Markovian kinetic equation

There have been a number of attempts to calculate time-correlation functions on the basis of simple models. Notable among these is the non-Markovian kinetic equation, the memory function equation for time-correlation functions first derived by Zwanzig33 and studied in great detail by Berne et al.34 This approach is reviewed in this article. Its relation to other methods is pointed out and its applicability is extended to other areas. The results of this theory are compared with the results of molecular dynamics. 

It is likely that there is only one exception to this rule given in the earliest DET [274]. There the non-Markovian kinetic equation for nonlinear (biexciton) annihilation after 8-pulse excitation was derived from first principles ... 

Doktorov, A.B., Kipriyanov, A.A. A many-particle derivation of the Integral Encounter Theory non-Markovian kinetic equations of the reversible reaction A -F B reversible arrow C in solutions. PhysicaA2003,319,253. 

The theory of diffusion-assisted noncontact reactions was developed [339] and applied to ET [31b] long ago. If the acceptor concentration c is sufficiently small, the normalized population of excited donors P t) obeys the non-Markovian kinetic equation [339],... 

If it cannot be guaranteed that the adsorbate remains in local equilibrium during its time evolution, then a set of macroscopic variables is not sufficient and an approach based on nonequihbrium statistical mechanics involving time-dependent distribution functions must be invoked. The kinetic lattice gas model is an example of such a theory [56]. It is derived from a Markovian master equation, but is not totally microscopic in that it is based on a phenomenological Hamiltonian. We demonstrate this approach... 

However, this equation still differs from a basic kinetic equation of the standard (Markovian) perturbation theory [39]. 

Stochastic equation (A8.7) is linear over SP and contains the operators La and V.co of differentiation over time-independent variables Q and co. Therefore, if we assume that the time fluctuations of the liquid cage axis orientation Z(t) are Markovian, then the method used in Chapter 7 yields a kinetic equation for the partially averaged distribution function P(Q, co, t, E). The latter allows us to calculate the searched averaged distribution function... 

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 exhaustive statistical description of living copolymers is provided in the literature [25]. There, proceeding from kinetic equations of the ideal model, the type of stochastic process which describes the probability measure on the set of macromolecules has been rigorously established. To the state Sa(x) of this process monomeric unit Ma corresponds formed at the instant r by addition of monomer Ma to the macroradical. To the statistical ensemble of macromolecules marked by the label x there corresponds a Markovian stochastic process with discrete time but with the set of transient states Sa(x) constituting continuum. Here the fundamental distinction from the Markov chain (where the number of states is discrete) is quite evident. The role of the probability transition matrix in characterizing this chain is now played by the integral operator kernel ... 

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... 

The dynamics associated with the Hamiltonian Eq. (8) or its variants Eq. (11) and Eq. (14) can be treated at different levels, ranging from the explicit quantum dynamics to non-Markovian master equations and kinetic equations. In the present context, we will focus on the first aspect - an explicit quantum dynamical treatment - which is especially suited for the earliest, ultrafast events at the polymer heterojunction. Here, the coherent vibronic coupling dynamics dominates over thermally activated events. On longer time scales, the latter aspect becomes important, and kinetic approaches could be more appropriate. 

The time-dependent (non-Markovian) rate constant k(t) determines the rate of energy quenching in the differential kinetic equation that constitutes the basis of this theory ... 

Using the same input data, we can now compare the kinetics of the charge accumulation/recombination obtained with the non-Markovian UT equations... 

The memory kernel in (2.59), recall that v] represents a nonlocal-in-time integral operator, is a clear indication that subdiffusive transport is non-Markovian. Incorporating kinetic terms into a non-Markovian transport equation requires great care and is best carried out at the mesoscopic level. We show in Sect. 3.4 how to proceed directly at the level of the mesoscopic balance equations for non-Markovian CTRWs. Here we pursue a different approach. As stated above, if all processes are Markovian, then contributions from different processes are indeed separable and simply additive. As is well known, processes often become Markovian if a sufficiently large and appropriate state space is chosen. For the case of reactions and subdiffusion, the goal of a Markovian description can be achieved by taking the age structure of the system explicitly into account as done by Vlad and Ross [460,461]. This approach is equivalent to Model B, see Sect. 3.4. 

In this section we consider CTRW models for which the waiting time distribution is not exponential. The main challenge is to incorporate nonlinear kinetic terms into non-Markovian transport equations. Several approaches exist in the literature about how to include kinetic terms in reaction-transport systems with anomalous diffusion. We discuss them in detail in the following. 

