Statistical mechanics books

Another relatively easy modeling is for hard rods confined between two walls. The mathematics is a little messier and will not be completely given here (see Davis [1] or other statistical mechanic books). The modeling can also include an external field, which is also instructive. Using y and z for the position of the walls, Q, the canonical partition function for this case is... 

Applications of the canonical ensemble to quantum mechanical systems other than dilute gases are beyond the scope of this book, and we omit them. These applications are discussed in some of the statistical mechanics books listed at the end of this volume. We will make some comments on the application of the canonical ensemble to systems obeying classical mechanics in the next section. 

In passing one should note that the metliod of expressing the chemical potential is arbitrary. The amount of matter of species in this article, as in most tliemiodynamics books, is expressed by the number of moles nit can, however, be expressed equally well by the number of molecules N. (convenient in statistical mechanics) or by the mass m- (Gibbs original treatment). 

Berendsen, H.J.C., Postma, J.P.M., Van Gunsteren, W.F. Statistical mechanics and molecular dynamics The calculation of free energy, in Molecular Dynamics and Protein Structure, J. Hermans, ed.. Polycrystal Book Service, PO Box 27, Western Springs, 111., USA, (1985) 43-46. 

An important though deman ding book. Topics include statistical mechanics, Monte Carlo sim illation s. et uilibrium and non -ec iiilibrium molecular dynamics, an aly sis of calculation al results, and applications of methods to problems in liquid dynamics. The authors also discuss and compare many algorithms used in force field simulations. Includes a microfiche containing dozens of Fortran-77 subroutines relevant to molecular dynamics and liquid simulations. 

These values are from Mayo, Olafson, and Goddard [J. Phys. Chem. 94, 8897 (1990)] with additional values from A. Bondi J. Phys. Chem 68, 441 (1964). The values for the rare gases are from Davidson s book. Statistical Mechanics, McGraw-Hill, 1962. 

In this review we put less emphasis on the physics and chemistry of surface processes, for which we refer the reader to recent reviews of adsorption-desorption kinetics which are contained in two books [2,3] with chapters by the present authors where further references to earher work can be found. These articles also discuss relevant experimental techniques employed in the study of surface kinetics and appropriate methods of data analysis. Here we give details of how to set up models under basically two different kinetic conditions, namely (/) when the adsorbate remains in quasi-equihbrium during the relevant processes, in which case nonequilibrium thermodynamics provides the needed framework, and (n) when surface nonequilibrium effects become important and nonequilibrium statistical mechanics becomes the appropriate vehicle. For both approaches we will restrict ourselves to systems for which appropriate lattice gas models can be set up. Further associated theoretical reviews are by Lombardo and Bell [4] with emphasis on Monte Carlo simulations, by Brivio and Grimley [5] on dynamics, and by Persson [6] on the lattice gas model. 

While writing the first chapters, the author discovered that there existed no book giving a full yet elementary treatment of solutions from the point of view of statistical mechanics. Feeling the need of such a book to refer to, he set aside the present project and proceeded to write one. It was published in 1949 by McGraw-Hill under the title, Introduction to Statistical Mechanics. Some familiarity with Chapter 7 of that book will be found helpful by the reader of this one. 

See, for example, Terrell L. Hill, Statistical Mechanics, Eq. (14.15), McGraw-Hill Book Co., New York, 1966. 

What Eqs. (10-70) and (10-71) state is this One can predict the ET rate constant, given the other parameters. Or, a missing EE rate constant can be obtained by determining if the other factors are known. We shall return to this point shortly, after considering how to obtain these equations. The original derivation from statistical mechanics is outside the scope of this book, so we shall consider two others. 

The manner in which a film is formed on a surface by CVD is still a matter of controversy and several theories have been advanced to describe the phenomena. ] A thermodynamic theory proposes that a solid nucleus is formed from supersaturated vapor as a result of the difference between the surface free energy and the bulk free energy of the nucleus. Another and newer theory is based on atomistic nucle-ation and combines chemical bonding of solid surfaces and statistical mechanics. These theories are certainly valuable in themselves but considered outside the scope of this book. 

In his book States of Matter, [(1985), Prentice Hall, Dover] David L. Goodstein writes Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously. ... 

He is the author of two other books. Nonequilibrium Thermodynamics (1962) and Vector Analysis in Chemistry (1974), and has published research articles on the theory of optical rotation, statistical mechanical theory of transport processes, nonequilibrium thermodynamics, molecular quantum mechanics, theory of liquids, intermolecular forces, and surface phenomena. 

In the remainder of this chapter, we review the fundamentals that underlie the theoretical developments in this book. We outline, in sequence, the concept of density of states and partition function, the most basic approaches to calculating free energies and the essential strategies for improving the efficiency of these calculations. The ideas discussed here are, most likely, known to the reader. They can also be found in classical books on statistical mechanics [132-134] and molecular simulations [135, 136]. Thus, we do not attempt to be exhaustive. On the contrary, we present the material in a way that is most directly relevant to the topics covered in the book. 

