Kinetic processes master equations

The fimdamental kinetic master equations for collisional energy redistribution follow the rules of the kinetic equations for all elementary reactions. Indeed an energy transfer process by inelastic collision, equation (A3.13.5). can be considered as a somewhat special reaction . The kinetic differential equations for these processes have been discussed in the general context of chapter A3.4 on gas kmetics. We discuss here some special aspects related to collisional energy transfer in reactive systems. The general master equation for relaxation and reaction is of the type [H, 12 and 13, 15, 25, 40, 4T ] ... 

For adsorbates out of local equilibrium, an analytic approach to the kinetic lattice gas model is a powerful theoretical tool by which, in addition to numerical results, explicit formulas can be obtained to elucidate the underlying physics. This allows one to extract simplified pictures of and approximations to complicated processes, as shown above with precursor-mediated adsorption as an example. This task of theory is increasingly overlooked with the trend to using cheaper computer power for numerical simulations. Unfortunately, many of the simulations of adsorbate kinetics are based on unnecessarily oversimplified assumptions (for example, constant sticking coefficients, constant prefactors etc.) which rarely are spelled out because the physics has been introduced in terms of a set of computational instructions rather than formulating the theory rigorously, e.g., based on a master equation. 

In general, the equations for the density operator should be solved to describe the kinetics of the process. However, if the nondiagonal matrix elements of the density operator (with respect to electron states) do not play an essential role (or if they may be expressed through the diagonal matrix elements), the problem is reduced to the solution of the master equations for the diagonal matrix elements. Equations of two types may be considered. One of them is the equation for the reduced density matrix which is obtained after the calculation of the trace over the states of the nuclear subsystem. We will consider the other type of equation, which describes the change with time of the densities of the probability to find the system in a given electron state as a function of the coordinates of heavy particles Pt(R, q, Q, s,...) and Pf(R, q, ( , s,... ).74,77 80... 

One can derive Eqn. (12,12) in a more fundamental way by starting the statistical approach with the (Markovian) master equation, assuming that the jump probabilities obey Boltzmann statistics on the activation saddle points. Salje [E. Salje (1988)] has discussed the following general form of a kinetic equation for solid state processes... 

Since the formal chemical kinetics operates with large numbers of particles participating in reaction, they could be considered as continuous variables. However, taking into account the atomistic nature of defects, consider hereafter these numbers N as random integer variables. The chemical reaction can be treated now as the birth-death process with individual reaction events accompanied by creation and disappearance of several particles, in a line with the actual reaction scheme [16, 21, 27, 64, 65], Describing the state of a system by a vector N = TV),..., Ns, we can use the Chapmen-Kolmogorov master equation [27] for the distribution function P(N, t)... 

In terms of the master equation for the Markov process the formal kinetics is nothing but the mean-field theory where the fluctuation terms like that on the r.h.s. of equation (2.2.43) are neglected. Strictly speaking, the macroscopic description, equation (2.1.2), were correct if the fluctuation terms vanished as V —> oo. In a general case the function P(N, t) does not satisfy the Poisson distribution [16, 27] in particular, °N (N> ... 

Note here that the relation between mesoscopic and microscopic approaches is not trivial. In fact, the former is closer to the macroscopic treatment (Section 2.1.1) which neglects the structural characteristics of a system. Passing from the micro- to meso- and, finally, to macroscopic level we loose also the initial statement of a stochastic model of the Markov process. Indeed, the disadvantages of deterministic equations used for rather simplified treatment of bimolecular kinetics (Section 2.1) lead to the macro- and mesoscopic models (Section 2.2) where the stochasticity is kept either by adding the stochastic external forces (Section 2.2.1) or by postulating the master equation itself for the relevant Markov process (Section 2.2.2). In the former case the fluctuation source is assumed to be external, whereas in the latter kinetics of bimolecular reaction and fluctuations are coupled and mutually related. Section 2.3.1.2 is aimed to consider the relation between these three levels as well as to discuss problem of how determinicity and stochasticity can coexist. 

Chapter 8 provides a unified view of the different kinetic problems in condensed phases on the basis of the lattice-gas model. This approach extends the famous Eyring s theory of absolute reaction rates to a wide range of elementary stages including adsorption, desorption, catalytic reactions, diffusion, surface and bulk reconstruction, etc., taking into consideration the non-ideal behavior of the medium. The Master equation is used to generate the kinetic equations for local concentrations and pair correlation functions. The many-particle problem and closing procedure for kinetic equations are discussed. Application to various surface and gas-solid interface processes is also considered. 

To develop the kinetic equations in condensed phases the master equation must be formulated. In Section 3 the master equation is used to generate the kinetic equations for local concentrations and pair correlation functions. The latter set of equations permits consideration of history of formation of the local solid structure as well as its influence on the subsequent elementary stages. The many-body problem and closing procedure for kinetic equations are discussed. The influence of fast and slow stages on a closed system of equations is demonstrated. The multistage character of the kinetic processes in condensed phase creates a problem of self-consistency in describing the dynamics of elementary stages and the equilibrium state of the condensed system. This problem is solved within the framework of a lattice-gas model description of the condensed phases. 

