Statistical Fluctuation Theory

The fluctuation theory has received attention because it avoids some of the serious assumptions involved in the rate theory. The beginnings of fluctuation theory were presented by Einstein. Various workers since 

The advantages of the fluctuation theory are that it does not require that clusters be spheres, they need not have sharply defined bounding surfaces, nor is an equilibrium between phases assumed. The disadvantage is a practical one how can the work term (defined later) be evaluated  

The density, temperature, energy, and other properties of a given mass of liquid are statistical averages. For a tiny volume within the liquid the properties are not constant. The density, for example, fluctuates rapidly about the mean value. 

Imagine the liquid to be divided into microscopic cells as shown in Fig. 22. The volume of a cell must be chosen carefully to have a particular, special value, described later. A cell is small enough so that occasionally it may be empty. At other times one or many molecules may occupy the cell. Consider all possible configurations of all the molecules in all the cells. The most probable configuration will exist for a large mass for a long time. In this respect the most probable distribution is a description of the steady-state condition. The problem is to calculate the most likely distribution. Once this is known, the rate of formation of any density in a cell can be calculated. 

If the liquid molecules follow the gas law, a configurational integral solution of the possible configurations can be deduced (R2). A highly condensed summary of the procedure follows. 

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From statistical fluctuation theory (32), the static structnre factor is given by... 

R. Kubo, Statistical-mechanical theory of irreversible processes. 1. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Japan 12, 570 (1957) R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29, 255 (1966). 

The acceptance criteria for the Gibbs ensemble were originally derived from fluctuation theory [17]. An approximation was implicitly made in the derivation that resulted in a difference in the acceptance criterion for particle transfers proportional to 1/N relative to the exact expressions given subsequently [18]. A full development of the statistical mechanics of the ensemble was given by Smit et al. [19] and Smit and Frenkel [20], which we follow here. A one-component system at constant temperature T, total volume V, and total number of particles N is divided into two regions, with volumes Vj and Vu = V - V, and number of particles Aq and Nu = N - N. The partition function, Q NVt is... 

The conference was divided into four parts to each of which a full day was devoted the first one treated Equilibrium Statistical Mechanics, with special regard to The Theory of Critical Phenomena the second part regarded Nonequilibrium Statistical Mechanics. Cooperative Phenomena the third one, The Macroscopic Approach to Coherent Behavior in Far Equilibrium Conditions and the fourth and last, Fluctuation Theory and Nonequilibrium Phase Transitions. ... 

Shankland and collaborators [56] thoroughly reviewed Miller s results, and applied formal statistical tests to Miller s data to conclude that [56, p. 171] there can be little doubt that statistical fluctuations alone cannot account for the periodic fringe-shifts observed by Miller. To any outsider, this remark highly commends the experimental quality of Miller s work. However, regarding the curves depicting seasonal variations, Shankland et al. also noted that, according to their (Shankland s) theory [56, p. 172] the four curves should have a common maximum (or minimum). .. only the amplitude may be different at different epochs. ... 

One cannot expect a molecule that follows a random migration path, full of frivolous excursion, to arrive after a fixed time at exactly the same point as its equally frivolous companions. There will be a mean distance of migration X, but the individual molecules will exhibit fluctuations about this mean due to the peculiarities of their own migration. These statistical fluctuations will lead to zone broadening. The statistical (stochastic) theory of zone broadening was first developed by Giddings and Eyring [4] and has been expanded subsequently by a number of authors [5-8]. 

STATISTICAL MECHANICS Principles and Applications, Terrell L. Hill. Standard text covers fundamentals of statistical mechanics, applications to fluctuation theory, imperfect gases, distribution functions, more. 448pp. 5X 8X. 

This chapter discusses several statistical mechanical theories that are strongly positioned in the historical sweep of the theory of liquids. They are chosen for inclusion here on the basis of their potential for utility in analyzing simulation calculations, and their directness in conneeting to the other fundamental topic discussed in this book, the potential distribution theorem. Therefore tentacles can be understood as tentacles of the potential distribution theorem. From the perspective of the preface discussion, the theories presented here might be useful for discovery of models such as those discussed in Chapter 4. These theories are a significant subset of those referred to in Chapter 1 as ... both difficult and strongly established. .. (Friedman and Dale, 1977), but the present chapter does not exhaust the interesting prior academic development of statistical mechanical theories of solutions. Sections 6.2 and 6.3 discuss alternative views of chemical potentials, namely those of density functional theory and fluctuation theory. 

By equation (193), the Kerr constant raises the value of Ae but lowers Ae". The general relation (194) states that in order to have the value — 2 resulting from Langevin s reorientation theory one has to neglect the quantity Cm due to statistical fluctuations and non-linear polarizabilities. On neglecting Cm and and taking into account that 2 = 5K, equation (194) yields + 3, in conformity with Voigt s non-linear deformation theory. 

Some interesting behavior in single-molecule spectroscopy involves the stochastic migration of lines. Usual statistical quantum theory describes only mean values or dispersions of observables, but not the actual fluctuations in the dynamics of single quantum systems. In an individual formalism of quantum mechanics, such fluctuations are of great importance. 

Another approach is to employ rigorous statistical thermodynamic theories. In this paper, the Kirkwood-Buff (KB) theory of solutions (fluctuation theory of solutions) is employed to analyze the thermodynamics of multicomponent mixtures, with the emphasis on quaternary mixtures. This theory connects the macroscopic properties of re-component solutions, such as the isothermal compressibility, the concentration deriva-... 

In contrast to the present treatment there are two types of earlier theories of refraction of light. Yvon32 has developed a statistical-mechanical theory of the refractive index. This theory is set up in such a way that an explicit expression is obtained for the index of refraction. It does not, however, contain an analysis of the optical phenomena (such as the extinction of the incident field) which are involved. These last aspects are considered very carefully in the other, electrodynamic, type of theory, which Hoek,8 following work done by a number of authors, has presented with great rigor. The disadvantage of this second method is that macroscopic quantities are not obtained by statistical-mechanical methods, but by averaging the microscopic quantities oVer physically infinitesimal volume elements. The result is that almost all the effect of density fluctuations is lost. Both of the theories mentioned assume furthermore thp molecular polarizability to be a constant independent of intermolecular distances. 

According to thermodynamic fluctuation theory, fluctuations in the density and temperature are statistically independent. This is demonstrated in Appendix (10. Q [cf. Eq. (10.C.27)] thus proving (a). Conditions (b) and (c) follow from the fact that... 

It is also interesting to consider the classical/quantal correspondence in the number of energized molecules versus time N(/, E), Eq. (8.22), for a microcanonical ensemble of chaotic trajectories. Because of the above zero-point energy effect and the improper treatment of resonances by chaotic classical trajectories, the classical and quantal I l( , t) are not expected to agree. For example, if the classical motion is sufficiently chaotic so that a microcanonical ensemble is maintained during the decomposition process, the classical N(/, E) will be exponential with a rate constant equal to the classical (not quantal) RRKM value. However, the quantal decay is expected to be statistical state specific, where the random 4i s give rise to statistical fluctuations in the k and a nonexponential N(r, E). This distinction between classical and quantum mechanics for Hamiltonians, with classical f (/, E) which agree with classical RRKM theory, is expected to be evident for numerous systems. 

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