Macroscopic theory

Flaving now developed some of the basic notions for the macroscopic theory of nonlinear optics, we would like to discuss how the microscopic treatment of the nonlinear response of a material is handled. Wliile the classical nonlinear... 

The focus of the present chapter is the application of second-order nonlinear optics to probe surfaces and interfaces. In this section, we outline the phenomenological or macroscopic theory of SHG and SFG at the interface of centrosymmetric media. This situation corresponds, as discussed previously, to one in which the relevant nonlinear response is forbidden in the bulk media, but allowed at the interface. 

Thus one must rely on macroscopic theories and empirical adjustments for the determination of potentials of mean force. Such empirical adjustments use free energy data as solubilities, partition coefficients, virial coefficients, phase diagrams, etc., while the frictional terms are derived from diffusion coefficients and macroscopic theories for hydrodynamic interactions. In this whole field of enquiry progress is slow and much work (and thought ) will be needed in the future. 

FLUID DYNAMICAL ASPECTS AND MACROSCOPIC THEORY. The following section shows that one can join statistical mechanics with fluid dynamics in the spirit of the global simulations this link is essential. The conceptual, intellectual and practical importance of this link is equally important and we are confident to have opened an important path to further understand physical phenomena. 

The EMD studies are performed without any external electric field. The applicability of the EMD results to useful situations is based on the validity of the Nemst-Planck equation, Eq. (10). From Eq. (10), the current can be computed from the diffusion coefficient obtained from EMD simulations. It is well known that Eq. (10) is valid only for a dilute concentration of ions, in the absence of significant ion-ion interactions, and a macroscopic theory can apply. Intuitively, the Nemst-Planck theory can be expected to fail when there is a significant confinement effect or ion-wall interaction and at high electric... 

This strategy permits one to test the validity of macroscopic theories on microscopic length scales, the reliability of experimental techniques and, vice versa, the appropriateness of the CFD treatment. Furthermore, having put the simulations on a safe basis also enables one to predict transport features outside the experimentally accessible parameter range with some confidence of reliability [8]. 

R. Graham, Macroscopic theory of activated decay of metastable states, J. Stat. Phys. 60, 675 (1990). 

Elastomers are solids, even if they are soft. Their atoms have distinct mean positions, which enables one to use the well-established theory of solids to make some statements about their properties in the linear portion of the stress-strain relation. For example, in the theory of solids the Debye or macroscopic theory is made compatible with lattice dynamics by equating the spectral density of states calculated from either theory in the long wavelength limit. The relation between the two macroscopic parameters, Young s modulus and Poisson s ratio, and the microscopic parameters, atomic mass and force constant, is established by this procedure. The only differences between this theory and the one which may be applied to elastomers is that (i) the elastomer does not have crystallographic symmetry, and (ii) dissipation terms must be included in the equations of motion. 

The relation between the modulus of elasticity and the smallest nonzero eigenvalue of the connectivity matrix established here lends support to the theory3 that has been developing in recent years. By utilizing some of the techniques that have been applied to the solid state one gains an important relation between macroscopic theory and statistical mechanics. 

This macroscopic theory justifies the complex nature of the dielectric permittivity for media with dielectric loss. The real part of the dielectric permittivity expresses the orienting effect of the electric field with the component of polarization which fol-... 

From a study of overall rate constant k(T) for a reaction in the bulk and its dependence on concentrations of reactants, catalyst/inhibitor, temperature etc., the kinetics come up with a mechanism by putting together a lot of direct and indirect evidences. The determination of the overall rate constant k(T) using transition state theory was a more sophisticated approach. But the macroscopic theories such as transition state theory in different versions are split to some extent in some cases, e.g. for very fast reactions. The experimental and theoretical studies in reaction dynamics have given the indications under which it becomes less satisfactory and further work in this direction may contribute much more to solve this problem. 

The fact that the molecules are adsorbed on a solid surface giv rise to a number of new effects compared to the gas phase situation. The experimental situation consists of a monolayer of molecules adsorbed on a metal surface, on which we shine infrared radiation and then detect the reflected light. The macroscopic theory for the electromagnetic response of such a system is reproduce in the previous reviews. A more microscopic treatment has been given by Persson, showing that the integrated infrared absorptance for p-polarized light is given by ... 

