Statistical-mechanic summary

The large deformability as shown in Figure 21.2, one of the main features of rubber, can be discussed in the category of continuum mechanics, which itself is complete theoretical framework. However, in the textbooks on rubber, we have to explain this feature with molecular theory. This would be the statistical mechanics of network structure where we encounter another serious pitfall and this is what we are concerned with in this chapter the assumption of affine deformation. The assumption is the core idea that appeared both in Gaussian network that treats infinitesimal deformation and in Mooney-Rivlin equation that treats large deformation. The microscopic deformation of a single polymer chain must be proportional to the macroscopic rubber deformation. However, the assumption is merely hypothesis and there is no experimental support. In summary, the theory of rubbery materials is built like a two-storied house of cards, without any experimental evidence on a single polymer chain entropic elasticity and affine deformation. 

Table 3.2. Summary of the most important results of statistical mechanics for diatomic gases.

The theory of electron-transfer reactions presented in Chapter 6 was mainly based on classical statistical mechanics. While this treatment is reasonable for the reorganization of the outer sphere, the inner-sphere modes must strictly be treated by quantum mechanics. It is well known from infrared spectroscopy that molecular vibrational modes possess a discrete energy spectrum, and that at room temperature the spacing of these levels is usually larger than the thermal energy kT. Therefore we will reconsider electron-transfer reactions from a quantum-mechanical viewpoint that was first advanced by Levich and Dogonadze [1]. In this course we will rederive several of, the results of Chapter 6, show under which conditions they are valid, and obtain generalizations that account for the quantum nature of the inner-sphere modes. By necessity this chapter contains more mathematics than the others, but the calculations axe not particularly difficult. Readers who are not interested in the mathematical details can turn to the summary presented in Section 6. 

Since the pioneer work of Mayer, many methods have become available for obtaining the equilibrium properties of plasmas and electrolytes from the general formulation of statistical mechanics. Let us cite, apart from the well-known cluster expansion 22 the collective coordinates approach, the dielectric constant method (for an excellent summary of these two methods see Ref. 4), and the nodal expansion method.23... 

The microscopic structure of water at the solution/metal interface has been the focus of a large body of literature, and excellent reviews have been published summarizing the extensive knowledge gained from experiments, statistical mechanical theories of varied sophistication, and Monte Carlo and molecular dynamics computer simulations. To keep this chapter to a reasonable size, we limit ourselves to a brief summary of the main results and to a sample of the type of information that can be gained from computer simulations. 

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. 

In summary, bond and group additivity rules, as well as the model compound approach, in conjunction with statistical mechanics, represent useful tools for the estimation of thermochemical properties. However, their utility for the determination of thermochemistry of new classes of compounds is limited, especially with regard to the determination of Aiff. For new classes of compounds, we must resort to experiments, as well as to computational quantum mechanical methods. 

Spherical nonpolar molecules obey an interaction potential which has the characteristic shape shown in Fig. 2. At large values of the separation r it is known that the potential curve has the shape — r 6, and at short distances the potential curve rises exponentially the exact shape of the bottom part of the curve is not very well known. Numerous empirical equations of the form of Eq. (78) have been suggested for describing the molecular interaction given pictorially in Fig. 2. The discussion here is restricted to the two most important empirical functions. A rather complete summary of the contributions to intermolecular potential energy and empirical intermolecular potential energy functions used in applied statistical mechanics may be found in (Hll, Sec. 1.3) ... 

We wish to acknowledge with many thanks the cooperation of a number of associates who took part in our series of investigations on the helix-coil transition of polypeptides. In fact, this article is a summary of their endeavors. We are particularly indebted to Dr. T. Norisuye, who not only made valuable contributions, both theoretical and experimental, to our research project but also gave us a good many pertinent comments in the course of the preparation of this manuscript. Finally, we must record that we learned much about the statistical mechanics of polypeptides through personal contact with the late Dr. K. NagaL... 

