Relaxation Processes in the Phenomenological Theory

The general form of transfer equations for a medium of arbitrary structure, including melts and solutions of polymers, is established on the basis of conservation laws of mass, momentum, angular momentum and energy (Landau and Lifshitz 1987a, Shliomis 1966). 

A continuous medium is characterised by its mean density, a function of co-ordinates and time 

The motion of a continuous medium is described by its velocity vector v, which is a certain mean macroscopic velocity and has three components -functions of the co-ordinates and time - 

The law of conservation of mass can be written in the form of the continuity equation 

We can use the above relations to rewrite the law of conservation of momentum density in the form 

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The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

The paper is organized in five sections. In Section 4.2 we describe the phenomenological theory for the heat release, based on the tunneling model and the influence of different relaxation rates, and the cooling process. In Section 4.3 we briefly describe the experimental procedures and samples. In Section 4.4 we show and discuss the experimental results. Conclusions are drawn in Section 4.5. 

Accordingly to the phenomenological theory of the solvent polarization (see Appendix A.3), the non-equilibrium solvation is required to describe the properties of the excites states immediately after a fast vertical excitation/de-excitation process, while an equilibrium solvation may be used to describe the changes in the the excited states properties of the solvated chromophores after the solvent relaxation which follows a vertical excitation process [20, 21]. 

The perturbation theory presented in Chapter 2 implies that orientational relaxation is slower than rotational relaxation and considers the angular displacement during a free rotation to be a small parameter. Considering J(t) as a random time-dependent perturbation, it describes the orientational relaxation as a molecular response to it. Frequent and small chaotic turns constitute the rotational diffusion which is shown to be an equivalent representation of the process. The turns may proceed via free paths or via sudden jumps from one orientation to another. The phenomenological picture of rotational diffusion is compatible with both... 

A detailed discussion of the statistical thermodynamic aspects of thermally stimulated dielectric relaxation is not provided here. It should suffice to state that kinetics of most of the processes are again complicated and that the phenomenological kinetic theories used to described thermally stimulated currents make use of assumptions that, being necessary to simplify the formalism, may not always be justified. Just as in the general case, TSL and TSC, the spectroscopic information may in principle be available from the measurement of thermally stimulated depolarization current (TSDC). However, it is frequently impossible to extract it unambiguously from such experiments. 

Abstract Contribution of the Jahn-Teller system to the elastic moduli and ultrasonic wave attenuation of the diluted crystals is discussed in the frames of phenomenological approach and on the basis of quantum-mechanical theory. Both, resonant and relaxation processes are considered. The procedure of distinguishing the nature of the anomalies (either resonant or relaxation) in the elastic moduli and attenuation of ultrasound as well as generalized method for reconstruction of the relaxation time temperature dependence are described in detail. Particular attention is paid to the physical parameters of the Jahn-Teller complex that could be determined using the ultrasonic technique, namely, the potential barrier, the type of the vibronic modes and their frequency, the tunnelling splitting, the deformation potential and the energy of inevitable strain. The experimental results obtained in some zinc-blende crystals doped with 3d ions are presented. 

In the frames of phenomenological theory we cannot discuss the resonant effects and will restrict ourselves with relaxation processes only. We will follow an approach proposed by Zener [7]. For a small-amplitude wave equation (6) represents Hooke s law in terms of stress, strain and stiffness... 

A simple phenomenological theory has been given (4-7) which appears to rationalize all the experimental observations for the a, B and (aB) dielectric relaxations in amorphous solid polymers. It is assumed that a reference dipolar group in the polymer system may partially reorientate via motions in a temporary local environment (B process) but at sufficiently long times may completely reorientate by cooperative rearrangements of that local environment (a process). When the time scales of the two processes are sufficiently different, i.e. for T < Tg or for T just above Tg, the dipole moment vector correlation function may be written as (4-6)... 

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