Ensemble average method

The proper response is dependent upon the data acquisition method used. The means to select a specific method will be discussed. If an ensemble averaging method is used, then the spectral rate is the number of spectra to be averaged. 

Batchelor (1988) developed a force balance equation for the solid phase in fluidized beds using the ensemble average method. With the help of the resulting equation and the theory of linear stability the following criterion was obtained by Ham et al. (1990) ... 

In this section we examine the ensemble averaging method in the framework of the generic single averaging procedure proposed by Drew [54] and further elaborated in [55, 137, 56, 58]. The paper by Enwald et al [67] summarizes this approach nicely in the setting of fluidized bed reactor simulations. 

A consideration of the transition probabilities allows us to prove that microscopic reversibility holds, and that canonical ensemble averages are generated. This approach has greatly extended the range of simulations that can be perfonned. An early example was the preferential sampling of molecules near solutes [77], but more recently, as we shall see, polymer simulations have been greatly accelerated by tiiis method. 

Due to the noncrystalline, nonequilibrium nature of polymers, a statistical mechanical description is rigorously most correct. Thus, simply hnding a minimum-energy conformation and computing properties is not generally suf-hcient. It is usually necessary to compute ensemble averages, even of molecular properties. The additional work needed on the part of both the researcher to set up the simulation and the computer to run the simulation must be considered. When possible, it is advisable to use group additivity or analytic estimation methods. 

All discussions of transport processes currently available in the literature are based on perturbation theory methods applied to kinetic pictures of micro-scattering processes within the macrosystem of interest. These methods do involve time-dependent hamiltonians in the sense that the interaction operates only during collisions, while the wave functions are known only before and after the collision. However these interactions are purely internal, and their time-dependence is essentially implicit the over-all hamiltonian of the entire system, such as the interaction term in Eq. (8-159) is not time-dependent, and such micro-scattering processes cannot lead to irreversible changes of thermodynamic (ensemble average) properties. 

To measure the strength of the forces exerted on particles, various analytical techniques have been developed [6, 7]. Unfortunately, since most of these techniques are based on hydrodynamics, assumption of the potential profiles is required and the viscosities of the fiuid and the particle sizes must be precisely determined in separate experiments, for example, using the viscous flow technique [8,9] and power spectrum analysis of position fluctuation [10]. Furthermore, these methods provide information on ensemble averages for a mass of many particles. The sizes, shapes, and physical and chemical properties of individual particles may be different from each other, which will result in a variety of force strengths. Thus, single-particle... 

In the case of being successful in calculating multiple conformations by using time- or ensemble-averaged MD restraints the solved molecular structures are presented as 3D models and can be deposited in an electronic structure database (17). Finally, it is recommended to provide an accurate explanation of the procedures used for the structure elucidation because the application of different methods (NMR, DG, MD, SA, Monte-Carlo calculations. X-ray crystallography) may result in varying conformational models which do not implicitly display the real state of a molecule. This aspect should be always kept in mind when dealing with structure determination methods. 

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]. 

In the complete Eulerian description of multiphase flows, the dispersed phase may well be conceived as a second continuous phase that interpenetrates the real continuous phase, the carrier phase this approach is often referred to as two-fluid formulation. The resulting simultaneous presence of two continua is taken into account by their respective volume fractions. All other variables such as velocities need to be averaged, in some way, in proportion to their presence various techniques have been proposed to that purpose leading, however, to different formulations of the continuum equations. The method of ensemble averaging (based on a statistical average of individual realizations) is now generally accepted as most appropriate. 

An alternative approach to free energy calculations is the thermodynamic integration (TI) method,18 20 which considers the ensemble average of the first derivative of the hybrid potential with respect to A at various values of A... 

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... 

To summarize, the results presented for five representative examples of nonadiabatic dynamics demonstrate the ability of the MFT method to account for a qualitative description of the dynamics in case of processes involving two electronic states. The origin of the problems to describe the correct long-time relaxation dynamics as well as multi-state processes will be discussed in more detail in Section VI. Despite these problems, it is surprising how this simplest MQC method can describe complex nonadiabatic dynamics. Other related approximate methods such as the quantum-mechanical TDSCF approximation have been found to completely fail to account for the long-time behavior of the electronic dynamics (see Fig. 10). This is because the standard Hartree ansatz in the TDSCF approach neglects all correlations between the dynamical DoF, whereas the ensemble average performed in the MFT treatment accounts for the static correlation of the problem. 

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