Stochastic equations nonlinearity

The argument by Zwanzig links the violation of condition (24) with a well-defined nonlinearity in the stochastic equations describing the system under consideration (an ensemble of molecules). One way of introducing this type of nonlinearity is to use the nonlinear itinerant oscillator... 

The subject of multiplicative fluctuations (in linear and especially nonlinear systems) is still deeply fraught with ambiguity. The authors of Chapter X set up an experiment that simulates the corresponding nonlinear stochastic equations by means of electric circuits. This allows them to shed light on several aspects of external multiplicative fluctuation. The results of Chapter X clearly illustrate the advantages resulting from the introduction of auxiliary variables, as recommended by the reduced model theory. It is shown that external multiplicative fluctuations keep the system in a stationary state distinct from canonical equilibrium, thereby opening new perspectives for the interpretation of phenomena that can be identified as due to the influence of multiplicative fluctuations. 

It should be noted that Eq. (5.63) may be derived from a microscopic model ° only for the special case where the friction kernel does not depend on the particle s velocity. This is not generally the case, and a rigorous derivation of reduced stochastic equations describing the motion of a subsystem coupled nonlinearly to its thermal environment leads to more complicated equations. (See the References for further discussions of this issue . ) Equation (5.63) may still be derived for special cases. An analysis very similar to that presented above leads to the energy diffusion equation (5.48) where now D(E) is given by... 

For information on how stochastic differential equations can be used to analyze the effect of the fluctuations in operating conditions, see Rao et al. (1974a). The actual method of solving nonlinear stochastic equations can be found in Rao et al. (1974b). Algorithms simpler than Rao et al. (1974b) are reported elsewhere (e.g., Pardoux and Talay, 1985). 

Let a limit cycle oscillator be exposed to some weak random forces which may depend on the state variable X. The governing equation is a nonlinear stochastic equation ... 

The stochastic equations of motion in a general case of a nonlinear oscillator and a parametric excitation are... 

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

This is the same equation of motion that is satisfied by the original coordinate qa(t), except that the stochastic driving term is absent. The relative dynamics is therefore deterministic. We have chosen the notation accordingly and left out the index a in the definition (41) of Aq (although, of course, we cannot expect the relative dynamics to remain noiseless in the full nonlinear system). Although noiseless, the relative dynamics is still dissipative because Eq. (43) retains the damping term. 

It may be useful to point out a few topics that go beyond a first course in control. With certain processes, we cannot take data continuously, but rather in certain selected slow intervals (c.f. titration in freshmen chemistry). These are called sampled-data systems. With computers, the analysis evolves into a new area of its own—discrete-time or digital control systems. Here, differential equations and Laplace transform do not work anymore. The mathematical techniques to handle discrete-time systems are difference equations and z-transform. Furthermore, there are multivariable and state space control, which we will encounter a brief introduction. Beyond the introductory level are optimal control, nonlinear control, adaptive control, stochastic control, and fuzzy logic control. Do not lose the perspective that control is an immense field. Classical control appears insignificant, but we have to start some where and onward we crawl. 

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. 

Of course, the macroscopic equations cannot actually be derived from the microscopic ones. In practice they are pieced together from general principles and experience. The stochastic mesoscopic description must be obtained in the same way. This semi-phenomenological approach is remarkably successful in the range where the macroscopic equations are linear, see chapter VIII. In the nonlinear case, however, difficulties appear, which can only be resolved by the improved, but still mesoscopic, method of chapter X. 

Summary. The special class of master equations characterized by (1.1) will be said to be of diffusion type. For such master equations the -expansion leads to the nonlinear Fokker-Planck equation (1.5), rather than to a macroscopic law with linear noise, as found in the previous chapter for master equations characterized by (X.3.4). The definition of both types presupposes that the transition probabilities have the canonical form (X.2.3), but does not distinguish between discrete and continuous ranges of the stochastic variable. The -expansion leads uniquely to the well-defined equation (1.5) and is therefore immune from the interpretation difficulties of the Ito equation mentioned in IX.4 and IX.5. 

