The Averaging Procedure

An important point for all these studies is the possible variability of the single molecule or single particle studies. It is not possible, a priori, to exclude bad particles from the averaging procedure. It is clear, however, that high structural resolution can only be obtained from a very homogeneous ensemble. Various classification and analysis schemes are used to extract such homogeneous data, even from sets of mixed states [69]. In general, a typical resolution of the order of 1-3 mn is obtained today. 

According to Eq. (86), the averaging procedure mentioned above leads to the following total reaction current J,... 

Then, after solving Eq. (91), according to the averaging procedure of Eq. (92), the fluctuation is averaged. At the initial stage, the following fluctuation-diffusion current equation is obtained,... 

When perturbation is fast enough in comparison with the molecular response to it, the averaging procedure proposed in [91] is justified. After substitution of a formal solution of Eq. (2.20)... 

After reducing this TBF to an effective, density dependent, two-body force by the averaging procedure described earlier, the resulting effective two-nucleon potential assumes a simple structure,... 

The leading parameter that characterizes the distributions of randomly branched samples is an exponent Tthat is defined in the next section. The average procedures for the z-average of the mean square radius of gyration and the weight average molar mass results in the relationship [7]... 

This expression includes the curvature corrections to the Gaussian function, which play an important role in the averaging procedure they ensure the smoothed spectrum g(E) to be approximated, through the above definition, by its own truncated Taylor expansion. This smoothing procedure of the one-electron spectrum is an application of the method of moments, also used in other systems [23]. 

This condition should be fulfilled, at least approximately, for a given range of the smearing width 7 = 70 and a given order M of the correction polynomial, in order to obtain an independent of the averaging procedure and of the formal parameter 7 smooth energy. 

In that way the averaging procedure is closed which allows us to determine the generalized occupation numbers Ui and 6rii connected with standard occupation numbers rii through the relation ... 

The value of a a) is always the same, but the averaging procedure differs in each case. The relations (65)-(67) are a simple consequence of the boson commutation relation [ , = 1 and the definition... 

Equation (65) also permits us to assign a physical interpretation to the diffusion coefficient in addition to the macroscopic meaning it has from Fick s laws. Rearranging and factoring in a way that admittedly ignores the averaging procedure, we write Equation (65) as... 

The average value of the square velocity has been used in Equation (12) to allow for the fact that a distribution of molecular velocities exists. The nature of the averaging procedure to be used in this case is well established from physical chemistry. We also know from physical chemistry that the average kinetic energy per molecule (KE) per degree of freedom is... 

Since the liquid circulates, there must be certain locations in the cell at which the forward and backward flows of the liquid are equal. An alternative to the averaging procedure suggested in the example is to do the particle tracking at a location at which the medium experiences no net flow. 

Let us proceed to define appropriate coordinate systems (see Fig. 1 and Table 1). We use four different levels of coordinate frames in the averaging procedures of our model construction scheme to describe the molecular arrangement in the scattering volume. On the first level, System M describes the real structure of a segment, whereby the origin is fixed to a well-defined molecular unit System MC (second level) is defined in cylindrical coordinates with the symmetry axis given by the long axis of the rod-like molecular... 

Now we must look at the consequences of the averaging procedures (1)—(5) with regard to the monodomain structure factor (see Table 2). First we calculate the intermolecular part Iimer.D °f the structure factor of a single domain and apply the averaging procedures (1)—(4) to it... 

Before concluding this section, we should briefly explain the averaging procedure which is implicit in the derivation of parabolic rate constants. In order to simplify without loss of generality, we assume that DA>DB. Integration of Eqn. (6.27) yields... 

There are no conceptual difficulties if the oxidizing system conforms to the conditions stated earlier. The only detail which needs to be discussed and clarified is the averaging procedure which has been performed in order to arrive at Eqns. (7.5) or (7.6). By definition... 

Here the average is taken over a whole volume with a fixed f = f — r2 distance. Due to the system s homogeneity and isotropy the Si and Sij are functions of r = r only. We illustrate below how the equations for the correlation functions could be derived from the set of equations (2.1.40) and the averaging procedure. First, let us define the initial conditions for Si(r, 0) and Sij(r, 0). 

According to Doi [107], Zeldovich and Ovchinnikov [35], the evolution of the state of a system given by the vector (t)) obeys the Schrodinger equation with imaginary time and non-Hermitian Hamiltonian. The averaging procedure also differs from that generally-accepted in quantum mechanics. 

Let us define now the averaging procedure over coordinates with the help of the normalised distribution... 

Collecting information and its processing will convert them in data. So, data represent agglomerated information, which are partially or finally processed. Examples of data can be found as a parameter, which describe evaluated information to be used for the specific purpose. In this respect the average inlet temperature of cooling water in the condenser is data obtained by the averaging procedure adapted for this purpose. Also heat transfer coefficient used in the design of condenser is the data obtained by the experimental procedure for the heat transfer evaluation. 

In point-like models, the averaging procedure is conducted over all the sites of a lattice, therefore diffusion flows are absent in the models. But the effects... 

Here the dimensionless time z=t/t is normalized by the characteristic relaxation time t, the time required for a charge carrier to move the distance equal to the size of one droplet, which is associated with the size of the unit cell in the lattice of the static site-percolation model. Similarly, we introduce the dimensionless time zs = ts/t where ts is the effective correlation time of the s-cluster, and the dimensionless time z = tm/t. The maximum correlation time t, is the effective correlation time corresponding to the maximal cluster sm. In terms of the random walker problem, it is the time required for a charge carrier to visit all the droplets of the maximum cluster sm. Thus, the macroscopic DCF may be obtained by the averaging procedure... 

The averaging procedure inherent in optical diffractometry, as well as the direct, real-space crystallographic power of high resolution electron microscopy combine to yield a uniquely powerful set of techniques for structural characterisation. 

---