart of technical simulation there must be unreality otherwise, there would not be need for simulation. The essence of the subject linder study may be represented by a model of it that serves a certain purpose, e.g., the use of a wind tunnel to simulate conditions to which an aircraft may be subjected. One uses the Monte Carlo method to study an artificial stochastic model of a physical or mathematical process, e.g., evaluating a definite integral by probability methods (using random numbers) using the graph of the function as an aid. 

For the case 5=1 and D = 1 the results of the stochastic model are in good agreement with the CA model y = 0.262). This is understandable because the different definition of the reaction which leads to a difference in the blocking of activated sites cannot play significant role because all sites are activated. The diffusion rate of D = 10 leads nearly to the same reactivity as if we define the reaction between the nearest-neighbour particles. If the diffusion rate is considerably lowered (D = 0.1), the behaviour of the system changes completely because of the decrease of the reaction probability. This leads to the disappearance of the kinetic phase transition at y because different types of particles may reside on the surface as the nearest neighbours without reaction, a case which does not occur at all in the CA approach. 

This definition of diffusion coefficients considers the non-isotropic diffusion behaviour in some materials. So, this stochastic modelling can easily be applied for the analysis of the oriented diffusion phenomena occurring in materials with designed properties for directional transport. 

In the Mint model, we have to take into account the following considerations (i) the initial filtration coefficient Xq, which is a parameter, presents a constant value after time and position (ii) the detachment coefficient, which is another constant parameter (iii) the quantity of the suspension treated by deep filtration depends on the quantity of the deposited solid in the bed this dependency is the result of the definition of the filtration coefficient (iv) the start of the deep bed filtration is not accompanied by an increase in the filtration efficiency. These considerations stress the inconsistencies of the Mint model 1. valid especially when the saturation with retained microparticles of the fixed bed is slow 2. unfeasible to explain the situations where the detachment depends on the retained solid concentration and /or on the flowing velocity 3. unfeasible when the velocity of the mobile phase inside the filtration bed, varies with time this occurrence is due to the solid deposition in the bed or to an increasing pressure when the filtration occurs with constant flow rate. Here below we come back to the development of the stochastic model for the deep filtration process. 

For the transformation of the stochastic model into a form, such as the Mint model, that allows the computation of Cx,g(x, r)/Cvo, we consider that this ratio gives a measure of the probability to locate the microparticle in the specified position P(x,t) = Pi(x,t) + P2(x,t). We can simplify our equations by eliminating probabilities Pi(x, r) and P2(x,t) with the use of this last definition and the rela-... 

For K 2kT and /Km, the collinear result of Eq. (5.13) differs from the three-dimensional model of this work by a factor of 3. This difference is interpreted in terms of the projection of forces along the internuclear axis. The slightly different kinematic factors arise, in part, from the definition of the collision frequency that is used to derive, Eq. (5.11). The hard-sphere model gave excellent agreement with simulations for a very steep exponential repulsive potential with exponent 2a = 256h, where b is that of the Morse oscillator. It is to be remembered that Eq. (5.12) was derived from a stochastic model with three major assumptions ... 

A. Stochastic Difference in Time Definition A Stochastic Model for a Trajectory Weights of Trajectories and Sampling Procedures Mean Field Approach, Fast Equilibration, and Molecular Labeling Stochastic Difference Equation in Length Fractal Refinement of Trajectories Parameterized by Length... 

Stimulation, environmental vs. task, 1357, 1358 STL (stereo lithography format), 208 Stochastic approximation, 2634-2635 Stochastic counterpart method, 2635 Stochastic decision trees, 2384, 2385 Stochastic models, 2146-2170 benefits of mathematical analysis of, 2146 definition of, 2146, 2150 Markov chains, 2150-2156 in continuous time, 2154-2156 and Markov property, 2150-2151 queueing model based on, 2153-2154... 

The exact definition of the model will be done very easily using the metalanguage of formal reaction kinetics. The stochastic model of the reaction to be defined below will be the stochastic model of the phenomena of eutrophication. Therefore the only questions worth treating here are ... 

