Stochastic description of chemical

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... 

An interesting, but probably incorrect, application of the probabilistic master equation is the description of chemical kinetics in a dilute gas.5 Instead of using the classical deterministic theory, several investigators have introduced single time functions of the form P(n1,n2,t) where P(nu n2, t) is the probability that there are nl particles of type 1 and n2 particles of type 2 in the system at time t. They use the transition rate A(nt, n2 n2, n2, t) from the state with particles of type 1 and n2 particles of type 2 to the state with nt and n2 particles of types 1 and 2, respectively, at time t. The rates that are used are obtained by assuming that only uncorrelated binary collisions occur in the system. These rates, however, are only correct in the thermodynamic limit for a low density system. In this limit, the Boltzmann equation is valid from which the deterministic theory follows. Thus, there is no reason to attach any physical significance to the differences between the results of the stochastic theory and the deterministic theory.6... 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

Chemical process rate equations involve the quantity related to concentration fluctuations as a kinetic parameter called chemical relaxation. The stochastic theory of chemical kinetics investigates concentration fluctuations (Malyshev, 2005). For diffusion of polymers, flows through porous media, and the description liquid helium, Fick s and Fourier s laws are generally not applicable, since these laws are based on linear flow-force relations. A general formalism with the aim to go beyond the linear flow-force relations is the extended nonequilibrium thermodynamics. Polymer solutions are highly relevant systems for analyses beyond the local equilibrium theory. 

Lemarchand, H. Fraikin, A. Stochastic description of various types of bifurcations in chemical systems. In Non-equilibrium dynamics in chemical systems, eds C, Vidal, A. Pacault (Springer Series in Synergetics, Vol. 27), pp. 205-9. Springer Verlag, Berlin. 

Methodologically, even if the diffusive stochastic approach has some theoretical advantages, it is more difficult to adapt and apply to the description of chemical reactions than TST. It requires notable mathematical knowledge and physical concepts that are not so familiar in chemistry. TST on the other hand, relying on the powerful means of quantum mechanics, produces more predictive results, although we have to apply phenomenological coefficients in some cases and make some arbitrary assumptions. 

In this section we have considered the two simplest examples using the deterministic description of chemical reactions. This approach is adequate but only in the so-called thermodynamic limit when we can neglect the discrete nature of the processes considered, as well as the fluctuations of the reactants. Rigorous consideration of these processes becomes possible within a stochastic approach to the description of chemical reactions (for references, see the excellent review by McQuarrie [20]). For the sequence of monomole-cular reactions in open systems with an arbitrary number of intermediates, the problem has been investigated in depth by Nicolis and Babloyantz [31], Ishida [32] and other authors (see, for references, [33]). The stochastic approach, however, faces serious analytical difficulties for more complex systems (for instance, the bimolecular reaction A BoC). Some unusual properties of this reaction in small volumes, associated with enormously large fluctuations, will be considered in Chapter 3. 

Stochastic Description of Various Types of Bifurcations in Chemical Systems... 

The stochastic description of a compartmental system is usually performed in terms of a multivariate master equation. For a chemical system this equation has the form... 

This selection process is then iterated, beginning from an initial state of the system, as defined by species populations, to simulate a chemical evolution. A statistical ensemble is generated by repeated simulation of the chemical evolution using different sequences of random numbers in the Monte Carlo selection process. Within limits imposed by computer time restrictions, ensemble population averages and relevant statistical information can be evaluated to any desired degree of accuracy. In particular, reliable values for the first several moments of the distribution can be obtained both inexpensively and efficiently via a computer algorithm which is incredibly easy to implement (21, 22), especially in comparison to now-standard techniques foF soTving the stiff ordinary differential equations (48, 49) which may arise in the deterministic description of chemical kinetics (53). Now consider briefly the essential features of a simple chemical model which illustrates well the attributes of stochastic chemical simulations. 

The brief review of the newest results in the theory of elementary chemical processes in the condensed phase given in this chapter shows that great progress has been achieved in this field during recent years, concerning the description of both the interaction of electrons with the polar medium and with the intramolecular vibrations and the interaction of the intramolecular vibrations and other reactive modes with each other and with the dissipative subsystem (thermal bath). The rapid development of the theory of the adiabatic reactions of the transfer of heavy particles with due account of the fluctuational character of the motion of the medium in the framework of both dynamic and stochastic approaches should be mentioned. The stochastic approach is described only briefly in this chapter. The number of papers in this field is so great that their detailed review would require a separate article. 

For many synthetic copolymers, it becomes possible to calculate all desired statistical characteristics of their primary structure, provided the sequence is described by a Markov chain. Although stochastic process 31 in the case of proteinlike copolymers is not a Markov chain, an exhaustive statistic description of their chemical structure can be performed by means of an auxiliary stochastic process 3iib whose states correspond to labeled monomeric units. As a label for unit M , it was suggested [23] to use its distance r from the center of the globule. The state of this stationary stochastic process 31 is a pair of numbers, (a, r), the first of which belongs to a discrete set while the second one corresponds to a continuous set. Stochastic process ib is remarkable for being stationary and Markovian. The probability of the transition from state a, r ) to state (/i, r") for the process of conventional movement along a heteropolymer macromolecule is described by the matrix-function of transition intensities... 

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. 

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. 

The jump from the description of the phenomenological process to its stochastic variant, which shows the process s elementary states and its connection procedure, is strongly dependent on the process cognition in terms of chemical engineering as well as on the researcher s ability and experience. 

Chapter 4 is devoted to the description of stochastic mathematical modelling and the methods used to solve these models such as analytical, asymptotic or numerical methods. The evolution of processes is then analyzed by using different concepts, theories and methods. The concept of Markov chains or of complete connected chains, probability balance, the similarity between the Fokker-Plank-Kolmogorov equation and the property transport equation, and the stochastic differential equation systems are presented as the basic elements of stochastic process modelling. Mathematical models of the application of continuous and discrete polystochastic processes to chemical engineering processes are discussed. They include liquid and gas flow in a column with a mobile packed bed, mechanical stirring of a liquid in a tank, solid motion in a liquid fluidized bed, species movement and transfer in a porous media. Deep bed filtration and heat exchanger dynamics are also analyzed. 

With turbulent combustion viewed as a random (or stochastic) process, mathematical bases are available for addressing the subject. A number of textbooks provide introductions to stochastic processes (for example, [55]). In turbulence, any stochastic variable, such as a component of velocity, temperature, or the concentration of a chemical species, which we might call v, is a function of the continuous variables of space x and time t and is, therefore, a stochastic function. A complete statistical description of a stochastic function would be provided by a probability-density functional, tf, defined by stating that the probability of finding the function in a small range i (x, t) about a particular function v(x, t) is [t (x, t)]<3t (x, t) ... 

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