Relaxational kinetic equations

The obvious ("model A") set of relaxational kinetic equations are (Flohenberg and Flalperin, 1977)... 

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 ] ... 

Equations (4-21) are linear first-order differential equations. We considered in detail the solution of such sets of rate equations in Section 3-2, so it is unnecessary to carry out the solutions here. In relaxation kinetics these equations are always solved by means of the secular equation, but the Laplace transformation can also be used. Let us write Eqs. (4-21) as... 

Relaxation kinetics. In the course of a study of hemiacetal formation with a phenol nucleophile, an equation was given for the relaxation time in the system ... 

In the impact approximation (tc = 0) this equation is identical to Eq. (1.21), angular momentum relaxation is exponential at any times and t = tj. In the non-Markovian approach there is always a difference between asymptotic decay time t and angular momentum correlation time tj defined in Eq. (1.74). In integral (memory function) theory Rotc is equal to 1/t j whereas in differential theory it is 1/t. We shall see that the difference between non-Markovian theories is not only in times but also in long-time relaxation kinetics, especially in dense media. 

The particular features of the problem, namely the properties of kernel (7.15) of the relaxational part of (7.13) and of Hamiltonian (7.12) in the dynamical part of (7.13), allow one to advance essentially in solving kinetic equation (7.13). 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

An accurate calculation of the heat conductivity requires solving a kinetic equation for the phonons coupled with the multilevel systems, which would account for thermal saturation effects and so on. We encountered one example of such saturation in the expression (21) for the scattering strength by a two-level system, where the factor of tanh((3co/2) reflected the difference between thermal populations of the two states. Neglecting these effects should lead to an error on the order of unity for the thermal frequencies. Within this single relaxation time approximation for each phonon frequency, the Fermi golden rule yields, for the scattering rate of a phonon with Ha kgT,... 

In a conventional relaxation kinetics experiment in a closed reaction system, because of mass conservation, the system can be described in a single equation, e.g., SCc(t) = SCc(0)e Rt where R = ((Ca) + ( C b)) + kh- The forward and reverse rate constants are k and k t, respectively. In an open system A, B, and C, can change independently and so three equations, one each for A, B, and C, are required, each equation having contributions from both diffusion and reaction. Consequently, three normal modes rather than one will be required to describe the fluctuation dynamics. Despite this complexity, some general comments about FCS measurements of reaction kinetics are useful. 

In the case where the arylsulfonate group is a benzene instead of a naphthalene the relaxation kinetics for guest complexation with a-CD measured by stopped-flow showed either one or two relaxation processes.185,190 When one relaxation process was observed the dependence of the observed rate constant on the concentration of CD was linear and the values for the association and dissociation rate constants were determined using Equation (3). When two relaxation processes were observed the observed rate constant for the fast process showed a linear dependence on the... 

In the right-hand part of Figure 10 are shown simulation results obtained by using the above kinetic equations and the rectangular cell model which divides the air/water interface into one hundred cells. In this simulation, the relative magnitudes of the rate of relaxation processes and the rate of compression were set up as follows. ... 

Time-course of enzyme-catalyzed reactions, ENZYME KINETIC EQUATIONS TIME-OF-FLIGHT MASS SPECTROMETRY Time of relaxation,... 

Multiscale ensembles of reaction networks with well-separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors ( modes ) is presented. In particular, we prove that for systems with well-separated constants eigenvalues are real (damped oscillations are improbable). For systems with modular structure, we propose the selection of such modules that it is possible to solve the kinetic equation for every module in the explicit form. All such solvable networks are described. The obtained multiscale approximations, that we call dominant systems are... 

The fast stage of relaxation of a complex reaction network could be described as mass transfer from nodes to correspondent attractors of auxiliary dynamical system and mass distribution in the attractors. After that, a slower process of mass redistribution between attractors should play a more important role. To study the next stage of relaxation, we should glue cycles of the first auxiliary system (each cycle transforms into a point), define constants of the first derivative network on this new set of nodes, construct for this new network an (first) auxiliary discrete dynamical system, etc. The process terminates when we get a discrete dynamical system with one attractor. Then the inverse process of cycle restoration and cutting starts. As a result, we create an explicit description of the relaxation process in the reaction network, find estimates of eigenvalues and eigenvectors for the kinetic equation, and provide full analysis of steady states for systems with well-separated constants. 

In the first case, the limit (for t- co) distribution for the auxiliary kinetics is the well-studied stationary distribution of the cycle A A , +2, described in Section 2 (ID-QS), (15). The set A j+], A . c+2, , n is the only ergodic component for the whole network too, and the limit distribution for that system is nonzero on vertices only. The stationary distribution for the cycle A i+] A t+2. ., A A t+i approximates the stationary distribution for the whole system. To approximate the relaxation process, let us delete the limiting step A A j+] from this cycle. By this deletion we produce an acyclic system with one fixed point, A , and auxiliary kinetic equation (33) transforms into... 

