Stochastic relaxation model

A. THE STOCHASTIC RELAXATION MODEL. The most general theories of magnetic relaxation in Mossbauer spectroscopy involve stochastic models see, for example. Ref. 283 for a review. A formalism using superoperators (Liouville operators) was introduced by Blume, who presented a general solution for the lineshape of radiation emitted (absorbed) by a system whose Hamiltonian jumps at random as a function of time between a finite number of possible forms that do not necessarily commute with one another. The solution can be written down in a compact form using the superoperator formalism. 

Fig. 23. Mossbauer spectra of Fe(J-mph)NO between 84 and 319 K. Solid lines result from a fit by a two-state relaxation model based on the stochastic theory of lineshapes. According to Ref. [164]...

The stochastic description of barrierless relaxations by Bagchi, Fleming, and Oxtoby (Ref. 195 and Section IV.I) was first applied by these authors to TPM dyes to explain the observed nonexponential fluorescence decay and ground-state repopulation kinetics. The experimental evidence of an activation energy obs < Ev is also in accordance with a barrierless relaxation model. The data presented in Table IV are indicative of nonexponential decay, too. They were obtained by fitting the experiment to a biexponential model, but it can be shown50 that a fit of similar quality can be obtained with the error-function model of barrierless relaxations. Thus, r, and t2 are related to r° and t", but, at present, we can only... 

For a diatomic close to thermal equilibrium the assumption of impulsive collisions with the bath molecules is realistic only when the vibrational jjeriod is longer than the duration of a collision, but this is rarely the case for diatomics vibrating near the potential minimum. However, for realistic potentials the vibrational period lengthens near the dissociation threshold, and it is not so clear that an impulsive model will be quantitatively inaccurate in modeling dissociation, even though it may fail badly in describing vibrational relaxation deep with the well. A stochastic impulsive model of dissociation of a diatomic AB, which uses the zero-frequency frictions describ-... 

Figure 9.18. (a) Long-time rate constant of forward reaction as a function of coupling in a stochastic Markovian model of reversible ET assisted by a fast vibrational mode for AG = — 2E E, = ISfegT and several values of Ef / (numbers attached to the curves), (b) The results for a 2D Markovian model with biexponential relaxation for E = El = 9k T, AG = —2E, and several values of t /t (numbers attached to the curves). The parameters Xj-and Tj denote the relaxation times of the fast and the slow mode, respectively. Note that here we have full , = - - E fixed. (Reproduced from [309].)... 

In this paper the outlines of anomalous stochastic kinetic models are discussed. These models are derived by relaxing certain traditional assumptions. Furthermore, a very specific example is given to illustrate that spatial inhomogenity can lead to time inhomogenity by some lumping procedure. 

Given a stochastic model for the turbulence frequency, it is natural to enquire how fluctuations in co will affect the scalar dissipation rate (Anselmet and Antonia 1985 Antonia and Mi 1993 Anselmet et al. 1994). In order to address this question, Fox (1997) extended the SR model discussed in Section 4.6 to account for turbulence frequency fluctuations. The resulting model is called the Lagrangian spectral relaxation (LSR) model. The LSR model has essentially the same form as the SR model, but with all variables conditioned on the current and past values of the turbulence frequency [ /(. ),. v < r. In order to simplify the notation, this conditioning is denoted by ( , e.g.,... 

Inner slip, between the solid wall and an adsorbed film, will also influence the surface-liquid boundary conditions and have important effects on stress propagation from the liquid to the solid substrate. Linked to this concept, especially on a biomolecular level, is the concept of stochastic coupling. At the molecular level, small fluctuations about the ensemble average could affect the interfacial dynamics and lead to large shifts in the detectable boundary condition. One of our main interests in this area is to study the relaxation time of interfacial bonds using slip models. Stochastic boundary conditions could also prove to be all but necessary in modeling the behavior and interactions of biomolecules at surfaces, especially with the proliferation of microfluidic chemical devices and the importance of studying small scales. 

The irradiation of water is immediately followed by a period of fast chemistry, whose short-time kinetics reflects the competition between the relaxation of the nonhomogeneous spatial distributions of the radiation-induced reactants and their reactions. A variety of gamma and energetic electron experiments are available in the literature. Stochastic simulation methods have been used to model the observed short-time radiation chemical kinetics of water and the radiation chemistry of aqueous solutions of scavengers for the hydrated electron and the hydroxyl radical to provide fundamental information for use in the elucidation of more complex, complicated chemical, and biological systems found in real-world scenarios. 