Note that the Markovian dissipative dynamical process is governed by a frequency -independent Il-dissipator in eqn (13.48) that also implies an 5-in-dependent /C-tensor here, while the Markovian kinetic rate process is governed by the constant rate matrix, Al(j) = iC(0). Equation (13.52) would indicate non-Markovian rates in general, even with Markovian dissipative dynamics. However, kinetic rates are physically concerned with post-coherence events, in which the coherence-to-coherence dynamics timescale, the magnitude of l ccl is short compared with the relevant of interest. Therefore, the kinetic rate matrix of eqn (13.52) in the kinetics regime is often of K s) K K 0) = - /Cpp -I- /Cpc cc cp, where /Cpp = 0 in the absence of level relaxations. 

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 non-equivalence of the statistical and kinetic methods Is given by the fact that the statistical generation Is always a Markovian process yielding a Markovian distribution, e.g. In case of a blfunc-tlonal monomer the most probable or pseudo-most probable distributions. The kinetic generation Is described by deterministic differential equations. Although the Individual addition steps can be Markovian, the resulting distribution can be non-Markovian. An Initiated step polyaddltlon can be taken as an example the distribution Is determined by the memory characterized by the relative rate of the Initiation step ( ). ... 

Equation (5.1) described the vibrational response of a single particle to an applied forceF(t). In a (crystalline) system of many mobile particles (ensemble), the problem is analogous but the question now is how the whole system responds to an external force or perturbation Let us define the system s state (a) as a particular configuration of its particles and the probability of this state as pa. In a thermodynamic system, transitions from an a to a p configuration occur as thermally activated events. If the transition frequency a- /5 is copa and depends only on a and / (Markovian), the time evolution of the system is given by a master equation which links atomic and macroscopic parameters (dynamics and kinetics)... 

Obviously, some of the probabilities may be zero. The two equations (56) and (57) may be considered as postulates or else they can be derived by assuming Markovian character for the mass transfer in the equilibrium in a way analogous to that given in reference (33). We now present an interpretation of such an approach from the point of view of chemical kinetics. 

The proper procedure should be quite the opposite. Rq has to be found by fitting the non-Markovian asymptote (3.59) to the experimental data, provided D has been measured or calculated from the Stokes equation. Only then can the true Markovian rate constant k be found as AkRqD. The pure exponential decay with this rate constant is shown by the dashed line in Figure 3.7 for comparison with the true nonstationary kinetics. 

They may be obtained by means of the matrix IET but only together with the kernel E(f) = F(t) specified by its Laplace transformation (3.244), which is concentration-independent. However, from the more general point of view, Eqs. (3.707) are an implementation of the memory function formalism in chemical kinetics. The form of these equations shows the essentially non-Markovian character of the reversible reactions in solution the kernel holds the memory effect, and the convolution integrals entail the prehistoric evolution of the process. In the framework of ordinary chemical kinetics S(/j = d(t), so that the system (3.707) acquires the purely differential form. In fact, this is possible only in the limit when the reaction is entirely under kinetic control. 

When the statistical moments of the distribution of macromolecules in size and composition (SC distribution) are supposed to be found rather than the distribution itself, the problem is substantially simplified. The fact is that for the processes of synthesis of polymers describable by the ideal kinetic model, the set of the statistical moments is always closed. The same closure property is peculiar to a set of differential equations for the probability of arbitrary sequences t//j in linear copolymers and analogous fragments in branched polymers. Therefore, the kinetic method permits finding any statistical characteristics of loopless polymers, provided the Flory principle works for all chemical reactions of their synthesis. This assertion rests on the fact that linear and branched polymers being formed under the applicability of the ideal kinetic model are Markovian and Gordonian polymers, respectively. 

Up to the early 1970s a kinetic approach to the time-dependent properties of fluids was synonymous with a framework based on the Boltzmann equation and its extension by Enskog, in which a central role is played by those dynamical events referred to as uncorrelated binary collisions [29]. Because of this feature the Boltzmann equation is in general not applicable to dense fluids, where the collisions are so frequent that they are likely to interfere with each other. The uncorrelation ansatz is clearly equivalent to a loss of memory, or to a Markov approximation. As a result, for dense fluids the traditional kinetic approach should be critically revised to allow for the presence of non-Markovian effects. 

Let us now incorporate this nonlinear kinetic process into a non-Markovian transport process described by a CTRW. We write the equations for the densities j (x, t) and p x, t) in the following forms ... 

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