It is often stated that MC methods lack real time and results are usually reported in MC events or steps. While this is immaterial as far as equilibrium is concerned, following real dynamics is essential for comparison to solutions of partial differential equations and/or experimental data. It turns out that MC simulations follow the stochastic dynamics of a master equation, and with appropriate parameterization of the transition probabilities per unit time, they provide continuous time information as well. For example, Gillespie has laid down the time foundations of MC for chemical reactions in a spatially homogeneous system.f His approach is easily extendable to arbitrarily complex computational systems when individual events have a prescribed transition probability per unit time, and is often referred to as the kinetic Monte Carlo or dynamic Monte Carlo (DMC) method. The microscopic processes along with their corresponding transition probabilities per unit time can be obtained via either experiments such as field emission or fast scanning tunneling microscopy or shorter time scale DFT/MD simulations discussed earlier. The creation of a database/lookup table of transition... 

In many cases, reliable theoretical descriptions of multi-rate processes can be obtained by using master equations in which individual rates are obtained from golden-rule type calculations (see Sections 8.3.3 and 10.4). A condition for the validity of such an approach is that individual rate processes will proceed independently. For example, after evaluating the rates ki 2 and 2 1, Eqs (12.55a) and (12.55b), a description of the overall dynamics of the coupled two-level system by the kinetic scheme 1 2 relies on the assumption that after each transition, say... 

In a typical sample with 10 °-coupled proton spins, a full quantum description is impractical and simplifications are needed. To this end, the polarization-transfer process in a many-spin system is modeled by a kinetic master equation for the polarizations p, = (S, ) ... 

Using a master equation to model the VER process as a multistep reaction, the excess energy flow kinetics in FeP was examined, where the third order Fermi resonance parameters served as approximate reaction rate constants [88]. It was found that the subsequent relaxation is slow relative to relaxation of the initially excited system mode, providing an explanation for the observed difference in relaxation timescales. 

Understanding the structure and function of biomolecules requires insight into both thermodynamic and kinetic properties. Unfortunately, many of the dynamical processes of interest occur too slowly for standard molecular dynamics (MD) simulations to gather meaningful statistics. This problem is not confined to biomolecular systems, and the development of methods to treat such rare events is currently an active field of research. - If the kinetic system can be represented in terms of linear rate equations between a set of M states, then the complete spectrum of M relaxation timescales can be obtained in principle by solving a memoryless master equation. This approach was used in the last century for a number of studies involving atomic... 

The use of the master equation to describe the relaxation of internal energy in molecules is, in fact, nothing more than the writing of a set of kinetic rate equations, one equation for each individual rotation-vibration state of the molecule. The simplest case we can consider is that of an assembly of diatomic molecules highly diluted in a monatomic gas under these conditions, we only need to consider the set of processes... 

The master equation, however, can only be solved analytically for very simple systems such as the gas-phase reaction A—>B. The analysis of these systems typically requires numerical simulation of a lattice-based kinetic Monte Carlo model. The lattice gas model can then be used to formulate the respective transition probabilities in order to solve the master equationThe groups of both Zhdanov[ ° ° ] and Kreuzerl ° l have been instrumental in demonstrating the application of lattice gas models to solve adsorption and desorption processed from surfaces. Once a lattice model has been formulated there are three types of solution ... 

In the present chapter we analyzed the dynamical and thermodynamical behavior of a ubiquitous process in biology the binding of one or more ligands to one receptor. We studied in detail the case when a single receptor molecule is present, as well as when there are several receptors. Not only we were able to prove once more the unity of the various approaches previously introduced, but also derived some interesting conclusions. For instance, we confirmed that the chemical kinetics equations govern the evolutions of the average molecular counts, as computed from the master equation approach. We also proved that, in the present system, the stationary state is compatible with chemical and thermodynamic equilibria, and showed that the stationary state is unique and stable. 

Over the last years it has become clear that the dynamics of most biological phenomena can be studied via the techniques of either nonlinear dynamics or stochastic processes. In either case, the biological system is usually visualized as a set of interdependent chemical reactions and the model equations are derived out of this picture. Deterministic, nonlinear dynamic models rely on chemical kinetics, while stochastic models are developed from the chemical master equation. Recent publications have demonstrated that deterministic models are nothing but an average description of the behavior of unicellular stochastic models. In that sense, the most detailed modeling approach is that of stochastic processes. However, both the deterministic and the stochastic approaches are complementary. The vast amount of available techniques to analytically explore the behavior of deterministic, nonlinear dynamical models is almost completely inexistent for their stochastic counterparts. On the other hand, the only way to investigate biochemical noise is via stochastic processes. 

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