Therefore, the simplest procedure to get the stochastic description of the reaction leads to the rather complicated set of equations containing phenomenological parameters / (equation (2.2.17)) with non-transparent physical meaning. Fluctuations are still considered as a result of the external perturbation. An advantage of this approach is a useful analogy of reaction kinetics and the physics of equilibrium critical phenomena. As is well known, because of their nonlinearity, equations (2.1.40) reveal non-equilibrium bifurcations [78, 113]. A description of diffusion-controlled reactions in terms of continuous Markov process - equation (2.2.15) - makes our problem very similar to the static and dynamic theory of critical phenomena [63, 87]. When approaching the bifurcation points, the systems with reactions become very sensitive to the environment fluctuations, which can even produce new nonequilibrium transitions [18, 67, 68, 90, 108]. The language developed in the physics of critical phenomena can be directly applied to the processes in spatially extended systems. 

The boundaries between different waste classes would be quantified in terms of limits on concentrations of hazardous substances using a quantity called the risk index, which is defined in Equation 6.1. The risk index essentially is the ratio of a calculated risk that arises from waste disposal to an allowable risk (a negligible or acceptable risk) appropriate to the waste class (disposal system) of concern. The risk index is developed taking into account the two types of hazardous substances of concern substances that cause stochastic responses and have a linear, nonthreshold dose-response relationship, and substances that cause deterministic responses and have a threshold dose-response relationship. The risk index for any substance can be expressed directly in terms of risk, but it is more convenient to use dose instead, especially in the case of substances that cause determinstic responses for which risk is a nonlinear function of dose and the risk at any dose below a nominal threshold is presumed to be zero. The risk index for mixtures of substances that cause stochastic or deterministic responses are given in Equations 6.4 and 6.5, respectively, and the simple rule for combining the two to obtain a composite risk index for all hazardous substances in waste is given in Equation 6.6 or 6.7 and illustrated in Equation 6.8. The risk (dose) that arises from waste disposal in the numerator of the risk index is calculated based on assumed scenarios for exposure of hypothetical... 

If the hazard rate of any single particle out of a compartment depends on the state of the system, the equations of the probabilistic transfer model are still linear, but we have nonlinear rate laws for the transfer processes involved and such systems are the stochastic analogues of nonlinear compartmental systems. For such systems, the solutions for the deterministic model are not the same as the solutions for the mean values of the stochastic model. 

An important property of the stochastic version of compartmental models with linear rate laws is that the mean of the stochastic version follows the same time course as the solution of the corresponding deterministic model. That is not true for stochastic models with nonlinear rate laws, e.g., when the probability of transfer of a particle depends on the state of the system. However, under fairly general conditions the mean of the stochastic version approaches the solution of the deterministic model as the number of particles increases. It is important to emphasize for the nonlinear case that whereas the deterministic formulation leads to a finite set of nonlinear differential equations, the master equation... 

The possible states for substrate are 11 and 6 for the complex. R is a 66-dimensional matrix and the initial condition for the master equation is pio,o (0) = 1. Figures 9.27 and 9.28 show the associated probabilities for each state as functions of time for the substrate and the complex, respectively. As previously, the full markers are the expected values and the solid lines the solution of the deterministic model. Notably, the expectation of the stochastic model does not follow the time profile of the deterministic system. This is the main characteristic of nonlinear systems. 

These equations are not equivalent to the deterministic formulation given by (8.8). The last term k+iKn involving the stochastic interaction in the previous equations expresses the main difference between deterministic and stochastic solutions for a nonlinear system. 

Basically, we shall apply the AEP, derived by the Zwanzig approach, to explore the long-time behavior of nonlinear stochastic processes, whereas the CFP derived from the Mori approach will be used as a calculation technique, which sometimes will prove useful for application to the reduced equations of motion provided by the AEP itself. 

---