Markovian

In the presence of some fomi of relaxation the equations of motion must be supplemented by a temi involving a relaxation superoperator—superoperator because it maps one operator into another operator. The literature on the correct fomi of such a superoperator is large, contradictory and incomplete. In brief, the extant theories can be divided into two kinds, those without memory relaxation (Markovian) Tp and those with memory... 

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... 

Straub J E and Berne B J 1986 Energy diffusion in many-dimensionai Markovian systems the oonsequenoes of oompetition between inter- and intramoieouiar vibrationai energy transfer J. Chem. Phys. 85 2999-3006... 

Straub J E and Berne B J 1986 Energy diffusion in many dimensional Markovian systems the consequences of the competition between inter- and intra-molecular vibrational energy transfer J. Chem. Phys. 85 2999 Straub J E, Borkovec M and Berne B J 1987 Numerical simulation of rate constants for a two degree of freedom system in the weak collision limit J. Chem. Phys. 86 4296... 

Bigot J-Y, Portella M T, Schoenlein R W, Bardeen C J, Migus A and Shank C V 1991 Non-Markovian dephasing of molecules in solution measured with three-pulse femtosecond photon echoes Phys. Rev. Lett. 66 1138 1... 

The analysis of Manousiouthakis and Deem [75] mentioned above has demonstrated that it is also correct to choose atoms sequentially rather than randomly it has been tacitly assumed for many years that this violation of the Markovian restriction is acceptable, so a proof of tiiis kind is very welcome. 

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... 

There is a fundamental relationship between d-dimensional PCA and d + 1)-dimensional Ising spin models. The simplest way to make the connection is to think of the successive temporal layers of the PCA as successive hyper-planes of the next higher-dimensional spatial lattice. Because the PCA rules (at least the set of PCA rules that we will be dealing with) are (1) Markovian (i.e. the probability of a state at time t + T depends only on a set of states at time t, and (2) local, one can always define a Hamiltonian on the higher-dimensioned spatial lattice such that the thermodynamic weight of a configuration 5j,( is equal to the probability of a corresponding space-time history Si t). ... 

Now consider an entire temporal history of this PCA. That is, consider the effective two-dimensional lattice that is formed by stacking successive one-dimensional layers on top of one another (see figure 7.4). Because of the Markovian nature of the evolution, the probability of this temporal history is given simply by... 

To introduce this principle, consider the following equation expressing the Markovian transition between the t -step probability distribution for state S, p S, t), and t -f l) -step probability distribution for state 5, p S,t -f 1) ... 

We also know that if there exists an equilibrium distribution, p, then it must remain invariant under the Markovian transition i.e. p o must satisfy the eigenvector equation... 

Chapter 3 is devoted to pressure transformation of the unresolved isotropic Raman scattering spectrum which consists of a single Q-branch much narrower than other branches (shaded in Fig. 0.2(a)). Therefore rotational collapse of the Q-branch is accomplished much earlier than that of the IR spectrum as a whole (e.g. in the gas phase). Attention is concentrated on the isotropic Q-branch of N2, which is significantly narrowed before the broadening produced by weak vibrational dephasing becomes dominant. It is remarkable that isotropic Q-branch collapse is indifferent to orientational relaxation. It is affected solely by rotational energy relaxation. This is an exceptional case of pure frequency modulation similar to the Dicke effect in atomic spectroscopy [13]. The only difference is that the frequency in the Q-branch is quadratic in J whereas in the Doppler contour it is linear in translational velocity v. Consequently the rotational frequency modulation is not Gaussian but is still Markovian and therefore subject to the impact theory. The Keilson-... 

The quantum theory of spectral collapse presented in Chapter 4 aims at even lower gas densities where the Stark or Zeeman multiplets of atomic spectra as well as the rotational structure of all the branches of absorption or Raman spectra are well resolved. The evolution of basic ideas of line broadening and interference (spectral exchange) is reviewed. Adiabatic and non-adiabatic spectral broadening are described in the frame of binary non-Markovian theory and compared with the impact approximation. The conditions for spectral collapse and subsequent narrowing of the spectra are analysed for the simplest examples, which model typical situations in atomic and molecular spectroscopy. Special attention is paid to collapse of the isotropic Raman spectrum. Quantum theory, based on first principles, attempts to predict the. /-dependence of the widths of the rotational component as well as the envelope of the unresolved and then collapsed spectrum (Fig. 0.4). 

Fig. 1.1. Time-dependence of the components of angular momentum J, (Markovian process) and the torque M, (white noise) in the impact approximation.

Markovian perturbation theory as well as impact theory describe solely the exponential asymptotic behaviour of rotational relaxation. However, it makes no difference to this theory whether the interaction with a medium is a sequence of pair collisions or a weak collective perturbation. Being binary, the impact theory holds when collisions are well separated (tc < to) while the perturbation theory is broader. If it is valid, a new collision may start before the preceding one has been completed when To < Tc TJ = t0/(1 - y). 

Judging by these results the angular momentum relaxation in a dense medium has the form of damped oscillations of frequency jRo = (Rctc/to)i and decay decrement 1/(2tc). This conclusion is quantitatively verified by computer experiments [45, 54, 55]. Most of them were concerned with calculations of the autocorrelation function of the translational velocity v(t). However the relation between v(t) and the force F t) acting during collisions is the same as that between e> = J/I and M. Therefore, the results are qualitatively similar. In Fig. 1.8 we show the correlation functions of the velocity and force for the liquid state density. Oscillations are clearly seen, which point to a regular character of collisions and non-Markovian nature of velocity changes. 

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... 

Although non-Markovian, the differential theory surely has Markovian asymptotics at sufficiently long times ... 

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. 

It is commonly believed that K (t ) may be carried outside the integral without lack of accuracy if inequality (2.23) is satisfied. This is the same way that was used in Chapter 1 to obtain the non-Markovian differential equation... 

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

The Markovian theory is obtained when the integration over time in Eq. (2.25) is extended to infinity ... 

Markovian theory of orientational relaxation implies that it is exponential from the very beginning but actually Eq. (2.26) holds for t zj only. If any non-Markovian equations, either (2.24) or (2.25), are used instead, then the exponential asymptotic behaviour is preceded by a short dynamic stage which accounts for the inertial effects (at t < zj) and collisions (at t < Tc). 

The first component in expression (2.53) corresponds to the long-time behaviour of K( t) described by Markovian perturbation theory, while the second term introduces a correction for times less than zj. Within this time interval (before the first collision occurs) the system should display the dynamic properties of free rotation ( inertial effects ). 

In Markovian approximation (zj =0) this quantity approaches the famous Debye plateau shown in Fig. 2.3 whereas non-Markovian absorption coefficient (2.56) tends to 0 when ft) — 0 as it is in reality. This is an advantage of the Rocard formula that eliminates the discrepancy between theory and experiment by taking into account inertial effects. As is seen from Eq. (2.56) and the Hubbard relation (2.28)... 

The mutual correspondence of non-Markovian and Markovian (impact) approximations becomes clear, if the second derivative of K/(t) is considered. It varies differently within three time intervals with the following bounds xc < xj < Tj1 (Fig. 2.5). Orientational relaxation occurs in times Fj1. The gap near zero has a scale of xj. A parabolic vertex of extent xc and curvature I4 > 0 is inscribed into its acute end. The narrower the vertex, the larger is its curvature, thus, in the impact approximation (tc = 0) it is equal to 00. In reality xc =j= 0, and the... 

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