Density, classical equilibrium

The density functional theory for classical(equilibrium) statistical mechanics is generalized to deal with various dynamical processes associated with density fluctuations in liquids and solutions. This is effected by deriving a Langevin-diffusion equation for the density field. As applications of our theory we consider density fluctuations in both supercooled liquids and molecular liquids and transport coefficients. 

This model of the liquid will be characterized by some macroscopic quantities, to be selected among those considered by classical equilibrium thermodynamics to define a system, such as the temperature T and the density p. This macroscopic characterization should be accompanied by a microscopic description of the collisions. As we are interested in chemical reactions, one is sorely tempted to discard the enormous number of non-reactive collisions. This temptation is strenghtened by the fact that reactive collisions often regard molecules constituting a minor component of the solution, at low-molar ratio, i.e. the solute. The perspective of such a drastic reduction of the complexity of the model is tempered by another naive consideration, namely that reactive collisions may interest several molecular partners, so that for a nominal two body reaction A + B —> products, it may be possible that other molecules, in particular solvent molecules, could play an active role in the reaction. 

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

In equilibrium statistical mechanics involving quantum effects, we need to know the density matrix in order to calculate averages of the quantities of interest. This density matrix is the quantum analog of the classical Boltzmann factor. It can be obtained by solving a differential equation very similar to the time-dependent Schrodinger equation... 

The implimentation of quantum statistical ensemble theory applied to physically real systems presents the same problems as in the classical case. The fundamental questions of how to define macroscopic equilibrium and how to construct the density matrix remain. The ergodic theory and the hypothesis of equal a priori probabilities again serve to forge some link between the theory and working models. 

The canonical ensemble was developed as the appropriate description of a system in thermal equilibrium with its surroundings by free exchange of energy. Following the discussion of classical systems the density operator of the canonical ensemble is introduced axiomatically as... 

It is clear that the strong form of the QCT is impossible to obtain from either the isolated or open evolution equations for the density matrix or Wigner function. For a generic dynamical system, a localized initial distribution tends to distribute itself over phase space and either continue to evolve in complicated ways (isolated system) or asymptote to an equilibrium state (open system) - whether classically or quantum mechanically. In the case of conditioned evolution, however, the distribution can be localized due to the information gained from the measurement. In order to quantify how this happ ens, let us first apply a cumulant expansion to the (fine-grained) conditioned classical evolution (5), resulting in the equations for the centroids (x = (t), P= (P ,... 

Nuclear spin relaxation is considered here using a semi-classical approach, i.e., the relaxing spin system is treated quantum mechanically, while the thermal bath or lattice is treated classically. Relaxation is a process by which a spin system is restored to its equilibrium state, and the return to equilibrium can be monitored by its relaxation rates, which determine how the NMR signals detected from the spin system evolve as a function of time. The Redfield relaxation theory36 based on a density matrix formalism can provide... 

The electrostatic Hellmann-Feynman theorem states that for an exact electron wave function, and also of the Hartree-Fock wave function, the total quantum-mechanical force on an atomic nucleus is the same as that exerted classically by the electron density and the other nuclei in the system (Feynman 1939, Levine 1983). The theorem thus implies that the forces on the nuclei are fully determined once the charge distribution is known. As the forces on the nuclei must vanish for a nuclear configuration which is in equilibrium, a constraint may be introduced in the X-ray refinement procedure to ensure that the Hellmann-Feynman force balance is obeyed (Schwarzenbach and Lewis 1982). 

Feynman first suggested - that the path centroid may be the most classical-like variable in an equilibrium quantum system, thus providing the basis for the formulation of a classical-like equilibrium density function. The path centroid... 

Below we present a well-known calculation of membrane potential based on the classical Teorell-Meyer-Sievers (TMS) membrane model [2], [3]. The essence of this model is in treating the ion-selective membrane as a homogeneous layer of electrolyte solution with constant fixed charge density and with local ionic equilibrium at the membrane/solution interfaces. In spite of the obvious idealization involved in the first assumption the TMS model often yields useful results and represents in fact the main tool for practical membrane calculations. We shall return to TMS once again in 4.4 when discussing the electric current effects upon membrane selectivity. In the case of our present interest, the simplest TMS model of membrane potential for a 1,2 valent electrolyte reads... 

It seems to me that we can scarcely progress in our understanding of the structural and kinetic effects of the H-bond without knowing the AG and AH terms involved, so I intend to discuss some methods of determining them. The references will provide simple examples of the methods mentioned. The most significant AG and AH values are those evaluated from equilibrium measurements in the gas phase—either by classical vapour density measurements, the second virial coefficient [1], or from, spectroscopic, specific heat or thermal conductance [2], or ultrasonic absorptions [3]. All these methods essentially measure departures from the ideal gas laws. The second virial coefficient provides a measure of the equilibrium constant for the formation of collision dimers in the vapour as was emphasized by Dr. Rowlinson in the discussion, this factor is particularly significant as only the monomer-dimer interaction contributes to it. 

Equilibrium of Adsorption, (a) The Soap Films. The isotherms of adsorption of the organic cations of the soaps are reproduced on the Figure 2. Classical methods (20) are used to obtain the surface densities of the soaps I and IV. 

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