Furthermore, it may be seen that for all the normal modes of relaxation, including the most rapid, the freely jointed chain model and the Rouse model are identical if we set n = N + 1 that is, the relaxation time xp of the pth normal mode of a freely-jointed chain is the same as that of a Rouse marcromolecule composed of N + 1 subchains, each of mean square end-to-end length b2. Moreover, for the special choice a = 0, Eq. (10) is true for arbitrarily large departures from equilibrium. We thus seem to have confirmed analytically the discovery of Verdier24 that quite short chains executing a stochastic process described by Eqs. (1) and (3) on a simple cubic lattice display Rouse relaxation behavior. Of course, Verdier s Monte Carlo technique permits study of excluded volume effects, quite beyond the range of our present efforts. 

First of all, we define the transition rates for our stochastic model using an ansatz of Kawasaki [39, 40]. In the following we use the abbreviation X for an initial state (07 for mono- and oion for bimolecular steps), Y for a final state (ct[ for mono- and a[a n for bimolecular steps) and Z for the states of the neighbourhood ( cr f 1 for mono- and a -1 a -1 for bimolecular steps). If we study the system in which the neighbourhood is fixed we observe a relaxation process in a very small area. We introduce the normalized probability W(X) and the corresponding rates 8.(X —tY Z). For this (reversible) process we write down the following Markovian master equation... 

The single relaxation time approximation corresponds to a stochastic model in which the fluctuating force on a molecule has a Lorentzian spectrum. Thus if the fluctuating force is a Gaussian-Markov process, it follows that the memory function must have this simple form.64 Of course it would be naive to assume that this exponential memory will accurately account for the dynamical behavior on liquids. It should be regarded as a simple model which has certain qualitative features that we expect real memory functions to have. It decays to zero and, moreover, is of a sufficiently simple mathematical form that the velocity autocorrelation function,... 

C. An analogous model was considered in Ref. 12b, but an important new step was made. Now it was assumed that the stochastic processes with two different relaxation times correspond to types of motion described by two wells. Two different complex susceptibilities were calculated, which have split Eq. (235) by two similar expressions for reorientation and vibration processes ... 

In Section V the reorientation mechanism (A) was investigated in terms of the only (hat curved) potential well. Correspondingly, the only stochastic process characterized by the Debye relaxation time rD was discussed there. This restriction has led to a poor description of the submillimeter (10-100 cm-1) spectrum of water, since it is the second stochastic process which determines the frequency dependence (v) in this frequency range. The specific vibration mechanism (B) is applied for investigation of the submillimetre and the far-infrared spectrum in water. Here we shall demonstrate that if the harmonic oscillator model is applied, the small isotope shift of the R-band could be interpreted as a result of a small difference of the masses of the water isotopes. 

Their theory, based on the classical Bloch equations, (31) describes the exchange of non-coupled spin systems in terms of their magnetizations. An equivalent description of the phenomena of dynamic NMR has been given by Anderson and by Kubo in terms of a stochastic model of exchange. (32, 33) In the latter approach, the spectrum of a spin system is identified with the Fourier transform of the so-called relaxation function. 

In Figure 2.10 we show a selection of results, in which experimental and calculated spectra are compared at 292 and 155K. The results are quite satisfactory, especially when considering that no fitted parameters, but only calculated quantities (via QM and hydrodynamic models) have been employed. The overall satisfactory agreement of the spectral line shapes, particularly at low temperatures, is a convincing proof that the simplified dynamic modelling implemented in the SLE through the purely rotational stochastic diffusive operator f, and the hydrodynamic calculation of the rotational diffusion tensor, is sufficient to describe the main slow relaxation processes. 

It is instructive to compare the system of equations (3.46) and (3.47) with the system (3.37). One can see that both the radius of the tube and the positions of the particles in the Doi-Edwards model are, in fact, mean quantities from the point of view of a model of underlying stochastic motion described by equations (3.37). The intermediate length emerges at analysis of system (3.37) and can be expressed through the other parameters of the theory (see details in Chapter 5). The mean value of position of the particles can be also calculated to get a complete justification of the above model. The direct introduction of the mean quantities to describe dynamics of macromolecule led to an oversimplified, mechanistic model, which, nevertheless, allows one to make correct estimates of conformational relaxation times and coefficient of diffusion of a macromolecule in strongly entangled systems (see Sections 4.2.2 and 5.1.2). However, attempts to use this model to formulate the theory of viscoelasticity of entangled systems encounted some difficulties (for details, see Section 6.4, especially the footnote on p. 133) and were unsuccessful. 

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