Statistical-Mechanical Definitions

In this section we first give some statistical-mechanical definitions of the fundamental quantities that are needed for the cluster theory of fluids and some of the elementary relationships among them. Then we will give formulas for the most important of these in terms of infinite series of graphs. 

In the canonical ensemble, the independent variables are the temperature r, the volume V, and the number of particles of type a in the sample, where 

It is convenient, for purposes of graph theory, to imagine a system with the same temperature, volume, and number of particles as the system of interest, but in which the particles do not interact with each other. Such a system is an ideal gas or ideal gas mixture, and its thermodynamic properties have a rather simple form. (Note, however, that if the original system is at high density, the imaginary system is a dense ideal gas. ) The partition function for this ideal gas will be denoted Oig(r, V, Ni. its Helmholtz free energy is Ajg(r, V, Ni. N ), and the relationship between these two quantities is the same as between A and Q. We now define the quantity pi, p2, .., p ), where 

Here A - Ajg is the excess Helmholtz free energy with respect to an ideal gas at the same temperature, volume, and number density of each species. Thus, because of the minus sign, the factor kT, and the factor V in the first equality, si can be regarded as a negative dimensionless excess free energy density for the system. Since both A and Aig are extensive thermodynamic properties of the system, A/V and A JV are functions only of the intensive independent variables. Thus si has been expressed as a function of only the temperature and the number density of each species. (Moreover, we have chosen to use j8 = l/Ztr, rather than T, as the independent temperature variable.) It is this quantity si which has a simple representation in terms of graphs, which will be given below. If si can be calculated (exactly or approximately), this leads to (exact or approximate) results for A and hence for all the thermodynamic properties. 

The pair correlation function g(xi,X2) is another quantity of interest because it is related to the X-ray and neutron scattering properties of the fluid and to many other measurable properties such as the dielectric constant. It is the basic quantity one uses to discuss the structure of a fluid. For atomic fluids, it is defined so that 

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To close this chapter we emphasize that Hie statistical mechanical definition of macroscopic parameters such as temperature and entropy are well designed to describe isentropic equilibrium systems, but are not immediately applicable to the discussion of transport processes where irreversible entropy increase is an essential feature. A macroscopic system through which heat is flowing does not possess a single tempera-... 

A few simple calculations would show that, unless both the energy levels are present, the statistical mechanical definition of temperature has no meaning. Also, the following relations must hold, or Eq. (5) will be violated ... 

In the past few years, development of new theories have led to completely new ways of determining free energy changes. Traditionally, the difference in the free energy of two equilibrium state is (AFi 2) and the free energy change of a process can be obtained directly from the statistical mechanical definition of the free energy, F, in terms of the partition function. For the canonical ensemble F = —k T In J = —ksTln Z, where ka is Boltzmann s constant, //(F) is the phase... 

Consider now the statistical mechanics definition of internal pressure ... 

From the scientific definition point of view, there is a slight difference between our continuum thermodynamics definition of the Second Law and its statistical mechanical version so that the continuum thermodynamics definition of the Second Law states that an observation of decreased universal entropy is impossible in isolated systems however the statistical mechanical definition says that an observation of universal increased entropy is not probable. 

In these equations, r is a so-called "partition coefficient , co is a degeneracy, k is Boltzsmann s constant, and T is in degrees Kelvin. The first equation in 2.5.3. is a statistical mechanical definition of work, whereas the last two describe total energy states. Having these definitions and equations allows us to define point defects firom a Statistical Mechanical viewpoint. 

A clearer insight may be gained by considering the statistical mechanical definition of internal pressure, valid for any state of matter ... 

In the statistical mechanical definition of anion-cation radial distribution function g/ c(r) we have that. 

This means that we can arrive at a quantum statistical mechanics definition of the internal energy as the thermally averaged energy of the various occupied states ... 

The name entropy is used here because of the similarity of Eq. (4-6) to the definition of entropy in statistical mechanics. We shall show later that H(U) is the average number of binary digits per source letter required to represent the source output. 

Another simple approach assumes temperature-dependent AH and AS and a nonlinear dependence of log k on T (123, 124, 130). When this dependence is assumed in a particular form, a linear relation between AH and AS can arise for a given temperature interval. This condition is met, for example, when ACp = aT" (124, 213). Further theoretical derivatives of general validity have also been attempted besides the early work (20, 29-32), particularly the treatment of Riietschi (96) in the framework of statistical mechanics and of Thorn (125) in thermodynamics are to be mentioned. All of the too general derivations in their utmost consequences predict isokinetic behavior for any reaction series, and this prediction is clearly at variance with the facts. Only Riietschi s theory makes allowance for nonisokinetic behavior (96), and Thorn first attempted to define the reaction series in terms of monotonicity of AS and AH (125, 209). It follows further from pure thermodynamics that a qualitative compensation effect (not exactly a linear dependence) is to be expected either for constant volume or for constant pressure parameters in all cases, when the free energy changes only slightly (214). The reaction series would thus be defined by small differences in reactivity. However, any more definite prediction, whether the isokinetic relationship will hold or not, seems not to be feasible at present. 

