Ensemble concept

The same pseudo-ensemble concept has been used by Camp and Allen [44] to obtain a pseudo-Gibbs method in which particle transfers are substituted by volume fluctuations of the two phases. The volume fluctuations are unrelated to the ones required for pressure equality (10.7) but are instead designed to correct imbalances in the chemical potentials of some of the components detected, for example, by test particle insertions. 

Metastability of some subset of the state space is characterized by the property that the Markov process is likely to remain within the subset for a long period of time, until it exits and a transition to some other region of the state space occurs. There are in fact several related but different definitions of metastability in the literature (see, e.g., [16-20]) we will focus on the so-called ensemble concept introduced in (6), for a comparison with, e.g., the exit time concept, see [2]. 

We must acknowledge that two-state is an ensemble concept. If we accept that a given single molecule does not tunnel from one conformation to another then the number of states we observe a single molecule in depends, ideally, only on the integration time used. 

The same pseudo-ensemble concept has been used by Camp and Allen [53] to obtain a pseudo jibbs method in which particle transfers are substituted by volume fluctuations of the two phases. The volume... 

The ability to define an ensemble does not necessarily mean that real granular packings conform to this ensemble. The assumption of factorizabilty could break down or the frequency of occurrence of a packing, cu could be history dependent, making the temperature or the ensemble concept not very useful. It is, therefore, essential to check the predictions of this ensemble against experiments and simulations. 

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

Many approaches and methods for the generation of multiple conformations have been developed and published since the early 1980s. Below we describe briefly some of the basic concepts and methods of automatic conformer generation. However, interested readers are referred to Chapter 11, Section 7.2 in the Handbook, where the approaches to automatic generation of ensembles of conformations and the program systems that have been developed are described in detail. 

This model was first introduced by Kauffman [kauff69] in a study of cellular differentiation in a biological system (binary sites were interpreted as elements of an ensemble of genes switching on and off according to some set of random rules). Since its original conception, however, related models have found wide application in an... 

Since early in this century the concept of the active site in catalysis [1] has been a focus of attention in this area of chemistry. This was proposed to be that ensemble of surface atoms/reactants which is responsible for the crucial surface reaction step involved in a catalytic conversion. Since those days much work has been done in the area, which cites the concept of the active site. However, no such ensemble has been positively identified due to the lack of availability of techniques which could image such a structure, which is of atomic dimensions. 

The ensemble of elements that are mutually conjugate form a class TJje concept of a class is most easily demonstrated by an example. The multiplication table for the group of matrices defined by Eq. (2) is given in Table 3. With its use the relations... 

The Boltzmann constant is represented by kB. It is more difficult to use Monte Carlo methods to investigate dynamic events as there is no intrinsic concept of time but an ensemble average over the generated states of the system should give the same equilibrium thermodynamic properties as the MD methods. A good review of both MD and the Monte Carlo methods can be found in the book by Frenkel and Smit [40]. 

The concept of machine at the molecular level is not new. Our body can be viewed as a very complex ensemble of molecular-level machines that power our motions, repair damage, and orchestrate our... 

Abstract Chemists may find it difficult to admit that their concepts and opportunities have always been strongly influenced by the available methods for characterization and analysis. Physics, has, of course, the lead when it comes to the visualization of single molecules in real space and to the detection of their specific, not ensemble-averaged properties. The challenge for chemistry is to provide molecules as objects of study which really disclose new concepts of structure and function. This chapter presents a chemical approach toward nanosciences which comprises (i) design and synthesis, (ii) immobilization, often using principles of self-assembly, (iii) visualization, e.g. by scanning probe... 

Since j = x, this approximation for /a reduces to the MuUiken definition for the electronegativity [16]. Within the ensemble approach, the hardness 17 as defined in Equation 34.3 would be zero for noninteger N and undefined for integer values of the electron number. One obtains quantitative values for this concept for integer N, using a finite difference approximation of the chemical potentials yu,1 and /a, i.e.,... 

Model of a supramolecular structure of polymolecular ensembles or clusters, determined by interaction and mutual arrangement of the forming molecules. At this level, the specific mechanisms of supramolecular chemistry, including molecular recognition, self-assembly, etc. [4] can be allocated. In most cases, it is possible to limit this area to objects with the sizes under 1 to 2 nm, since further increase in the sizes admits application of statistical concepts like phase and interphase surface. 

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