Statistical mechanical basis

It is of interest in the present context (and is useful later) to outline the statistical mechanical basis for calculating the energy and entropy that are associated with rotation [66]. According to the Boltzmann principle, the time average energy of a molecule is given by... 

J. G. Kirkwood and J. Boss, The Statistical Mechanical Basis of the Boltzmann Equation, in I. Frigogine, ed., Transport Processes in Statistical Mechanics, pp. 1-7, Interscience Publishers, Inc., New York, 1958. Also, J. G. Kirkwood, The Statistical Mechanical Theory of Transport Processes I. General Theory, J, Chem, Phys, 14, 180 (1946) II. Transport in Gases, J, Chem. Phys, 15, 72 (1947). 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

Kirkwood, J. G. Poirier, J. C., The statistical mechanical basis of the Debye-Htickel theory of strong electrolytes, J. Phys. Chem. 1954, 86, 591-596... 

These mixing rules, when joined with the Redlich-Kwong equation of state, will constitute the Redlich-Kwong equation of state for mixtures that is consistent with the statistical mechanical basis of the van der Waals mixing rules. 

How the equation of state is actually employed to compute /i depends on the type of problem being solved If AG is an input parameter, Vb is fixed throughout the simulation and the gas mixture is further specified by its number density />a/b+a = (Mb + )/Kb. If, on the other hand, the pressure is fixed, its value defines the state of the mixture. The statistical mechanical basis for Eqs. (14) and (15) is discussed elsewhere. ... 

Transition path sampling, based on a statistical mechanics in trajectory space, is a set of computational methods for the simulation of rare events in complex systems. In this chapter we give an overview of these techniques and describe their statistical mechanical basis as well as their application. 

Marcus R. A. (I960), Theory of oxidation-reduction reactions involving electron transfer. Part 4. A statistical-mechanical basis for treating contributions from solvent, ligands and inert salt , Faraday Discuss. Chem. Soc. 29, 21-31. 

One of the earliest attempts to interpret the anomalous properties of aqueous solutions was based on splitting the solvation process into two parts first, we form a cavity in the solvent, then introduce the solute into the cavity. In the present section, we shall discuss the statistical-mechanical basis for such a split of the solvation process. For more details see Eley (1939, 1944) Ben-Naim (1992). 

Chapter 11 reviews the statistical mechanical basis of hydrodynamics and discusses theories that may be used to extend hydrodynamics beyond the classical equations discussed in Chapter 10. Chapter 12 applies the statistical mechanical theory to the calculation of depolarized light-scattering spectra from dense liquids where interactions between anisotropic molecules are important. 

Song, Y., and Mason, E. A., Statistical-mechanical basis for accurate analytical equations of state for fluids. Fluid Phase Equilih, 75, 105-115 (1992). 

In the next section we give the statistical mechanical basis for binding polynomials. 

The mean field density functional theory (DFT) approach was primarily developed by Evans and co-workers [67,68] for studying the interactions of fluids in pores at the molecular level. Recently, DFT methods have been developed specifically with the objective of the estimation of PSD of carbon-based as well as other types of microporous materials. This technique was first proposed by Seaton et al. [16], who used the local-DFT approximation. Later, the theory was modified by Lastoskie et al. [17,18] to incorporate the smoothed or nonlocal DFT approach. The rigorous statistical mechanics basis behind the DFT model has been recently reviewed by Gubbins [34]. Some sahent features of the theory are discussed later in this subsection. The DFT method initially proceeds by estimating the properties of a fluid directly from intermolecular forces such as that between sorbate-sorbent and sorbate-sorbate molecules. The interactions are divided into a short-ranged repulsive part and a long-ranged attractive part, which are both determined separately. 

The expressions whose limits give s and <7 are found to converge rapidly and the estimates for s and a so obtained are close to the values obtained experimentally. While improvement of the calculated values of s and <7 will undoubtedly be possible in the future, these results, together with the results of the theory of the one-dimensional Ising lattice, place the description of helix-coil transitions of polyfa-amino acids) on a firm combined molecular and statistical mechanical basis. 

In this book, we demonstrate the use of transition-state theory to describe catalytic reactions on surfaces. In order to do this we start by treating the kinetics of catalytic reactions (Chapter 2) and provide some background information on important catalytic processes (Chapter 3). In Chapter 4 we introduce the statistical mechanical basis of transition-state theory and apply it to elementary surface reactions. Chapter 5 deals with the physical justification of the transition-state theory. We also discuss the consequences of media effects and of lateral interactions between adsorbates on surfaces for the kinetics. In the final chapter we present the principles of catalytic kinetics, based on the application of material given in earlier chapters. 

QSM theory provides a firm statistical mechanical basis for the phenomenological theory of irreversible processes as formulated by Onsager and the version of classical Brownian motion discussed in Section 5.2. Moreover, it gives a number of formulas that can be employed in the investigation of the role of microscopic interactions in a diversity of nonequilibrium phenomena. 

Zisman was one of the first to recognize that the critical surface tension concept is strictly empirical and to suggest that needs to be replaced by parameters having a thermodynamic or statistical mechanical basis [26]. Fox and Zisman [30] have cautioned that jc varies between liquid types and that it is not a measure of the surface energy of the solid yso-... 

So far we have concentrated on the analysis of detailed rate data, as distinct from their synthesis. It has been implied that vibrational surprisal plots are frequently linear because of some, qualitatively common, dynamic constraint, but this has not been identified. Bernstein and Levine have elegantly reviewed the statistical mechanical basis of the relationship between real distributions and the constraints which lead to them. The general principle is that a system will adopt the distribution with maximum entropy, which is also consistent with all the constraints. Consequently, a complete distribution could be synthesized if the constraints could be independently determined. Alternatively, it should be possible, in principle, to deduce the constraints by observing the distribution. 

Kirkwood, J. G. Poirier, J. C. (1954) The Statistical Mechanical Basis of the Debye-Hiickel Theory of Strong Electrolytes. J. Phys. Chem. 58, 8, 591-5%, ISSN 0022-3654 Kjellander, R Mitchell, D.J. (1992) An exact but linear and Poisson—Boltzmann-like theory for electrolytes and coUoid dispersions in the primitive model. Chem. Phys. Lett. 200,1-2, 76-82, ISSN 0009-2614... 

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