Thermodynamic probability statistics

The present theory can be placed in some sort of perspective by dividing the nonequilibrium field into thermodynamics and statistical mechanics. As will become clearer later, the division between the two is fuzzy, but for the present purposes nonequilibrium thermodynamics will be considered that phenomenological theory that takes the existence of the transport coefficients and laws as axiomatic. Nonequilibrium statistical mechanics will be taken to be that field that deals with molecular-level (i.e., phase space) quantities such as probabilities and time correlation functions. The probability, fluctuations, and evolution of macrostates belong to the overlap of the two fields. 

We shall begin (Section II) by assembling the basic equipment. Section II.A formulates the problem in the complementary languages of thermodynamics and statistical mechanics. The shift in perspective—from free energies in the former to probabilities in the latter—helps to show what the core problem of phase behavior really is a comparison of the a priori probabilities of two regions of configuration space. Section II.B outlines the standard portfolio of MC tools and explains why they are not equal to the challenge posed by this core problem. 

At a conceptual level, Eq. (10) provides a helpful link between the languages of thermodynamics and statistical mechanics. According to the familiar mantra of thermodynamics, the favored phase will be that of minimal free energy, from a statistical mechanics perspective the favored phase is the one of maximal probability, given the probability partitioning implied by Eq. (1). 

Planck s hypothesis is based on the statistical thermodynamics in that the entropy is related to the number of possible energy states for a given energy or thermodynamic probability W ... 

As with the Fermi-Dirac statistics, this is a product of terms, one for each of the G cells to find the whole number of complexions, or the thermodynamic probability, we have... 

Boltzmann6 proposed that at the temperature T = 0, all thermal motion stops (except for zero-point vibration), and the entropy function S can be evaluated by a statistical function W, called the thermodynamic probability W (or, as we will learn in Section 5.2, the partition function Q for a microcanonical ensemble) ... 

To truly appreciate how thermodynamic principles apply to chemical systems, it is of great value to see how these principles arise from a statistical treatment of how microscopic behavior is reflected on the macroscopic scale. While this appendix by no means provides a complete introduction to the subject, it may provide a view of thermodynamics that is refreshing and exciting for readers not familiar with the deep roots of thermodynamics in statistical physics. The primary goal here is to provide rigorous derivations for the probability laws used in Chapter 1 to introduce thermodynamic quantities such as entropy and free energies. 

Data analysis options include the preparation of isoenergy contour maps (CMAPS), linear energy vs. rotation angle plots (LINPLOT) and tables of local energy minima, statistical thermodynamic probabilities, and entropy terms to enable the reporting of "free" energies in addition to "conformational" energies. 

In order to determine the statistical thermodynamic probabilities and entropies for the conformational energy surface, a set of "dots" is plotted indicating the angular values of the set of conformers which define the surface. The joystick curser control 1s used to select the set of conformers which occupy a given low energy region. The chosen "dots" are replaced by "asterisks" (to avoid duplication) and the probability and entropy terms are tabulated. Tables of probabilities and entropies may also be produced. 

Note the qualitative — not merely quantitative — distinction between the thermodynamic (Boltzmann-distribution) probability discussed in Sect. 3.2. as opposed to the purely dynamic (quantum-mechanical) probability Pg discussed in this Sect. 3.3. Even if thermodynamically, exact attainment of 0 K and perfect verification [22] that precisely 0 K has been attained could be achieved for Subsystem B, the pure dynamics of quantum mechanics, specifically the energy-time uncertainty principle, seems to impose the requirement that infinite time must elapse first. [This distinction between thermodynamic probabilities as opposed to purely dynamic (quantum-mechanical) probabilities should not be confused with the distinction between the derivation of the thermodynamic Boltzmann distribution per se in classical as opposed to quantum statistical mechanics. The latter distinction, which we do not consider in this chapter, obtains largely owing to the postulate of random phases being required in quantum but not classical statistical mechanics [42,43].]... 

Now, if we contrast thermodynamics with statistical mechanics, we find that they confront completely different problems. The problem of statistical mechanics is to compute the probabilities of the various possible results of a variety of measurements one might choose to make on a system, (a) knowing it consists of particles with known mechanical properties, and (b) knowing certain macroscopic information about the system. The problem is approached by building the appropriate information into an ensemble which represents the system and generates all... 

