Stochastic description

Stochastic Description of Copolymerization and Network Formation in a Six-Component, Three-Stage Process... 

This model enables us to investigate the character of the motion of the system in the course of the transition in various limits and to analyze under what conditions the stochastic description is applicable. 

Wet-weather processes are subject to high variability. A simple deterministic model result in terms of the impacts on the water quality is out of scope. From a modeling point of view, a stochastic description is a realistic solution for producing relevant results. Furthermore, an approach based on a historical rainfall series as model input is needed to establish extreme event statistics for a critical CSO impact that can be compared to a water quality criterion. In terms of CSO design including water quality, this approach is a key point. 

Still there exists a large gap between the quantum-mechanical description, Eq. (9), in terms of amplitudes and the classical or stochastic description. It seems to me that the basic reason is that the ordering in Eq. (9) does not proceed according to the macroscopic time but according to microscopic, specifically quantum time. Indeed the time evolution of... 

Internal noise is described by a master equation. When this equation cannot be solved exactly it is necessary to have a systematic approximation method - rather than the naive Fokker-Planck and Langevin approximations. Such a method will now be developed in the form of a power series expansion in a parameter Q. In lowest order it reproduces the macroscopic equation and thereby demonstrates how a deterministic equation emerges from the stochastic description. 

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 understanding of the photoprocess responsible for the dual fluorescence can be developed in two steps. The first step is the static and structural nature and involves estimates of energy surfaces, mainly the location of minima and barriers. It is based on traditional quantum chemistry. The second step deals with dynamics of the process and requires the use of additional theoretical tools of chemical dynamics such as stochastic description. 

For this case, the stochastic description (Section IV) predicts that the rate constants to be observed can be time dependent [fc(f)]. Moreover,... 

A class of continuous-time Markov processes with integer state space and with transitions allowed only between adjacent states plays a central role in the stochastic description of chemical kinetics.122 134 135... 

The stochastic description of barrierless relaxations by Bagchi, Fleming, and Oxtoby (Ref. 195 and Section IV.I) was first applied by these authors to TPM dyes to explain the observed nonexponential fluorescence decay and ground-state repopulation kinetics. The experimental evidence of an activation energy obs < Ev is also in accordance with a barrierless relaxation model. The data presented in Table IV are indicative of nonexponential decay, too. They were obtained by fitting the experiment to a biexponential model, but it can be shown50 that a fit of similar quality can be obtained with the error-function model of barrierless relaxations. Thus, r, and t2 are related to r° and t", but, at present, we can only... 

Kramers idea was to give a more realistic description of the dynamics in the reaction coordinate by including dynamical effects of the solvent. Instead of giving a deterministic description, which is only possible in a large-scale molecular dynamics simulation, he proposed to give a stochastic description of the motion similar to that of the Brownian motion of a heavy particle in a solvent. From the normal coordinate analysis of the activated complex, a reduced mass pi has been associated with the motion in the reaction coordinate, so the proposal is to describe the motion in that coordinate as that of a Brownian particle of mass g in the solvent. 

Cover illustration Left panel Stochastic description of the kinetics of a population of particles, Fig 9.15. Middle panel Dissolution in topologically restricted media, Fig. 6.8B (reprinted with permission from Springer). Right panel A pseudophase space for a chaotic model of cortisol kinetics, Fig.11.11. 

Although it was not shown here, a general cluster size distribution in equilibrium can be obtained using a different approach [18, 19]. It involves a stochastic description for the aggregation-fragmentation system given by the master equation of a probability balance. The equilibrium probability then follows from the detailed balance. That work is under way. 

Our starting point in the stochastic description is the overdamped Langevin equation [54,59]6... 

The relevance of stochastic descriptions brings out the issue of their theoretical and numerical evaluation. Instead of solving the equations of motion for 6x102 degrees of freedom we now face the much less demanding, but still challenging need to construct and to solve stochastic equations of motion for the few relevant variables. The next section describes a particular example. 

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