Fluctuation-controlled

Asymmetrical fluctuations controlling progress in pitting, 299 in pitting dissolution, 251 and unstable systems, 255... 

Electrochemical iron generation is a site-specific technology that is pH dependent. Process pH should be from 6 to 9. Optimal removal efficiencies require electrochemical treatment in combination with an ideal precipitation pH for the metals being removed. Nearly all fuU-scale systems include a pH control system. Andco performs lab and pilot-scale testing to evaluate the ability of the process to treat a particular waste stream. If flow rates or contaminant loads fluctuate, control equipment is required to compensate for changes in influent. 

To sum up, the Kurizki-Shapiro-Brumer photocurrent coherent control exhibits resilience to noise and fluctuations. Nevertheless, excessive noise will degrade or completely destroy the effect. We therefore turn to noise/fluctuation control in what follows. 

The insertion of a protein within a membrane not only stabilizes the optimal conformation for the enzyme activity, but also limits the possible conformations it may undergo. Thus, assuming a mechanism for the enzymatic activity, where the underlying assumption is that the conformational fluctuation controls the chemical reaction, a decrease in the number of degrees of freedom for the enzyme increases its probability for... 

Therefore, we tried to develop the adequate mathematical formalism of the fluctuation-controlled chemical kinetics based on a concept of active particles. Simultaneously, the mesoscopic theory of concentration field fluctuations was developed by a number of investigators (see Chapter 2) having more qualitative character. Undoubtedly, these two approaches - microscopic and mesoscopic - overlap, since a lot of fundamental results like asymptotic... 

A careful study of the fluctuation-controlled kinetics performed in recent years has led us to numerous deviations from the results of generally-accepted standard chemical kinetics. To prevent readers from getting lost in details of different formalisms and the ocean of equations presented in this book, we present in this introductory Chapter a brief summary, explain the necessity of developing the fluctuation kinetics and demonstrate its peculiarities compared with techniques presented earlier. We will use here the simplest mathematical formalism and focus on basic ideas which will be discussed later on in full detail. 

It is convenient to divide a set of fluctuation-controlled kinetic equations into two basic components equations for time development of the order parameter n (concentration dynamics) and the complementary set of the partial differential equations for the joint correlation functions x(r, t) (correlation dynamics). Many-particle effects under study arise due to interplay of these two kinds of dynamics. It is important to note that equations for the concentration dynamics coincide formally with those known in the standard kinetics... 

The transition from a stable steady-state solution observed at large p to the oscillatory regime assumes the existence of the critical value of the parameter pc, which defines the point of the kinetic phase transition as p > pc, the fluctuations of the order parameter are suppressed and the standard chemical kinetics (the mean-field theory) could be safely used. However, if p < pc, these fluctuations are very large and begin to dominate the process. Strictly speaking, the region p pc at p > pc is also fluctuation-controlled one since here the fluctuations of the order parameter are abnormally high. 

The scope of this book is as follows. Chapter 2 gives a general review of different theoretical techniques and methods used for description the chemical reactions in condensed media. We focus attention on three principally different levels of the theory macroscopic, mesoscopic and microscopic the corresponding ways of the transition from deterministic description of the many-particle system to the stochastic one which is necessary for the treatment of density fluctuations are analyzed. In particular, Section 2.3 presents the method of many-point densities of a number of particles which serves us as the basic formalism for the study numerous fluctuation-controlled processes analyzed in this book. 

Chapter 5 deals with derivation of the basic equations of the fluctuation-controlled kinetics, applied mainly to the particular bimolecular A + B 0 reaction. The transition to the simplified treatment of the density fluctuation spectrum is achieved by means of the Kirkwood superposition approximation. Its accuracy is estimated by means of a comparison of analytical results for some test problems of the chemical kinetics with the relevant computer simulations. Their good agreement permits us to establish in the next Chapters the range of the applicability of the traditional Waite-Leibfried approach. 

Lastly, non-elementary several-stage reactions are considered in Chapters 8 and 9. We start with the Lotka and Lotka-Volterra reactions as simple model systems. An existence of the undamped density oscillations is established here. The complementary reactions treated in Chapter 9 are catalytic surface oxidation of CO and NH3 formation. These reactions also reveal undamped concentration oscillations and kinetic phase transitions. Their adequate treatment need a generalization of the fluctuation-controlled theory for the discrete (lattice) systems in order to take correctly into account the geometry of both lattice and absorbed molecules. As another illustration of the formalism developed by the authors, the kinetics of reactions upon disorded surfaces is considered. 

We would like to conclude this introductory Chapter by the following general comment. Most of the papers dealing with the fluctuation-controlled reactions, focus their attention on the simplest bimolecular A + B —> B and A + B —> 0 reactions. To our mind, main results in this field are already obtained and the situation is quite clear. In the nearest future the most prospective direction of kinetic theory seems to be many-stage catalytic processes the first results are discussed in Chapters 8 and 9. Their study (stimulated also by the technological importance) should be continued using in parallel both refined mathematical formalisms of the fluctuation-controlled kinetics and full-scale computer simulations. 

The Fluctuation-Controlled Kinetics The Basic Formalism of Many-Point Particle Densities... 

The non-linearity of the equations (5.1.2) to (5.1.4) prevents us from the use of analytical methods for calculating the reaction rate. These equations reveal back-coupling of the correlation and concentration dynamics - Fig. 5.1. Unlike equation (4.1.23), the non-linear terms of equations (5.1.2) to (5.1.4) contain the current particle concentrations n (t), n t) due to which the reaction rate K(t) turns out to be concentration-dependent. (In particular, it depends also on initial reactant concentration.) As it is demonstrated below, in the fluctuation-controlled kinetics (treated in the framework of all joint densities) such fundamental steady-state characteristics of the linear theory as a recombination profile and a reaction rate as well as an effective reaction radius are no longer useful. The purpose of this fluctuation-controlled approach is to study the general trends and kinetics peculiarities rather than to calculate more precisely just mentioned actual parameters. 

D = Da + Db, t-o is a trapping radius and nB is trap concentration) is valid only if one neglects fluctuations of the volume of the Wiener sausage. In the opposite case at long times the kinetics for mobile donors A becomes fluctuation-controlled and as t -a oo obeys finally equation (2.1.106). Of our special interest here are arguments and results for the case of mobile traps B and d 3 based on simple estimates similar to those which resulted in equation (2.1.106). 

It should be stressed that the reversible chemical reactions give us better chance to observe many-particle effects since there is no need here to monitor vanishing particle concentrations over many orders of magnitude. Indeed, the fluctuation-controlled law of the approach to the reaction equilibrium similar to (2.1.61) was observed recently experimentally [85] for the pseudo-first-order reaction A + B AB of laser-excited ROH dye molecules which dissociate in the excited state to create a geminate proton-excited anion pair. The solvated proton is attracted to the anion and recombines with it reversibly. After several dissociation-association cycles it finally diffuses to long distances and further recombination becomes unobservable. 

Studies of the fluctuation-controlled asymptotics of the A + A-type reactions have led to the universal decay law [63, 90]... 

---