The equilibrium state, which is denoted x, is by definition both the most likely state, p(x E) > p(x E), and the state of maximum constrained entropy, iS,(T (x /ij > iS 0(x j. This is the statistical mechanical justification for much of the import of the Second Law of Equilibrium Thermodynamics. The unconstrained entropy, as a sum of positive terms, is strictly greater than the maximal constrained entropy, which is the largest term, S HE) >. S(1 (x j. However, in the thermodynamic limit when fluctuations are relatively negligible, these may be equated with relatively little error, S HE) . S(1 (x j. 

Thus, Prigogine and Petrosky (PP) introduced the model of a Large Poincare system (EPS). As stated above, the latter is, in fact, a large system, to which the operation of Thermodynamic limit is applied. Clearly, there exists no real system satisfying strictly the definition of a EPS This infinite system is an idealization, on which, by the way, all of statistical mechanics is based. One should thus be more specific about the statement The irreversible processes... cannot be interpreted as approximations of the fundamental laws (statement 1). Quite explicitly, the approximations that are avoided in the PP theory are (a) the arbitrary coarse-graining and (b) the restriction to small parameters. 

I now consider statement 3 How should an extension of dynamics be understood In the MPC theory the problem does not exist For the intrinsically stochastic systems there is no need for modifying the laws of dynamics. As for the LPS theory, one notes the presence of two essentially new concepts. The introduction of non-Hilbert functional spaces only concerns the definition of the states of the dynamical system, and not at all the law governing their evolution. It is an important precision introduced in statistical mechanics. The extension of dynamics thus only appears in the operation of regularization of the resonances. This step is also the one that is most difficult to justify rigorously it is related to the (practical) necessity to use perturbation calculus (see Appendix). 

One may then try to generalize this operation, by defining another decomposition, different from the previous one. Two new projection operators n and n, were first timidly defined by C. George in 1967 and were later introduced definitively in the toolbox of statistical mechanics in 1969 by I. Prigogine, C. George, and F. Henin (MSN.60). The new objects possess the usual properties of projectors ... 

To interpret the phase diagram in Fig. 7.1 quantitatively, we must return to Eq. (7.3) and more fully define the chemical potential. For ideal gases, the chemical potential can be rigorously derived from statistical mechanics. A useful definition of the ideal-gas chemical potential for O2 is... 

As for the theory of this phenomenon, it was first observed by Onsager [27a] that, since in the limit a — 0 an LCD a is expected to yield a singularity of the type —surface potential, the statistical-mechanical phase integral for counterions should diverge for a greater than some critical value, characteristic of a given valency. Indeed consider a counterion (for definiteness anion) of valency z. The appropriate phase integral is of the form... 

From the point of view of statistical mechanics there are many problems, such as strongly anharmonic lattices, to which the theory can be applied.14 It appears as a natural generalization of Landau s theory of quasi-particles in the case when dissipation can no longer be neglected. The most interesting feature is that equilibrium and nonequilibrium properties appear linked. The very definition of the strongly coupled anharmonic phonons depends on their lifetime. 

The usefulness of spectral densities in nonequilibrium statistical mechanics, spectroscopy, and quantum mechanics is indicated in Section I. In Section II we discuss a number of known properties of spectral densities, which follow from only the form of their definitions, the equations of motion, and equilibrium properties of the system of interest. These properties, particularly the moments of spectral density, do not require an actual solution to the equations of motion, in order to be evaluated. Section III introduces methods which allow one to determine optimum error bounds for certain well-defined averages over spectral densities using only the equilibrium properties discussed in Section II. These averages have certain physical interpretations, such as the response to a damped harmonic perturbation, and the second-order perturbation energy. Finally, Section IV discusses extrapolation methods for estimating spectral densities themselves, from the equilibrium properties, combined with qualitative estimates of the way the spectral densities fall off at high frequencies. 

The material covered in this chapter is self-contained, and is derived from well-known relationships such as Newton s second law and the ideal gas law. Some quantum mechanical results and the statistical thermodynamics definition of entropy are given without rigorous derivation. The end result will be a number of practical formulas that can be used to calculate thermodynamic properties of interest. 

We can now utilize some of the statistical mechanics relationships derived in Chapter 8 to find expressions for the free energy and the equilibrium constant in term of the molecular partition functions. From the definition of the free energy (Eq. 9.1) the expression for the enthalpy of an ideal gas (Eq. 8.121), and recalling that Ho = Eq (for an ideal gas), we obtain... 

Generalize this definition to dots in three dimensions and verify that it is the pair distribution function of statistical mechanics. 

At 315°C. the rate constant ki has a value of 7.0 X 1016 molecules/sec.-cm.2-atm. From the definition of kh this represents the rate of adsorption of methylcyclohexane per cm.2 of bare platinum surface at a methylcyclohexane partial pressure of 1 atm. From kinetic theory and statistical mechanics, one can calculate the number of molecules striking a unit area of surface per unit time with activation energy Ea. This is given by... 

In the general approach to classical statistical mechanics, each particle is considered to occupy a point in phase space, i.e., to have a definite position and momentum, at a given instant. The probability that the point corresponding to a particle will fall in any small volume of the phase space is taken proportional to die volume. The probability of a specific arrangement of points is proportional to the number of ways that the total ensemble of molecules could be permuted to achieve the arrangement. When this is done, and it is further required that the number of molecules and their total energy remain constant, one can obtain a description of the most probable distribution of the molecules in phase space. Tlie Maxwell-Boltzmann distribution law results. 

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