Stochastic system

We should stress that the temperature T has nothing to do with the real temperature of either a brain or neural circuit. Its sole purpose is to act as a control parameter regulating the amount of noise in the stochastic system. 

The form of the stochastic transfer function p x) is shown in figure 10.7. Notice that the steepness of the function near a - 0 depends entirely on T. Notice also that this form approaches that of a simple threshold function as T —> 0, so that the deterministic Hopfield net may be recovered by taking the zero temperature limit of the stochastic system. While there are a variety of different forms for p x) satisfying this desired limiting property, any of which could also have been chosen, this sigmoid function is convenient because it allows us to analyze the system with tools borrowed from statistical mechanics. 

Kulkarni, V.G. 1999. Modeling, Analysis, Design and Control of Stochastic Systems. Springer, Berlin. 

Dixon, L. C. W. and L. James. On Stochastic Variable Metric Methods. In Analysis and Optimization of Stochastic Systems. Q. L. R. Jacobs et al. eds. Academic Press, London (1980). 

But the major physical problem remained open Could one prove rigorously that the systems studied before 1979—that is, typically, systems of N interacting particles (with N very large)—are intrinsically stochastic systems In order to go around the major difficulty, Prigogine will take as a starting point another property of dynamical systems integrability. A dynamical system defined as the solution of a system of differential equations (such as the Hamilton equations of classical dynamics) is said to be integrable if the initial value problem of these equations admits a unique analytical solution, weekly sensitive to the initial condition. Such systems are mechanically stable. In order to... 

I now consider statement 3 How should an extension of dynamics be understood In the MPC theory the problem does not exist For the intrinsically stochastic systems there is no need for modifying the laws of dynamics. As for the LPS theory, one notes the presence of two essentially new concepts. The introduction of non-Hilbert functional spaces only concerns the definition of the states of the dynamical system, and not at all the law governing their evolution. It is an important precision introduced in statistical mechanics. The extension of dynamics thus only appears in the operation of regularization of the resonances. This step is also the one that is most difficult to justify rigorously it is related to the (practical) necessity to use perturbation calculus (see Appendix). 

Let us consider a stochastic system described by a generic variable C. This variable may stand for the position of a bead in an optical trap, the velocity field of a fluid, the current passing through a resistance, of the number of native contacts in a protein. A trajectory or path V in configurational space is described by a discrete sequence of configurations in phase space. 

Thus, a deterministic answer assumes that the laws of physics and chemistry have causally and sequentially determined the obligatory series of events leading from inanimate matter to life - that each step is causally linked to the previous one and to the next one by the laws of nature. In principle, in a strictly deterministic situation, the state of a system at any point in time determines the future behavior of the system - with no random influences. In contrast, in a non-deterministic or stochastic system it is not generally possible to predict the future behavior exactly and instead of a linear causal pathway the sequence of steps may be determined by the set of parameters operating at each step. 

In this Section we introduce a stochastic alternative model for surface reactions. As an application we will focus on the formation of NH3 which is described below, equations (9.1.72) to (9.1.76). It is expected that these stochastic systems are well-suited for the description via master equations using the Markovian behaviour of the systems under study. In such a representation an infinite set of master equations for the distribution functions describing the state of the surface and of pairs of surface sites (and so on) arises. As it was told earlier, this set cannot be solved analytically and must be truncated at a certain level. The resulting equations can be solved exactly in a small region and can be connected to a mean-field solution for large distances from a reference point. This procedure is well-suited for the description of surface reaction systems which includes such elementary steps as adsorption, diffusion, reaction and desorption.The numerical part needs only a very small amount of computer time compared to MC or CA simulations. 

Figure 9.20 Block diagram of the deterministic plus stochastic system.

As concerns the former, statistical tests on the measured data are usually adopted to detect any abnormal behavior. In other words, an industrial process is considered as a stochastic system and the measured data are considered as different realizations of the stochastic process. The distribution of the observations in normal operating conditions is different from those related to the faulty process. Early statistical approaches are based on univariate statistical techniques, i.e., the distribution of a monitored variable is taken into account. For instance, if the monitored variable follows a normal distribution, the parameters of interest are the mean and standard deviation that, in faulty conditions, may deviate from their nominal values. Therefore, fault diagnosis can be reformulated as the problem of detecting changes in the parameters of a stochastic variable [3, 30],... 

G. Adomian, Stochastic Systems, Academic Press, New York, 1983. 

Braun, H.A., Huber, M.T., Dewald, M., Schafer, K., and Voigt, K. The neuromodulatory properties of noisy neuronal oscillators . In J.B. Kadke, A. Bulsara, eds., Applied nonlinear dynamics and stochastic systems near the millenium, The American Institute of Physics, Washington, D.C., 1998, pp 281-286. 

In order to elucidate some of the issues in SA of stochastic systems, the gene-expression model proposed in Thattai and van Oudenaarden (2001) and Ozbudak et al. (2002) for transcription and translation, shown schematically in... 

Part II of this book represents the bulk of the material on the analysis and modeling of biochemical systems. Concepts covered include biochemical reaction kinetics and kinetics of enzyme-mediated reactions simulation and analysis of biochemical systems including non-equilibrium open systems, metabolic networks, and phosphorylation cascades transport processes including membrane transport and electrophysiological systems. Part III covers the specialized topics of spatially distributed transport modeling and blood-tissue solute exchange, constraint-based analysis of large-scale biochemical networks, protein-protein interactions, and stochastic systems. 

