A Classical Statistical Mechanics

In classical statistical mechanics, the Hamiltonian (p, q) = K(p) + (r) of a system of N particles in a fixed volume V is a sum of the kinetic energy K(p) and the potential energy E(r) of the particles here p and r represent the collective momenta p and positions q of the particles, respectively. The fact that the potential energy is taken to be a function of coordinates only is not always true as happens if we have charged particles in a magnetic field. These cases will not be considered here. The dimensionless total canonical PF Zr(T, V) of the system (we revert back to exhibiting the dependence on V in this section) can be written as a product of two independent integrals 

The prefactor in terms of h is used to expUdtly show the correspondence of Zt with the corresponding PF in the quantum statistical mechanics in the classical limit h O. Despite the classical limit requirement h O.we are not allowed to set h = 0 in the final result, but keep its actual nonzero value. Accordingly, some problems remain such as Wigner s distribution function not being a classical probability distribution, which we do not discuss any further but refer the reader to the Uterature [117]. Keeping h at its nonzero value avoids infinities as we will see below but in no way implies that we are dealing with quantum effects. In particular, it does not imply that the entropy is nonnegative, as we have discussed elsewhere [75]. We 

The contribution Ske(T) is the same for all systems (that have the same vq). regardless of their potential energy of interaction and volume. It is most certainly the same for all phases of any system such as SCL and CR at the same temperature, and we do not have to specifically take it into account. Thus, in general, we can focus on the configurational entropy without any loss of generality. It is obtained by subtracting Ske(T) from Sr(T) [36]  

I thank Andrea Corsi, Sagar Rane, and Fedor Semerianov with whom I have had numerous discussions and have closely collaborated. I also thank Martin Goldstein who inspired me to delve into the field of glass transition and Alexei Sokolov for various interesting discussions. 

1 Penrose, O. and Lebowitz, J.L. (1979) Fluctuation Phenomena (eds E.W. MontroD and J.L. Lebowitz), North-Holland. 

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The Marcus treatment uses a classical statistical mechanical approach to calculate the activation energy required to surmount the barrier. It assumes a weakly adiabatic electron transfer process and non-equilibrium dielectric polarization of the solvent (continuum) as the source of activation. This model also considers the vibrational contributions of the inner solvation sphere. The Hush treatment considers ion-dipole and ligand field concepts in the treatment of inner coordination sphere contributions to the energy of activation [55, 56]. 

We describe a classical statistical mechanical theory of solvation dynamics, formulated for general molecular interaiction site models (ISM) of the solute and solvent species. Ba-... 

Outer-sphere reactions occur with minimal electronic interaction through space. Their rates can be predicted by a classical statistical-mechanical approximation (see 12.2.3). Insuring a minimum of electronic interaction between the reagents often re-... 

It is well known that direct calculation of Helmholtz free energy is a nontrivial problem, which is readily seen from a classical statistical mechanical formula ... 

Density functional theory (DFT) as applied to adsorption is a classical statistical mechanic technique. For a discussion of DFT and classical statistical mechanics, with specific applications to surface problems, the text book by Davis [1] is highly recommended. (Here the more commonly used symbol for number density p(r) is used. Davis uses n(r) so one will have to make an adjustment for this text.) The calculations at the moment may be useful for modeling but are questionable for analysis with unknown surfaces. The reason for this is that the specific forces, or input parameters, required for a calculation are dependent upon the atoms assumed to be present on the surface. For unknown surfaces, a reversion to the use of the Brunaver, Emmett and Teller (BET) equation is often employed. 

Another important accomplislnnent of the free electron model concerns tire heat capacity of a metal. At low temperatures, the heat capacity of a metal goes linearly with the temperature and vanishes at absolute zero. This behaviour is in contrast with classical statistical mechanics. According to classical theories, the equipartition theory predicts that a free particle should have a heat capacity of where is the Boltzmann constant. An ideal gas has a heat capacity consistent with tliis value. The electrical conductivity of a metal suggests that the conduction electrons behave like free particles and might also have a heat capacity of 3/fg,... 

The set of microstates of a finite system in quantum statistical mechanics is a finite, discrete denumerable set of quantum states each characterized by an appropriate collection of quantum numbers. In classical statistical mechanics, the set of microstates fonn a continuous (and therefore infinite) set of points in f space (also called phase space). 

We have so far ignored quantum corrections to the virial coefficients by assuming classical statistical mechanics in our discussion of the confignrational PF. Quantum effects, when they are relatively small, can be treated as a perturbation (Friedman 1995) when the leading correction to the PF can be written as... 

As discussed above, to identify states of the system as those for the reactant A, a dividing surface is placed at the potential energy barrier region of the potential energy surface. This is a classical mechanical construct and classical statistical mechanics is used to derive the RRKM k(E) [4]. 

Only in the high-energy limit does classical statistical mechanics give accurate values for the sum and density of states tenns in equation (A3.12.15) [3,14]. Thus, to detennine an accurate RRKM lc(E) for the general case, quantum statistical mechanics must be used. Since it is difficult to make anliannonic corrections, both the molecule and transition state are often assumed to be a collection of hannonic oscillators for calculating the... 

Fixman M 1974 Classical statistical mechanics of constraints a theorem and application to polymers Proc. Natl Acad. Sc/. 71 3050-3... 

Is the temperature 1/0 related to the variance of the momentum distribution as in the classical equipartition theorem It happens that there is no simple generalization of the equipartition theorem of classical statistical mechanics. For the 2N dimensional phase space F = (xi. .. XN,pi,.. -Pn) the ensemble average for a harmonic system is... 

