Relaxation stochastic description

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... 

The friction coefficient y defines the timescale, y of thermal relaxation in the system described by (8.13). A simpler stochastic description can be obtained for a system in which this time is shorter than any other characteristic timescale of our system. This high friction situation is often referred to as the overdamped limit. In this limit of large y, the velocity relaxation is fast and it may be assumed to quickly reaches a steady state for any value of the applied force, that is, v = x = 0. This statement is not obvious, and a supporting (though not rigorous) argument is provided below. If true then Eqs (8.13) and (8.20) yield... 

In the stochastic approach, the Markovian random process is usually used for the description of the solvent, and it is assumed that the velocity relaxation is much faster than the coordinate relaxation.74 Such a description is applicable at long time intervals which considerably exceed the characteristic times of the electron... 

Following the solid-state approach, equations have been derived [8,9] also for the electron spin relaxation of 5 = V2 ions in solution determined by the aforementioned processes. Instead of phonons, collisions with solvent should be taken into consideration, whose correlation time is usually in the range 10"11 to 10 12 s. However, there is no satisfactory theory that unifies relaxation in the solid state and in solution. The reason for this is that the solid state theory was developed for low temperatures, while solution theories were developed for room temperature. The phonon description is a powerful one when phonons are few. By increasing temperature, the treatment becomes cumbersome, and it is more convenient to use stochastic theory (see Section 3.2) instead of analyzing the countless vibrational transitions that become active. 

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. 

Gao J, Weiner JH (1994) Monomer-level description of stress and birefringence relaxation in polymer melts. Macromolecules 27(5) 1201-1209 Gardiner CW (1983) Handbook of stochastic methods for physics, chemistry, and the natural sciences. Springer, Berlin... 

Chemical process rate equations involve the quantity related to concentration fluctuations as a kinetic parameter called chemical relaxation. The stochastic theory of chemical kinetics investigates concentration fluctuations (Malyshev, 2005). For diffusion of polymers, flows through porous media, and the description liquid helium, Fick s and Fourier s laws are generally not applicable, since these laws are based on linear flow-force relations. A general formalism with the aim to go beyond the linear flow-force relations is the extended nonequilibrium thermodynamics. Polymer solutions are highly relevant systems for analyses beyond the local equilibrium theory. 

It is important to point out that this does not imply that Markovian stochastic equations cannot be used in descriptions of condensed phase molecular processes. On the contrary, such equations are often applied successfully. The recipe for a successful application is to be aware of what can and what cannot be described with such approach. Recall that stochastic dynamics emerge when seeking coarsegrained or reduced descriptions of physical processes. The message from the timescales comparison made above is that Markovian descriptions are valid for molecular processes that are slow relative to environmental relaxation rates. Thus, with Markovian equations of motion we cannot describe molecular nuclear motions in detail, because vibrational periods (10 " s) are short relative to environmental relaxation rates, but we should be able to describe vibrational relaxation processes that are often much slower, as is shown in Section 8.3.3. 

As mentioned in the introduction, the stochastic approach can also provide a description of the effects of external pressine on spectral holes. In our experiments, however, the pressure change is always accompanied by a simultaneous change in temperature, see Ref. [20]. Therefore, the observed hole shifts and broadenings will not only be due to pressure changes, but also to the thermal expansion of the matrix. There may also be dynamical effects such as phonon scattering and (fast) tunneling systems (TLS) relaxations, which we will not treat in this contribution. 

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