When mechanistic information is available or obtainable for the components of a system, it is possible to develop detailed analyses and simulations of that system. Such analyses and simulations may be deterministic or stochastic in nature. (Stochastic systems are the subject of Chapter 11.) In either case, the overriding philosophy is to apply mechanistic rules to predict behavior. Often, however, the information required to develop mechanistic models accounting for details such as enzyme and transporter kinetics and precisely predicting biochemical states is not available. Instead, all that may be known reliably about certain large-scale systems is the stoichiometry of the participating reactions. As we shall see in this chapter, this stoichiometric information is sometimes enough to make certain concrete determinations about the feasible operation of biochemical networks. 

Deterministic dynamics of biochemical reaction systems can be visualized as the trajectory of (ci(t), c2(t), , c v(0) in a space of concentrations, where d(t) is the concentration of ith species changing with time. This mental picture of path traced out in the N-dimensional concentration space by deterministic systems may prove a useful reference when we deal with stochastic chemical dynamics. In stochastic systems, one no longer thinks in terms of definite concentrations at time t rather, one deals with the probability of the concentrations being xu x2, , Wy at time t ... 

As a generic description of a stochastic system, consider a system with N possible states, labeled 1, 2, , N. Since the system is stochastic, we cannot define equations that determine the specific state that the system adopts at a specific time. Rather, we look for equations that govern the probability pm(l) that the system is in state in at time t. 

By comparision with the property transport equation the advantage of a stochastic system of equations (SDE) is the capacity for a better adaptation for the numerical integration. 

The upper bound 2. 0 is to be regarded as an artifact of the perturbation criterion we adopted for calculating Q2(x) in Eq. (5.25). In order to check the rehability of our treatment, we carried out a numerical simulation of the stochastic system (5.2). A detailed description of the numerical algorithm is available elsewhere. The comparison between the analytical expression for P(x) and the result of our simulation is illustrated in Fig. 3. We note that the agreement with our predictions is fairly close. The lower bound for j(f) is correctly recovered, while a long tail lingers over the limiting value 2sq. Such a constraint is expected to disappear as we proceed further with our perturbation method. 

For deterministic systems, the FR takes on the form (1.1) and the dissipation function is given by (2.1). However, in the case of stochastic dynamics, the same process might be modelled at different levels with different dynamics, and for each model a different fluctuation relation may be obtained. Therefore there are more papers on stochastic systems than on deterministic dynamics, as derivations for new dynamics allow new systems to be treated. This is particularly true for the Jarzynski relation, as discussed in section 3.2. 

Extensive theoretical work on FR for stochastic systems has been carried out by van Zon et In particular they considered fluctuations in the properties work,... 

In deterministic systems, it is the dissipation function that is the subject of the FR (1.1). In contrast, it has been demonstrated that for stochastic systems, there can be more than one property that satisfies a fluctuation relation. ... 

A number of approaches have been used to try to improve the convergence of the JE in cases where it is problematic. Some of these are analogues of processes introduced to improve sampling in the FEP approaches. For example, Adjanor et al introduce path-biasing schemes to improve convergence. They consider a stochastic system, and carry out tests with simulations on clusters of LJ particles. 

A further discussion of the Schlogl reaction provides an illustration of an important difference in the behavior of steady states in deterministic systems and in systems subject to stochastic fluctuations. In contrast to the deterministic Schlogl system analyzed earlier in terms of conventional chemical concentrations, the analysis of the stochastic system is carried out in terms of the probability that there are n x t) molecules in the system at time t in addition to nA = a and n-Q = b molecules of A and B that are held constant. Figure 6.2 illustrates the individual pathways and rates by which a system with n molecules could undergo a transition to a system containing either n -f 1 or n — 1 molecules. This leads to the master equation for each of the... 

An important contrast illustrated by Figs. 6.1 and 6.3 has previously been discussed [6] the state in which a deterministic system settles is governed entirely by its initial state. Once it has reached its stable steady state, it does not undergo transitions to another steady state. In contrast, the initial state of the stochastic system has no effect on its ultimate long-time behavior. Rather the system passes between the available steady states dwelling in each steady state for times related to the overall stability of the state. 

Nowadays, for reasons of safety and performance, monitoring and supervision have an important role in process control. The complexity and the size of industrial systems induce an increasing number of process variables and make difficult the work of operators. In this context, a computer aided decision-making tool seems to be wise. Nevertheless the implementation of fault detection and diagnosis for stochastic system remains a challenging task. Various methods have been proposed in different industrial contexts (Venkatasubramanian et al., 2003). 

Can chaotic systems be differentiated from random fluctuations Yes, even though the dynamics are complex and resemble a stochastic system, they can be differentiated from a truly stochastic system. Figure 11.15 compares the plots of N = 10,001 and a selection of points chosen randomly from 13,000 to 0. Note that after approximately 10 time intervals, the dynamics of both are quite wild and it would be difficult to distinguish one from another as far as one is deterministic and the other chaotic. However, there is a simple way to differentiate these two alternatives the phase-space plot. 

As discussed in Section 8.2.1, the Langevin equation (8.13) describes a Markovian stochastic process The evolution of the stochastic system variable x(Z) is determined by the state of the system and the bath at the same time t. The instantaneous response of the bath is expressed by the appearance of a constant damping coefficient y and by the white-noise character of the random force 7 (Z). 

Deterministic models, simple error models, stochastic systems... 

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