Non-Markovian generalization

The theory of Section 1.8 is sometimes qualified as non-Markovian since it accounts for non-exponential angular momentum relaxation, unlike impact theory which is Markovian in this sense. However, it is not a unique non-Markovian generalization of impact theory. Not less known is a differential version of the theory... 

We will show below when and how the line interference and its special case, spectral exchange , appear in spectral doublets considered as an example of the simplest system. It will be done in the frame of conventional impact theory as well as in its modern non-Markovian generalization. Subsequently we will concentrate on the impact theory of rotational structure broadening and collapse with special attention to the shape of a narrowed Q-branch. 

The simplest Markovian result (3.86) was recently approved and subjected to a non-Markovian generalization by means of IET107 [see Eq. (3.372) and Section XI.E], IET is usually used instead of DET when the occasion requires accounting for the reversibility of the transfer reaction. 

The initial condition for N is prepared by instantaneous excitation, after which the annihilation rate constant k/(t) decreases with time, approaching its stationary (Markovian) value kt as t —> oo. The non-Markovian generalization of another equation, (3.761), became possible only in the framework of the unified theory, where it takes the integral form. Unfortunately, the system response to the light pulses of finite duration or permanent illumination remains a problem for either UT or DET. The convolution recipes such as (3.5) or (3.437) are inapplicable to annihilation, which is bilinear in N. Therefore we will start from IET, which is solely capable of consistent consideration of stationary absorbtion and conductivity [199]. Then we will turn to UT and the Markovian theories applied to the relaxation of the instantaneously excited system described in Ref. 275. 

Equation (8.54) is a stochastic equation of motion similarto Eq. (8.13). However, we see an important difference Eq. (8.54) is an integro-differential equation in which the term yx of Eq. (8.13) is replaced by the integral /J drZ t — r)x(r). At the same time the relationship between the random force R t) and the damping, Eq. (8.20), is now replaced by (8.59). Equation (8.54) is in fact the non-Markovian generalization of Eq. (8.13), where the effect of the thermal environment on the system is not instantaneous but characterized by a memory—at time t it depends on the past interactions between them. These past interactions are important during a memory time, given by the lifetime of the memory kernel Z t). The Markovian limit is obtained when this kernel is instantaneous... 

Consider now the barrier crossing problem in the barrier controlled regime discussed in Section 14.4.3. The result, the rate expressions (14.73) and (14.74), as well as its non-Markovian generalization in which cor is replaced by k ofEq. (14.90), has the structure of a corrected TST rate. TST is exact, and the correction factor becomes 1, if all traj ectories that traverse the barrier top along the reaction coordinate (x ofEq. (14.39)) proceed to a well-defined product state without recrossing back. Crossing back is easily visualized as caused by collisions with solvent atoms, for example, by solvent-induced friction. 

Onsager s theory can also be used to detemiine the fomi of the flucUiations for the Boltzmaim equation [15]. Since hydrodynamics can be derived from the Boltzmaim equation as a contracted description, a contraction of the flucUiating Boltzmann equation detemiines fluctuations for hydrodynamics. In general, a contraction of the description creates a new description which is non-Markovian, i.e. has memory. The Markov... 

Because of the general difficulty encountered in generating reliable potentials energy surfaces and estimating reasonable friction kernels, it still remains an open question whether by analysis of experimental rate constants one can decide whether non-Markovian bath effects or other influences cause a particular solvent or pressure dependence of reaction rate coefficients in condensed phase. From that point of view, a purely... 

As an introduction to the peculiar properties of the spin Hamiltonians, we first give a short summary of the theory of spin relaxation in liquids where the problem is in fact a Brownian motion one. Then we consider the many-spin problem in solids and apply the general formalism of the theory of irreversible processes developed by Prigogine and his co-workers. We also analyse some aspects of the recent work of Caspers and Tjon on this subject. Finally, we indicate the special interest of spin-spin relaxation phenomena in connection with non-Markovian processes. 

At the same time, Prigogine and his co-workers14 15,17 developed a general theory of non-equilibrium statistical mechanics. They derived a non-Markovian evolution equation for the velocity distribution function. Their results contain a generalization of the Boltzmann equation for arbitrary concentration and coupling parameter. This generalization is the long-time limit of their evolution equation. 

Section III is devoted to Prigogine s theory.14 We write down the general non-Markovian master equation. This expression is non-instantaneous because it takes account of the variation of the velocity distribution function during one collision process. Such a description does not exist in the theories of Bogolubov,8 Choh and Uhlenbeck,6 and Cohen.8 We then present two special forms of this general master equation. On the one hand, when one is far from the initial instant the Variation of the distribution functions becomes slower and slower and, in the long-time limit, the non-Markovian master equation reduces to the Markovian generalized Boltzmann equation. On the other hand, the transport coefficients are always calculated in situations which are... 

