Some relations from statistical thermodynamics

The relationships between defect concentrations and the independent thermodynamic variables which were derived in sections 4.1 and 4.2 are based upon the assumption that the chemical potentials of the defects are of the form Pi fJi -j- RT n Ni, and that the configurational entropy is of the form - Niln N where the summation is over all structural elements. 

This presupposes that Boltzmann statistics can be applied to the structural elements of the crystal, and expresses the assumption that the defect centers obey the laws of ideal dilute solutions. 

However, electrons in solids obey Fermi statistics. Now, Boltzmann statistics makes no assertions regarding the number of particles which are permitted in the cells in phase space, but rather, it examines the probability of finding a given number of particles in the individual cells. Fermi statistics, on the other hand, asserts from the start that, because of the Pauli principle, only two particles are permitted in any cell of phase space [7]. In order to show how Fermi statistics affects the formulation of the chemical potentials of electronic defects in crystals, and how the mass action laws are thereby affected, let us take the compound silver sulfide Aga+ S as an example [15]. 

Silver sulfide is a purely electronic conductor. Hall effect measurements show that conduction is by excess electrons. Above 180 (a-Ag2S), a high degree of cation disorder is 

Stoichiometric composition as a function of the silver activity is very accurately known. This is shown in Fig. 4-5. 5 is the excess of silver above the stoichiometric composition of the sulfide. Since the silver exists in the form of ions, d is equivalent to the concentration of excess electrons. 

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From the thermodynamic viewpoint, the basic statistical theory is still too complex to provide useful working equations, but it does suggest forms of equations with some purely theoretical terms, and other terms including parameters to be evaluated empirically. In general, the theoretical terms arise from the electrostatic interactions which are simple and well-known while the empirical, terms relate to short-range interionic forces whose characteristics are qualitatively but not quantitatively known from independent sources. But, as we shall see, this division is not complete - there are interactions between the two categories. 

The skeptical reader may reasonably ask from where we have obtained the above rules and where is the proof for the relation with thermodynamics and for the meaning ascribed to the individual terms of the PF. The ultimate answer is that there is no proof. Of course, the reader might check the contentions made in this section by reading a specialized text on statistical thermodynamics. He or she will find the proof of what we have said. However, such proof will ultimately be derived from the fundamental postulates of statistical thermodynamics. These are essentially equivalent to the two properties cited above. The fundamental postulates are statements regarding the connection between the PF and thermodynamics on the one hand (the famous Boltzmann equation for entropy), and the probabilities of the states of the system on the other. It just happens that this formulation of the postulates was first proposed for an isolated system—a relatively simple but uninteresting system (from the practical point of view). The reader interested in the subject of this book but not in the foundations of statistical thermodynamics can safely adopt the rules given in this section, trusting that a proof based on some... 

Some of the most valuable results we have obtained from chemical thermodynamics are those that relate the position of chemical equilibrium to the thermodynamic properties of the reactants and products. With the aid of statistical thermodynamics we can go one step further and relate the position of equilibrium to the masses, dimensions, and vibrational frequencies of the molecules involved. 

Abstract Fluctuation Theory of Solutions or Fluctuation Solution Theory (FST) combines aspects of statistical mechanics and solution thermodynamics, with an emphasis on the grand canonical ensemble of the former. To understand the most common applications of FST one needs to relate fluctuations observed for a grand canonical system, on which FST is based, to properties of an isothermal-isobaric system, which is the most common type of system studied experimentally. Alternatively, one can invert the whole process to provide experimental information concerning particle number (density) fluctuations, or the local composition, from the available thermodynamic data. In this chapter, we provide the basic background material required to formulate and apply FST to a variety of applications. The major aims of this section are (i) to provide a brief introduction or recap of the relevant thermodynamics and statistical thermodynamics behind the formulation and primary uses of the Fluctuation Theory of Solutions (ii) to establish a consistent notation which helps to emphasize the similarities between apparently different applications of FST and (iii) to provide the working expressions for some of the potential applications of FST. 

This book is intended to provide a few asymptotic methods which can be applied to the dynamics of self-oscillating fields of the reaction-diffusion type and of some related systems. Such systems, forming cooperative fields of a large number of interacting similar subunits, are considered as typical synergetic systems. Because each local subunit itself represents an active dynamical system functioning only in far-from-equilibrium situations, the entire system is capable of showing a variety of curious pattern formations and turbulencelike behaviors quite unfamiliar in thermodynamic cooperative fields. I personally believe that the nonlinear dynamics, deterministic or statistical, of fields composed of similar active (i.e., non-equilibrium) elements will form an extremely attractive branch of physics in the near future. 

In making models, we must respect the principles of equilibrium statistical mechanics, but cannot wholly rely on them, since we have every reason to believe that in polymer crystallization we have only a frustrated approach towards thermodynamic equilibrium. Most of the models are in some way or another founded on their authors conceptions of the nature of the process of high-polymer crystallization from the melt. That is a process on which direct detailed information is hard to get, and some imaginative extrapolation from what one knows about related problems is almost unavoidable. 

The partition function and the sum or density of states are functions which are to statistical mechanics what the wave function is to quantum mechanics. Once they are known, all of the thermodynamic quantities of interest can be calculated. It is instructive to compare these two functions because they are closely related. Both provide a measure of the number of states in a system. The partition function is a quantity that is appropriate for thermal systems at a given temperature (canonical ensemble), whereas the sum and density of states are equivalent functions for systems at constant energy (microcanonical ensemble). In order to lay the groundwork for an understanding of these two functions as well as a number of other topics in the theory of unimolecular reactions, it is essential to review some basic ideas from classical and quantum statistical mechanics. 

Assume that a finite system is in contact with a heat bath at constant temperature and driven away from equilibrium by some external time-dependent force. Many nonequilibrium statistical analyses are available for the systems in the vicinity of equilibrium. The only exception is the fluctuation theorems, which are related to the entropy production and valid for systems far away from global equilibrium. The systems that are far from global equilibrium are stochastic in nature with varying spatial and timescales. The fluctuation theorem relates to the probability distributions of the time-averaged irreversible entropy production a. The theorem states that, in systems away from equilibrium over a finite time t, the ratio between the probability that CT takes on a value A and the probability that it takes the opposite value, —A, will be exponential in At. For nonequilibrium system in a finite time, the fluctuation theorem formulates that entropy will flow in a direction opposite to that dictated hy the second law of thermodynamics. Mathematically, the fluctuation theorem is expressed as ... 

MD simulations provide the means to solve the equations of motion of the particles and output the desired physical quantities in the term of some microscopic information. In a MD simulation, one often wishes to explore the macroscopic properties of a system through the microscopic information. These conversions are performed on the basis of the statistical mechanics, which provide the rigorous mathematical expressions that relate macroscopic properties to the distribution and motion of the atoms and molecules of the N-body system. With MD simulations, one can study both thermodynamic properties and the time-dependent properties. Some quantities that are routinely calculated from a MD simulation include temperature, pressure, energy, the radial distribution function, the mean square displacement, the time correlation function, and so on (Allen and Tildesley 1989 Rapaport 2004). 

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