The concept of corresponding states was based on kinetic molecular theory, which describes molecules as discrete, rapidly moving particles that together constitute a fluid or soHd. Therefore, the theory of corresponding states was a macroscopic concept based on empirical observations. In 1939, the theory of corresponding states was derived from an inverse sixth power molecular potential model (74). Four basic assumptions were made (/) classical statistical mechanics apply, (2) the molecules must be spherical either by actual shape or by virtue of rapid and free rotation, (3) the intramolecular vibrations are considered identical for molecules in either the gas or Hquid phases, and (4) the potential energy of a coUection of molecules is a function of only the various intermolecular distances. 

From the experimental results and theoretical approaches we learn that even the simplest interface investigated in electrochemistry is still a very complicated system. To describe the structure of this interface we have to tackle several difficulties. It is a many-component system. Between the components there are different kinds of interactions. Some of them have a long range while others are short ranged but very strong. In addition, if the solution side can be treated by using classical statistical mechanics the description of the metal side requires the use of quantum methods. The main feature of the experimental quantities, e.g., differential capacitance, is their nonlinear dependence on the polarization of the electrode. There are such sophisticated phenomena as ionic solvation and electrostriction invoked in the attempts of interpretation of this nonlinear behavior [2]. 

However, in all the rest of their approach, Robertson and Yarwood consider the slow mode Q as a scalar obeying simply classical mechanics, because they neglect the noncommutativity of Q with its conjugate momentum P. As a consequence, the logic of their approach is to consider the fluctuation of the slow mode as obeying classical statistical mechanics and not quantum statistical mechanics. Thus we write, in place of Eq. (138), the corresponding classical formula ... 

There now exist several methods for predicting the free energy associated with a compositional or conformational change.7 These can be crudely classified into two types "exact" and "approximate" free energy calculations. The former type, which we shall discuss in the following sections, is based directly on rigorous equations from classical statistical mechanics. The latter type, to be discussed later in this chapter, starts with statistical mechanics, but then combines these equations with assumptions and approximations to allow simulations to be carried out more rapidly. 

The most commonly reported exact free energy simulations are based on the following equation, which can be derived in a straightforward fashion from elementary classical statistical mechanics ... 

Although historically less common, free energy calculations based on a different equation from classical statistical mechanics have grown in popularity in recent years. These calculations, termed Thermodynamic Integration (TI), are based on the integral... 

The theory of electron-transfer reactions presented in Chapter 6 was mainly based on classical statistical mechanics. While this treatment is reasonable for the reorganization of the outer sphere, the inner-sphere modes must strictly be treated by quantum mechanics. It is well known from infrared spectroscopy that molecular vibrational modes possess a discrete energy spectrum, and that at room temperature the spacing of these levels is usually larger than the thermal energy kT. Therefore we will reconsider electron-transfer reactions from a quantum-mechanical viewpoint that was first advanced by Levich and Dogonadze [1]. In this course we will rederive several of, the results of Chapter 6, show under which conditions they are valid, and obtain generalizations that account for the quantum nature of the inner-sphere modes. By necessity this chapter contains more mathematics than the others, but the calculations axe not particularly difficult. Readers who are not interested in the mathematical details can turn to the summary presented in Section 6. 

Beyond the clusters, to microscopically model a reaction in solution, we need to include a very big number of solvent molecules in the system to represent the bulk. The problem stems from the fact that it is computationally impossible, with our current capabilities, to locate the transition state structure of the reaction on the complete quantum mechanical potential energy hypersurface, if all the degrees of freedom are explicitly included. Moreover, the effect of thermal statistical averaging should be incorporated. Then, classical mechanical computer simulation techniques (Monte Carlo or Molecular Dynamics) appear to be the most suitable procedures to attack the above problems. In short, and applied to the computer simulation of chemical reactions in solution, the Monte Carlo [18-21] technique is a numerical method in the frame of the classical Statistical Mechanics, which allows to generate a set of system configurations... 

A well defined theory of chemical reactions is required before analyzing solvent effects on this special type of solute. The transition state theory has had an enormous influence in the development of modern chemistry [32-37]. Quantum mechanical theories that go beyond the classical statistical mechanics theory of absolute rate have been developed by several authors [36,38,39], However, there are still compelling motivations to formulate an alternate approach to the quantum theory that goes beyond a theory of reaction rates. In this paper, a particular theory of chemical reactions is elaborated. In this theoretical scheme, solvent effects at the thermodynamic and quantum mechanical level can be treated with a fair degree of generality. The theory can be related to modern versions of the Marcus theory of electron transfer [19,40,41] but there is no... 

The reader can see now that experimental conditions are progressing in such a way that would allow for verifications of the quantum theories of solvent effects. The important theoretical fact is the possibility of recasting the standard theory of solvent effects, based upon classical statistical mechanics, into a more complete quantum mechanical approach. 

Thus the first correction to the classical statistical mechanics at high temperature goes as h2. There are higher order corrections. The result obtained here is identical to that found by J. Kirkwood for a harmonic oscillator. The approach to the... 

In addition to the study of atomic motion during chemical reactions, the molecular dynamics technique has been widely used to study the classical statistical mechanics of well-defined systems. Within this application considerable progress has been made in introducing constraints into the equations of motion so that a variety of ensembles may be studied. For example, classical equations of motion generate constant energy trajectories. By adding additional terms to the forces which arise from properties of the system such as the pressure and temperature, other constants of motion have been introduced. 

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