In summary, we have seen that the application of microscopic reversibility for the forward and reverse cross-sections and the use of complete equilibrium distributions for the evaluation of the statistical rate constant lead to the usual results known from equilibrium statistical mechanics. If one knows the cross-section for a forward reaction, one can always determine the inverse cross-section through the principle of microscopic reversibility. Also, if one knows the cross-section for the forward reaction, and in addition one knows that the translational and internal distribution functions of reactants and products have reached equilibrium, one can calculate the rate constant. Detailed balance then permits the calculation of the reverse rate constant. 

A summary of the relevant statistical mechanics is presented in the next section. 

The calculation of entropies for gaseous species generally requires detailed knowledge of geometry, bond distances and vibrational frequencies. We have developed a statistical mechanical model which allows an estimation of the entropy of an unknown molecule using as input only the atomic masses and Interatomic distances. Details of the development of the model have been given in previous publications (4). A brief summary of the important assumptions and equations is given below. 

Summary of Applicable Results of Thermodynamics and Statistical Mechanics... 

Two important objectives of statistical mechanics are (1) to verify the laws of thermodynamics from a molecular viewpoint and (2) to make possible the calculation of thermodynamic properties from the molecular structure of the material composing the system. Since a thorough discussion of the foundations, postulates, and formal development of statistical mechanics is beyond the scope of this summary, we shall dispose of objective (1) by merely stating that for all cases in which statistical mechanics has successfully been developed, the laws quoted in the preceding section have been found to be valid. Furthermore, in discussing objective (2), we shall merely quote results the reader is referred to the literature [3-7] for amplification. 

Several excellent reviews on the statistical mechanics of the electrical double layer have been published (Camle and Torrie, Blum, Blum and Henderson, see sec. 3.15c). In this section we give a summary of the most Importcint elements of the statistical mechanical approach and indicate the improvements with respect to the Gouy-Chapman approach. Our treatment follows the review of Camie and Torrie. ... 

This section is devoted to a brief summary of the pertinent principles of d mamics and classical statistical mechanics. Hence, it establishes much of the notation used later presenting the kinetic theory concepts. 

The statistical mechanics formalism is probably the most effrcient way to connect molecular models with experimental data. We present here a brief summary of the most important equations used for numerical simulations. Of all the statistical ensembles that can be employed, the canonical and grand canonical... 

The rest of Section I contains a summary of the general concepts used throughout this article. Section II contains a statistical-mechanical derivation of Maxwell s equations in matter. Section III deals with the problem of the ponderomotive force in a dielectric. In Section IV we give the theory of refraction of light in a medium of isotropic molecules, having a polarizability which is a function of intermolecular distances. We also calculate in this section the polarizability as a function of intermolecular distances in a very simple case (helium gas). Finally, Section V is concerned with the theory of light scattering. 

For geochemical purposes, the dependence of isotope fractionation factors on temperature is the most important property. In principle, fractionation factors for isotope exchange reactions are also slightly pressure-dependent, but experimental studies have shown the pressure dependence to be of no importance within the outer earth environments (Hoefs 2004). Occasionally, the fractionation factors can be calculated by means of partition functions derivable from statistical mechanics. However, the interpretation of observed variations of the isotope distribution in nature is largely empirical and relies on observations in natural environments or experimental results obtained in laboratory studies. A brief summary of the theory of isotope exchange reactions is given by Hoefs (2004). 

In this chapter, we begin with some remarks on the technological and scientific importance of complex materials and interfaces and motivate the study of interface and surface properties. We then review some of the physical and mathematical methods that are used in the subsequent discussions of interface and membrane statistical thermodynamics. Many of these topics are discussed more fully in the references and throughout this chapter. We begin with a review of classical statistical mechanics ", including a description of fluctuations about equilibrium and of binary mixtures. The mathematical description of an interface is then presented (using only vector calculus) and the calculation of the area and curvature of an interface wifli an arbitrary shape is demonstrated. Finally, the chapter is concluded by a brief summary of hydrodynamics. ... 

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