Statistical dynamics, phenomenological

Aiming to establish the fundaments of the chemical reactivity on quantum mathematical-physical concepts, a new way can be approached, by considering the statistical quantum phenomenology of the multi-electronic processes. In this context, the relationship between the Heisenberg and Schrodinger dynamic formalisms is constituted to be the starting point. 

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. 

Equation (139) was already discussed elsewhere [22,23,31] as a phenomenological representation of the dynamic equation for the CC law. Thus, Eq. (139) shows that since the fractional differentiation and integration operators have a convolution form, it can be regarded as a consequence of the memory effect. A comprehensive discussion of the memory function (137) properties is presented in Refs. 22 and 23. Accordingly, Eq. (139) holds for some cooperative domain and describes the relaxation of an ensemble of microscopic units. Each unit has its own microscopic memory function m (t), which describes the interaction between this unit and the surroundings (interaction with the statistical reservoir). The main idea of such an interaction was introduced in Refs. 22 and 23 and suggests that mg(f) JT 8(f,- — t) (see Fig. 50). 

Consideration will be restricted to dilute sprays, so that the statistical fluctuations in the flow, which are induced by the random motion of individual particles, may be neglected. Therefore, our objective is to obtain the hydrodynamic equations for the (local) average properties of the gas. These equations will be derived by phenomenological reasoning and will be shown to be equivalent to the ordinary equations of fluid dynamics, with suitably added source terms accounting for the average effect of the spray. For the sake of generality, allowance will be made for M different kinds of droplets and N different chemical species in the gas. 

In this example the master equation formalism is appliedto the process of vibrational relaxation of a diatomic molecule represented by a quantum harmonic oscillator In a reduced approach we focus on the dynamics of just this oscillator, and in fact only on its energy. The relaxation described on this level is therefore a particular kind of random walk in the space of the energy levels of this oscillator. It should again be emphasized that this description is constructed in a phenomenological way, and should be regarded as a model. In the construction of such models one tries to build in all available information. In the present case the model relies on quantum mechanics in the weak interaction limit that yields the relevant transition matrix elements between harmonic oscillator levels, and on input from statistical mechanics that imposes a certain condition (detailed balance) on the transition rates. 

These results provide a strong phenomenological link between the property of exponential divergence and statistical reaction dynamics. More work is required, however, to explain why the specific numerical value of 103 divergence proves useful. 

On a modest level of detail, kinetic studies aim at determining overall phenomenological rate laws. These may serve to discriminate between different mechanistic models. However, to it prove a compound reaction mechanism, it is necessary to determine the rate constant of each elementary step individually. Many kinetic experiments are devoted to the investigations of the temperature dependence of reaction rates. In addition to the obvious practical aspects, the temperature dependence of rate constants is also of great theoretical importance. Many statistical theories of chemical reactions are based on thermal equilibrium assumptions. Non-equilibrium effects are not only important for theories going beyond the classical transition-state picture. Eventually they might even be exploited to control chemical reactions [24]. This has led to the increased importance of energy or even quantum-state-resolved kinetic studies, which can be directly compared with detailed quantum-mechanical models of chemical reaction dynamics [25,26]. 

Dynamic DFT method is usitally used to model the dynamic behavior of polymer systems and has been implemented in the software package Mesodyn TM from Accelrys [41]. The DFT models the behavior of polymer fluids by combining Gaussian mean-field statistics with a TDGL model for the time evolution of conserved order parameters. However, in contrast to traditional phenomenological free-energy expan-... 

Kinetic theory, non-equilibrium statistical mechanics and non-equilibrium molecular dynamics (NEMD) have proved to be useful in estimating both straight and cross-coefficients such as thermal conductivity, viscosity and electrical conductivity. In a typical case, cross-coefficient in case of electro-osmosis has also been estimated by NEMD. Experimental data on thermo-electric power has been analysed in terms of free electron gas theory and non-equilibrium thermodynamic theory [9]. It is found that phenomenological coefficients are temperature dependent. Free electron gas theory has been used for estimating the coefficients in homogeneous conductors and thermo-couples. 

For the dynamic critical phenomena, Debye s phenomenological approach, developed for mean-statistical critical phenomena (see siib.section 2.3.5), has turned out to be helpful. Introducing a gradient term into AF heis led (Fixman, 1966 Chu et al., 1972 Tscharnuter et al., 1972) to... 

The above dimensional analysis is at rather phenomenological and/or macroscopic levels, although it has succeeded in explaining the simulation results. Indeed, in the above analysis, we have neglected mesoscopic effects such as the system inhomogeneity, density fluctuations, and collective vortex motion. Thus, to understand the annihilation dynamics in a mesoscopic level, the statistical-mechanical treatment is needed. Here, as one of such attempts, we review the hydrodynamical approach for the annihilation dynamics due to Ginzberg et al. [7]. 

All previously discussed methods are primarily based on phenomenological considerations, in contrast to chapter six (K. Binder et al.), which starts from statistical thermodynamics. This section reviews the state of the art in fields of Mraite-Carlo and Molecular Dynamics simulations. These methods are powerful tools for the prediction of macroscopic properties of matter from suitable models for effective interactions between atoms and molecules. The final chapter (G. Sadowski) makes use of the results obtained with simulation tools for the establishment of molecular-based equations of state for engineering applications. This approach enables the description and in some cases even the prediction of the phase behavior as a function of pressure, temperature, molecular weight distribution and for copolymers also as a function of chemical composition. 

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