The temporal change of , that is, the relaxational behavior of , is governed by the kinetic equation... 

This is the kinetic equation for a simple A/AX interface model and illustrates the general approach. The critical quantity which will be discussed later in more detail is the disorder relaxation time, rR. Generally, the A/AX interface behaves under steady state conditions similar to electrodes which are studied in electrochemistry. However, in contrast to fluid electrolytes, the reaction steps in solids comprise inhomogeneous distributions of point defects, which build up stresses at the boundary on a small scale. Plastic deformation or even cracking may result, which in turn will influence drastically the further course of any interface reaction. 

In conclusion, we observe that the crossing of crystal phase boundaries by matter means the transfer of SE s from the sublattices of one phase (a) into the sublattices of another phase (/ ). Since this process disturbs the equilibrium distribution of the SE s, at least near the interface, it therefore triggers local SE relaxation processes. In more elaborated kinetic models of non-equilibrium interfaces, these relaxations have to be analyzed in order to obtain the pertinent kinetic equations and transfer rates. This will be done in Chapter 10. 

Equation (4.145) represents a first order relaxation. We found the same result in Section 1.3.1 by linearizing the kinetic equations of the bimolecular point defect reaction (Eqn. (1.7)). 

To complete the set of kinetic equations we observe that ub = (A/ /Ac)b where Acb can be expressed in terms of <5 ,b. Finally, the requirement of mass conservation yields a further equation. Considering the inherent nonlinearities, this problem contains the possibility of oscillatory solutions as has been observed experimentally. Let us repeat the general conclusion. Reactions at moving boundaries are relaxation processes between regular and irregular SE s. Coupled with the transport in the untransformed and the transformed phases, the nonlinear problem may, in principle, lead to pulsating motions of the driven interfaces. 

The expression for the time course (equation 4.20) may be divided into various factors. There is first the exponential term with the rate constant, or in terms of relaxation kinetics, the reciprocal relaxation time 1/r given by... 

The results of the numerical experiment for system (20) necessitated a general mathematical investigation of slow relaxations in chemical kinetic equations. This study was performed by Gorban et al. [226-228] who obtained several theorems permitting them to associate the existence of slow relaxations in a system of chemical kinetic equations (and, in general, in dynamic systems) with the qualitative changes in the phase portrait with its parameters (see Chap. 7). 

Finally, we can suggest a third explanation fast steps can compose a mechanism with slow relaxations. Indeed, nothing suggests that the relaxation time for a set of chemical kinetic equations is directly dependent on the characteristic times of the individual steps. But it cannot be treated as a reason for slow relaxations. It is only a simple indication for the possibility of finding such reasons here. Let us now indicate the reasons according to which fast steps can compose a mechanism with slow relaxations. 

The kinetics captured in disordered systems like polymers, glasses and poly-cristalline structures has been often described in terms of continuous relaxation times and exciton diffusion at recombination centers [10]. Assuming a <5— pulse function, the temporal data are best fitted by a monomolecular kinetic equation,... 

To obtain the closed system of kinetic equations one should divide all variables on slow (with the characteristics time xm) and rapid (with the characteristics time xr) variables with respect to the chosen time interval x, which is presented in the left sides of the differential Eqs. (31) and (32). As rule, in condensed reaction systems inner degrees of freedoms for reactants can reach the local relaxation state during less time by comparison with rapid variables. In opposite cases, various inner states of reactants should be considered as independent kinds of species. 

The kinetic equation (4.27) with the energy function (4.52), first studied by Brown [47], has since been investigated extensively [48,54-59]. However, due to mathematical difficulties, the case of arbitrary orientation of the external and anisotropy fields (i.e., vectors h and n) has been addressed only relatively recently. The numerical solution of the relaxation problem for h and n crossed under an arbitrary angle for the first time was given in Ref. 60. 

As mentioned in Section II, the magnetodynamic equation underlying the Brown kinetic equation (4.125) can be either that by Landau and Lifshitz or that by Gilbert. To be specific, we adopt the former one, noting their equivalence established by formulas (4.5). Thence, the reference relaxation time in Eq. (4.125) is given by Eq. (4.24). 

However, the case of il being exactly zero is to some extent an exception. At any finite Q, one has to remember that the time X is exponential in a that is, it grows infinitely with cooling the system. Therefore, at low temperatures the interwell transition is completely frozen, and the situation is governed by intrawell relaxation. The latter is sensitive to the details of the potential near the bottom of the well, and for the system in question is determined by the infinite eigenvalue spectrum A of the kinetic equation (4.225) for k > 3. 

An example of the phase relaxation curve observed for the alkyl radical in irradiated polyethylene has already been shown in Fig. 5. Regardless of the radiation dose, the relaxation kinetics monitored at the center of the ESR spectrum can be expressed empirically by the following equation ... 

---