We illustrate the general expressions (4.189)-(4.191) for a typical non-Markovian Lorentzian bath spectrum, that is, an exponentially decaying correlation function d>(t) = being the correlation (memory) time. 

In this review, we have expounded our universal approach to the dynamical control of qubits subject to noise or decoherence. It is based on a general non-Markovian ME valid for weak System-bath coupling and arbitrary modulations, since it does not invoke the RWA. The resulting universal convolution formula provide intuitive clues as to the optimal tailoring of modulation and noise spectra. 

Clearly, Eqs. (5) and (6) yield different results for p,(x, t) and the form for vv2 in Eq. (4) is not suitable. This is a reflection of a general property of the conditional probabilities for non-Markovian processes that we shall prove below. 

The rate theory of Grote and Hynes [149] included the non-Markovian (memory) effects by considering the following generalized Langevin equation (GLE) for the dynamics along the reaction coordinate ... 

It should be stressed that for the double-well reaction model in the non-Markovian case a general result similar to the Kramers expression (4.160) cannot be found. To evaluate the thermally activated escape rate, the motion within the barrier region is described by means of a GLE in which the potential near the barrier is linearized, that is,... 

Be aware of the fact that we have to consider the non-Markovian version of the quantum master equation to stay at a level of description where the emission rate, Eq. (39), can be deduced. Moreover, to be ready for a translation to a mixed quantum classical description a variant has been presented where the time evolution operators might be defined by an explicitly time-dependent CC Hamiltonian, i.e. exp(—iHcc[t — / M) has been replaced by the more general expression Ucc(t,F). 

If the phonon distribution of the model Eq. (8) spans a dense spectrum - as is generally the case for the extended systems under consideration, which are effectively infinite-dimensional - the dynamics induced by the Hamiltonian will eventually exhibit a dissipative character. However, the effective-mode construction demonstrates that the shortest time scales are fully determined by few effective modes, and by the coherent dynamics induced by these modes. The overall picture thus corresponds to a Brownian oscillator type dynamics, and is markedly non-Markovian [81,82],... 

The coupling parameters df. are sampled according to a specified spectral density, which is here taken to be Ohmic [89-91], More generally, the external bath itself can be taken to be non-Markovian. An example of this scheme is given in Fig. 8 of Sec. 5.1, for an Ohmic bath at zero temperature, i.e., exhibiting no thermal fluctuations. Here, the damping effect is generated by quantum fluctuations at T = 0 [90,91],... 

At the same time we do not have to assume that A -C c as we did previously. Under this condition the acceptor concentration A = [A] remained almost constant, approximately equal to its initial value c. In what follows we will eliminate this restriction and account for the expendable neutral acceptors whose concentration A(t) decreases in the course of ionization. When there is a shortage of acceptors, the theory becomes nonlinear in the concentration, even in absence of bulk recombination. Under such conditions only general encounter theories are appropriate for a full timescale (non-Markovian) description of the system relaxation. We will compare them against each other and with the properly generalized Markovian and model theories of the same phenomena. 

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. 

In the case where the correlation function <3> (f) has the form of Eq. (148), with p fitting the condition 2 < p < 3, the generalized diffusion equation is irreducibly non-Markovian, thereby precluding any procedure to establish a Markov condition, which would be foreign to its nature. The source of this fundamental difficulty is that the density method converts the infinite memory of a non-Poisson renewal process into a different type of memory. The former type of memory is compatible with the occurrence of critical events resetting to zero the systems memory. The second type of memory, on the contrary, implies that the single trajectories, if they exist, are determined by their initial conditions. 

The consistent kinetic analysis of the copolymerization with the simultaneous occurrence of the reactions (2.1) and (2.5) leads to the conclusion that the probabilities of the sequences of the monomer units M, and M2 in the macromolecules can not be described by a Markov chain of any finite order. Consequently, in this very case we deal with non-Markovian copolymers, the general theory for which is not yet available [6]. However, a comprehensive statistical description of the products of the complex-radical copolymerization within the framework of the Seiner-Litt model via the consideration of the certain auxiliary Markov chain was carried out [49, 59, 60]. 

Dekker has studied multiplicative stochastic processes. In his work the stochastic Liouville equation was solved explicitly through first order in an expansion in terms of correlation times of the multiplicative Gaussian colored noise for a general multidimensional weakly non-Markovian process. He followed the suggestions of refs. 17 and 18 and applied, Novikov s theorem. In the general multidimensional case, however, he improved the earlier work by San Miguel and Sancho. ... 

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