Fluctuational Subsystems

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 

The pair correlation function of the velocities and the pair correlation functions of some time derivatives of the velocity are sometimes taken into account.75 However, the validity of this description in the nonadiabaticity regions also has to be proved. The dynamic description or the description using the differentiable random process is more rigorous in this region.76 

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We assume that the o-Ps hole-size distribution above Tg directly mirrors the thermal density fiuctuation. This allows us to extract information on the length scale of the dynamic heterogeneity in polymers. Using a fluctuation approach, the temperature dependency of the volume of the smallest representative freely fluctuating subsystem can be estimated. Limits of this interpretation for polymers with a high structural disorder, which already appears in the glass, are discussed. 

FIGURE 11.19 Mean volume ( Vsv >, considered as the volume of the smallest representative freely fluctuating subsystem, from the PALS data as a function of the temperature T at zero pressure for PMPhS. For comparison the volume that contains one hole, l/A, is shown. The lines through the data are a visual aid. (From Dlubek et al. [2007a].)... 

Region II occurs between Tg and the knee in the temperature dependency of <7/ ,themeanhole-sizedispersion,at7]tCT.Both (u/,) anda increaselinearlywith the temperature, (u varies parallel to the specific free volume Vf, so that the specific number of o-Ps holes, N h = Vf vh), is constant. From (Vsv> oc hh it follows that the mean size of the fluctuation subsystem decreases almost quadrat-ically with T. At a certain temperature, here 1.3 Tg, the subvolume (Vsv) becomes small and constant, resulting in the knee in a. At Tg the subvolume is occupied by several small holes which are needed for the cooperative rearrangement of mers (the a-process). 

It is important to note that in this method, the dynamic fluctuations associated with the QM subsystem are assumed to be independent of the fluctuations from the MM subsystem. Also, in this method we assume that the contributions of the fluctuations of the QM subsystem to the total free energy are the same along the reaction coordinate. We have recently addressed these approximations by developing a novel reaction path potential method where the dynamics of the system are sampled by employing an analytical expression of the combined QM/MM PES along the MEP [40],... 

Polarization fluctuations of a certain type were considered in the configuration model presented above. In principle, fluctuations of a more complicated form may be considered in the same way. A more general approach was suggested in Refs. 23 and 24, where Eq. (16) for the transition probability has been written in a mixed representation using the Feynman path integrals for the nuclear subsystem and the functional integrals over the electron wave functions of the initial and final states t) and t) for the electron ... 

The physical mechanism of entirely nonadiabatic and partially adiabatic transitions is as follows. Due to the fluctuation of the medium polarization, the matching of the zeroth-order energies of the quantum subsystem (electrons and proton) of the initial and final states occurs. In this transitional configuration, q, the subbarrier transition of the proton from the initial potential well to the final one takes place followed by the relaxation of the polarization to the final equilibrium configuration. 

The brief review of the newest results in the theory of elementary chemical processes in the condensed phase given in this chapter shows that great progress has been achieved in this field during recent years, concerning the description of both the interaction of electrons with the polar medium and with the intramolecular vibrations and the interaction of the intramolecular vibrations and other reactive modes with each other and with the dissipative subsystem (thermal bath). The rapid development of the theory of the adiabatic reactions of the transfer of heavy particles with due account of the fluctuational character of the motion of the medium in the framework of both dynamic and stochastic approaches should be mentioned. The stochastic approach is described only briefly in this chapter. The number of papers in this field is so great that their detailed review would require a separate article. 

Changes in the degrees of freedom in a reaction can be classified in two ways (1) classical over the barrier for frequencies o) such that hot) < kBT and (2) quantum mechanical through the barrier for two > kBT. In ETR, only the electron may move by (1) all the rest move by (2). Thus, the activated complex is generated by thermal fluctuations of all subsystems (solvent plus reactants) for which two < kBT. Within the activated complex, the electron may penetrate the barrier with a transmission coefficient determined entirely by the overlap of the wavefunctions of the quantum subsystems, while the activation energy is determined entirely by the motion in the classical subsystem. 

Now we consider thermodynamic properties of the system described by the Hamiltonian (2.4.5) it is a generalized Hamiltonian of the isotropic Ashkin-Teller model100,101 expressed in terms of interactions between pairs of spins lattice site nm of a square lattice. Hamiltonian (2.4.5) differs from the known one in that it includes not only the contribution from the four-spin interaction (the term with the coefficient J3), but also the anisotropic contribution (the term with the coefficient J2) which accounts for cross interactions of spins a m and s m between neighboring lattice sites. This term is so structured that it vanishes if there are no fluctuation interactions between cr- and s-subsystems. As a result, with sufficiently small coefficients J2, we arrive at a typical phase diagram of the isotropic Ashkin-Teller model,101 102 limited by the plausible values of coefficients in Eq. (2.4.6). At J, > J3, the phase transition line... 

Alternatively, the subsystem may simply be part of a larger system that constitutes a reservoir. In such a case the fluctuations are said to be local within a nominally homogeneous system. 

For a system similar to the one described above, the second-order term of Eq. (12.6) due to fluctuations in the volume of a subsystem is... 

In essence, dispersion forces arise from the correlation between dynamic charge density fluctuations in two different systems or in distant parts of one system. The difficulty [228] in describing vdW forces in the static LDA or gradient approaches is therefore not surprising since in a highly inhomogeneous system (exemplified by, but not limited to, a pair of separated subsystems) these correlations may be quite different from those in the uniform or near-uniform electron gas upon which the LDA and the various gradient approximations are bas. ... 

A subsystem is an open system, free to exchange charge and momentum with its environment. Thus the current density jg for any observable G is of particular importance in the mechanics of a subsystem, since a non-vanishing flux in this current implies a fluctuation in the subsystem average value of the property G. 

The non-vanishing of the subsystem average of the commutator implies a fluctuation in the value of the observable G over the subsystem as measured by the flux of its vector current density through the surface of the subsystem. Thus one anticipates and finds non-vanishing fluctuations in subsystem expectation values for observables which do not commute with H. 

Recently (Ruckenstein and Shulgin, 2003c), a method was suggested to calculate the activity coefficient of a poorly soluble solid in an ideal multicomponent solvent in terms of its activity coefficients at infinite dilution in some subsystems of the multicomponent solvent. The method, based on the fluctuation theory of solutions (Kirkwood and Buff, 1951), provided the following expression for the activity coefficient of a poorly soluble solid solute in an ideal multicomponent solvent ... 

Figure 7. The power spectrum density of energy fluctuation of (a) the acoustic subsystem h and (b) an individual mode in optical and acoustic subsystem. In the simulation, the ratio of perturbation strength is taken as X = /1 / K — 5.0.

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