Kinetics is a macroscopic theory. Dynamics is particle physics. Statistical theory relates both fields and goes beyond statistical thermodynamics. It is not the aim of this book to enter the Field of statistical theory. However, a number of its concepts are needed for a correct understanding of kinetic parameters and for constructing appropriate models. In this sense, the following sections will be presented. 

Described in Section 2.1.1 the formal kinetic approach neglects the spatial fluctuations in reactant densities. However, in recent years, it was shown that even formal kinetic equations derived for the spatially extended systems could still be employed for the qualitative treatment of reactant density fluctuation effects under study in homogeneous media. The corresponding equations for fluctuational diffusion-controlled chemical reactions could be derived in the following way. As any macroscopic theory, the formal kinetics theory operates with physical quantities which are averaged over some physically infinitesimal volumes vq = Aq, neglecting their dispersion due to the atomistic structure of solids. Let us define the local particle concentrations... 

Of special interest in the recent years was the kinetics of defect radiation-induced aggregation in a form of colloids-, in alkali halides MeX irradiated at high temperatures and high doses bubbles filled with X2 gas and metal particles with several nanometers in size were observed [58] more than once. Several theoretical formalisms were developed for describing this phenomenon, which could be classified as three general categories (i) macroscopic theory [59-62], which is based on the rate equations for macroscopic defect concentrations (ii) mesoscopic theory [63-65] operating with space-dependent local concentrations of point defects, and lastly (iii) discussed in Section 7.1 microscopic theory based on the hierarchy of equations for many-particle densities (in principle, it is infinite and contains complete information about all kinds of spatial correlation within different clusters of defects). 

Let the system contain at the initial time a certain number N of nuclei of exactly the same size. The distribution function Z is equal to zero everywhere except at one specific point (curve 1, Fig. 3). In the macroscopic theory, each nucleus changes with time in a quite definite way, depending on its size and external conditions N nuclei which were identical at the initial moment will remain identical even after a certain time interval t, and curve 1 will be shifted as a whole to another place (curve 2) that corresponds to the change in the size of the nucleus, in accordance with the kinetics equation of the form... 

Let us compare the macroscopic theory of such a regime with the theory of the normal mechanism (NM) of detonation, presented in 11.4 and II.5. In both cases a substance which is compressed by a shock wave and whose parameters are determined by its velocity enters into the reaction. 

Such a comparison between experiment and theory would be interesting insofar as Vr is computed at distances of ca. 10 A, where the validity of the macroscopic theories used may be questionable. However, using the Arrhenius equation (see Equation 16) is not necessarily valid as it is written. It is assumed implicitly in writing Equation 16 that E is temperature independent. If rewriting the functional form of fci as given in Equation 14 into the form given in Equation 16 embeds a temperature dependence into E, then the Arrhenius plots will yield anomalous results. 

In the microscopic calculation pairwise additivity was assumed. We ignored the influence of neighboring molecules on the interaction between any pair of molecules. In reality the van der Waals force between two molecules is changed by the presence of a third molecule. For example, the polarizability can change. This problem of additivity is completely avoided in the macroscopic theory developed by Lifshitz [118,119]. Lifshitz neglects the discrete atomic structure and the solids are treated as continuous materials with bulk properties such as the... 

Friction is one of the oldest subjects in the history of science and technology and its importance in everyday life is obvious. Nevertheless, there is no precise macroscopic theory of friction that would allow us to predict the frictional force between two given bodies. For nearly every rule there are exceptions. An introduction to friction phenomena is Ref. [454], a detailed book on sliding friction is Ref. [455],... 

In the macroscopic theory, a state of a physical (chemical) system is described by a set of thermodynamic parameters ,-,/= 1,..., n. These parameters and their derivatives determine the values of the thermodynamic fluxes/,- and forces Xj, i= 1,.. 

This function shall provide the basis of the link between the "macroscopic" theory and the "microscopic" theory that deals with the properties (such as membrane transport) of a single cell. 

See R. Podgomik, G. Cevc, and B. Zeks, "Solvent structure effects in the macroscopic theory of van der Waals forces," J. Chem. Phys., 87, 5957-66 (1987), for a systematic exposition of problems in general formulation and solutions of specific cases involving (o> k). 

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