This contribution reviews the basic tools which are currently employed for interpreting ESR and NMR observables in condensed phases, with an emphasis on stochastic modeling as key for the prediction of continuous-wave ESR (cw-ESR) lineshapes and NMR relaxation times of proteins. Section 12.2 is therefore devoted to the definition of reduced (effective) magnetic Hamiltonians and the stochastic (Liouville) approach to spin/molecular dynamics in order to clarify the basic stochastic approach to cw-ESR observables. Section 12.3 provides a short overview of rotational stochastic models for the evaluation of relaxation NMR data in biomolecules. Conclusions are briefly summarized in Section 12.4. 

Deterministic models or elements of models are those in which each variable and parameter can be assigned a definite fixed number, or a series of numbers, for any given set of conditions, i.e. the model has no components that are inherently uncertain. In contrast, the principle of uncertainty is introduced in stochastic or probabilistic models. The variables or parameters used to describe the input-output relationships and the structure of the elements (and the constraints) are not precisely known. A stochastic model involves parameters characterized by probability distributions. Due to this the stochastic model will produce different results in each reahzation. 

Clearly, the proper definition of the power-spectral density functirai is the crucial step to address the stochastic modeling of the seismic action. In the context of defining spectrum-compatible ground motion processes, the issue consists in determining which is the inverse relationship between the power-spectral density function and the target response spectrum. It has to be emphasized that the evaluation of the response-spectrum-compatible power-spectral... 

Clearly, the proper definition of the power-spectral density matrix is the cmcial step to address for the stochastic modeling of the... 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

This chapter covers a variety of topics related to the class of probabilistic CA (PCA) i.e. CA that involve some elements of probability in their state-space definition and/or time-evolution. We begin with a physicist s overview of critical phenomena, then move on to discuss the equivalence between PCA and spin models, critical behavior of PCA, mean-field theory, and CA simulation of conventional spin models. The chapter concludes with a discussion of a stochastic version of Conway s Life rule. 

The Monte Carlo method permits simulation, in a mathematical model, of stochastic variation in a real system. Many industrial problems involve variables which are not fixed in value, but which tend to fluctuate according to a definite pattern. For example, the demand for a given product may be fairly stable over a long time period, but vary considerably about its mean value on a day-to-day basis. Sometimes this variation is an essential element of the problem and cannot be ignored. 

In his interesting paper Professor Nicolis raises the question whether models can be envisioned which lead to a spontaneous spatial symmetry breaking in a chemical system, leading, for example, to the production of a polymer of definite chirality. It would be even more interesting if such a model would arise as a result of a measure preserving process that could mimic a Hamiltonian flow. Although we do not have such an example of a chiral process, which imbeds an axial vector into the polymer chain, several years ago we came across a stochastic process that appears to imbed a polar vector into a growing infinite chain. 

Along with the isomerism of linear copolymers due to various distributions of different monomeric units in their chains, other kinds of isomerisms are known. They can appear even in homopolymer molecules, provided several fashions exist for a monomer to enter in the polymer chain in the course of the synthesis. So, asymmetric monomeric units can be coupled in macromolecules according to "head-to-tail" or "head-to-head"—"tail-to-tail" type of arrangement. Apart from such a constitutional isomerism, stereoisomerism can be also inherent to some of the polymers. Isomers can sometimes substantially vary in performance properties that should be taken into account when choosing the kinetic model. The principal types of such an account are analogous to those considered in the foregoing. The only distinction consists in more extended definition of possible states of a stochastic process of conventional movement along a polymer chain. 

The definition of the hi hazard rates and the model of (9.33) are the only required hypotheses to formulate the stochastic movement or reaction of particles in a spatial homogeneous mixture of m-particle populations interacting through m0 reactions. 

Dhar modelled the stretching of a polymer using the stochastic Rouse model, for which distributions of various definitions of the work can be obtained. Two mechanisms for the stretching were considered one where the force on the end of the polymer was constrained and the other where its end was constrained. Dhar commented that the variable selected for the work was only clearly identified as the entropy production in the latter case. In the former case they argue that the average work is non-zero for an adiabatic process, and therefore should not be considered as an entropy production, however we note that the expression is consistent with a product of flux and field as used in linear irreversible thermodynamics. 

---