In addition, there is a relation between entropy and disorder disordered states have higher probabilities than ordered states. In general, the changes that are accompanied by an increase in entropy result in increased molecular disorder. Thus, entropy is also a measure of the molecular disorder of the state. Although disorder may be related to entropy qualitatively, the amount of disorder is a subjective concept and it is much better to relate entropy to probability rather than to disorder. Such concepts can be described in terms of thermodynamic probabilities (Q) in statistical mechanics. The entropy of a system is a function of the probability of the thermodynamic state of this system, S = /( 2). We know from statistical mathematics that only logarithmic functions satisfy probabilistic equations, so that we may use... 

Readers not familiar with polymer science and/or statistical thermodynamics probably should have the following reference books in hand when reading this chapter Textbook of Polymer Science, F. W. Bill-meyer, Jr., John Wiley, New York (1971) and Introduction to Statistical Thermodynamics, T. Hill, Addison-Wesley, Reading, Mass. (1960). 

Using the statistical mechanical approach, we have been able to rederive equations (6.18)-(6.20) without any mention of steam engines or idealized Carnot cycles. These equations form the basis for much of the rest of thermodynamics, as we have already begtm to see in Chapter 5. These few relationships are so useful because they serve as pointers or criteria for the spontaneous direction of any process. Hopefully the statistical approach clarifies much of this, in the sense that we conceive of entropy as a measure of disorder or randomness. The most random permissible state is also the most probable statistically. It is self-evident that spontaneous processes head in the most probable direction by doing so, they maximize entropy. 

The objective of a statistical approach is to find a distribution function of particles over the different states i, taking into account that the probabihty to find particles in these states is proportional to the number of ways in which the distribution can be arranged. Thermodynamic probability W(ni, ri2, , rii) is the probabihty to have i particles in the state 1, 2 particles in the state 2, etc. ... 

The Boltzmann relation provides the bridge from a statistical mechanical description of molecular structure to experimentally determined thermodynamic quantities. It is an elegant yet simple statement of entropy (5) in terms of thermodynamic probability (W, the number of a priori equally probable states accessible to the system) (Eyring et al, 1964), i.e.,... 

The standard construction requires that the thermodynamic function of the macrostate it will be written by combining the statistical information contained within the thermodynamic probability (1.157) with the Langrange constraints of particle and energy conservation... 

The molecular theory of water and aqueous solutions has only recently emerged as a new entity of research, although its roots may be found in age-old works. The purpose of this book is to present the molecular theory of aqueous fluids based on the framework of the general theory of liquids. The style of the book is introductory in character, but the reader is presumed to be familiar with the basic properties of water [for instance, the topics reviewed by Eisenberg and Kauzmann (1969)] and the elements of classical thermodynamics and statistical mechanics [e.g., Denbigh (1966), Hill (I960)] and to have some elementary knowledge of probability [e.g.. Feller (1960), Papoulis (1965)]. No other familiarity with the molecular theory of liquids is presumed. 

Historically, statistical mechanics has been framed in terms of the frequency interpretation. JW Gibbs (1839-1903), American mathematical physicist and a founder of chemical thermodynamics and statistical mechanics, framed statistical mechanics as a counting problem. He envisioned an imaginary collection of all possible outcomes, called an ensemble, which was countable and could be taken to the limit of a large number of imaginary repetitions. To be specific, for die rolls, if you want to know the probability of each of the six outcomes on a roll, the ensemble approach would be to imagine that you had rolled the die N times. You would then compute the number of sequences that you expect to observe. This number will depend on N. On the other hand, if the outcomes can instead be described in terms of probabilities, the quantity N never appears. For example, the grid of cells described in Table 6.2 describes probabilities that bear no trace of the information that N = 1000 die roUs w-ere used to obtain Table 6.1. 

Given the enormous number of configurations for even short chains, polymers are best characterized by averages, such as are determined through the principles of probability theory, thermodynamics, and statistical mechanics. 

The contenet of the third law of thermodynamics is summarized in Fig. 2.4. The third law is particularly easy to understand if one combines the macroscopic entropy definition of entropy with its statistical, microscopic interpretation through the Boltzmann equation, Eq. (11). The symbol k is the Boltzmaim constant, the gas constant R divided by Avogadro s number and W is the thermodynamic probability, representing the number of ways a system can be arranged on a microscopic level. One can state the third law, as proposed by Nernst and formulated by Lewis and Randall, as follows "If... 

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