The book consists of 14 chapters, in which we attempt to summarize the current state of the art in the field. We also offer a look into the future by including descriptions of several methods that hold great promise, but are not yet widely employed. The first six chapters form the core of the book. In Chap. 1, we define the context of the book by recounting briefly the history of free energy calculations and presenting the necessary statistical mechanics background material utilized in the subsequent chapters. 

On several occasions, the reader will notice a direct connection between the topics covered in the book and other, related areas of statistical mechanics, such as the methodology of computer simulations, nonequilibrium dynamics or chemical kinetics. This is hardly a surprise because free energy calculations are at the nexus of statistical mechanics of condensed phases. 

Many other approaches for finding a correct structural model are possible. A short description of ab-initio, density functional, and semiempirical methods are included here. This information has been summarized from the paperback book Chemistry with Computation An Introduction to Spartan. The Spartan program is described in the Computer Software section below.65 Another description of computational chemistry including more mathematical treatments of quantum mechanical, molecular mechanical, and statistical mechanical methods is found in the Oxford Chemistry Primers volume Computational Chemistry,52... 

Even though the van der Waals equation is not as accurate for describing the properties of real gases as empirical models such as the virial equation, it has been and still is a fundamental and important model in statistical mechanics and chemical thermodynamics. In this book, the van der Waals equation of state will be used further to discuss the stability of fluid phases in Chapter 5. 

Hill, T. L., Statistical Mechanics, McGraw-Hill Book Company, New York, 1956. 

No attempt will be made here to extend our results beyond the simple lowest-order limiting laws the often ad hoc modifications of these laws to higher concentrations are discussed in many excellent books,8 11 14 but we shall not try to justify them here. As a matter of fact, for equilibrium as well as for nonequilibrium properties, the rigorous extension of the microscopic calculation beyond the first term seems outside the present power of statistical mechanics, because of the rather formidable mathematical difficulties which arise. The main interests of a microscopic theory lie both in the justification qf the assumptions which are involved in the phenomenological approach and in the possibility of extending the mathematical techniques to other problems where a microscopic approach seems necessary in the particular case of the limiting laws, obvious extensions are in the direction of other transport coefficients of electrolytes (viscosity, thermal conductivity, questions involving polyelectrolytes) and of plasma physics, as well as of quantum phenomena where similar effects may be expected (conductivity of metals and semi-... 

While details of the solution of the quantum mechanical eigenvalue problem for specific molecules will not be explicitly considered in this book, we will introduce various conventions that are used in making quantum calculations of molecular energy levels. It is important to note that knowledge of energy levels will make it possible to calculate thermal properties of molecules using the methods of statistical mechanics (for examples, see Chapter4). Within approximation procedures to be discussed in later chapters, a similar statement applies to the rates of chemical reactions. 

In Equations 4.51 and 4.52 k is Boltzmann s constant, T is the absolute temperature and the Eis s are the energy states of the molecules i. The statistical mechanical considerations in this book will refer to an ideal gas unless explicit mention is made to the contrary. For an ideal gas, a gas of non-interacting molecules, one can express the partition function Q of a collection of N molecules of species i in terms of the single molecule partition functions q as follows1... 

The molecular approach, adopted throughout this book, starts from the statistical mechanical formulation of the problem. The interaction free energies are identified as correlation functions in the probability sense. As such, there is no reason to assume that these correlations are either short-range or additive. The main difference between direct and indirect correlations is that the former depend only on the interactions between the ligands. The latter depend on the maimer in which ligands affect the partition function of the adsorbent molecule (and, in general, of the solvent as well). The argument is essentially the same as that for the difference between the intermolecular potential and the potential of the mean force in liquids. 

In 1985 I was glad to see T. L. Hill s volume entitled Cooperativity Theory in Biochemistry, Steady State and Equilibrium Systems. This was the first book to systematically develop the molecular or statistical mechanical approach to binding systems. Hill demonstrated how and why the molecular approach is so advantageous relative to the prevalent phenomenological approach of that time. On page 58 he wrote the following (my italics) ... 

In general, the statistical mechanical approach may also be appUed to systems where the adsorbent molecules are not necessarily independent. However, in this book we shall always assume independence of the adsorbent molecules. 

While there are several books that deal with the subject matter of this volume, the only one that develops the statistical mechanical approach is T. L. Hill s monograph (1985), which includes equilibrium as well as nonequilibrium aspects of cooperativity. Its style is quite condensed, formal, and not always easy to read. The emphasis is on the effect of cooperativity on the form of the PF and on the derived binding isotherm (BI). Less attention is paid to the sources of cooperativity and to the mechanism of communication between ligands, which is the main subject of the present volume. 

This volume is addressed mainly to anyone interested in the life sciences. There are, however, a few minimal prerequisites, such as elementary calculus and thermodynamics. A basic knowledge of statistical thermodynamics would be useful, but for understanding most of this book (except Chapter 9 and some appendices), there is no need for any knowledge of statistical mechanics. 

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