Average, definition

For example, the time average definition of the Lyapunov exponent for one-dimensional maps, A = lim v->oo (which is often difficult to calculate in prac-... 

In other words, it must be possible to choose an averaging volume that is arbitrarily small compared with the macroscale L while still remaining very much larger than the microscale 8. Although the condition (2 2) will always be sufficient for validity of the continuum hypothesis, it is unnecessarily conservative because of the use of volume averaging in the definition (2-1) rather than the more fundamental ensemble average definition of macroscopic variables. Nevertheless, the preceding discussion is adequate for our present purposes. 

Average Definition A Iternative form Integral form... 

Published averages and ranges of soil macrofauna occupation depths were obtained for three types of earthworms, ants, termites, and four types of vertebrates (Rodriguez, 2006). The depths reported for each were characterized as definite or likely. The former designation refers to the penetration depth below the surface observed in all the reports studied. The average definite depths for nine species appear in Table 13.2a. It is assumed that these data also represent the mixing length for biodiffusion. 

Y. Shimoni and D. J. Kouri, Quantum mechanical close coupling approach to molecular collisions Averaged definite parity approximation with Clebsch-Gordan weights, J. Chem. Phys. 66 ... 

Other SFA studies complicate the picture. Chan and Horn [107] and Horn and Israelachvili [108] could explain anomalous viscosities in thin layers if the first layer or two of molecules were immobile and the remaining intervening liquid were of normal viscosity. Other inteipretations are possible and the hydrodynamics not clear, since as Granick points out [109] the measurements average over a wide range of surface separations, thus confusing the definition of a layer thickness. McKenna and co-workers [110] point out that compliance effects can introduce serious corrections in constrained geometry systems. 

Note that the sums are restricted to the portion of the frill S matrix that describes reaction (or the specific reactive process that is of interest). It is clear from this definition that the CRP is a highly averaged property where there is no infomiation about individual quantum states, so it is of interest to develop methods that detemiine this probability directly from the Scln-ddinger equation rather than indirectly from the scattering matrix. In this section we first show how the CRP is related to the physically measurable rate constant, and then we discuss some rigorous and approximate methods for directly detennining the CRP. Much of this discussion is adapted from Miller and coworkers [44, 45]. 

The sunmration averages to n. Using the definition of the difhision coefficient, D=e /(2Xj), and the difhision time, equation B 1,14,7 gives... 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

This definition is based on and proportional to the g-expectation value. However, it is more useful since it is not necessary to evaluate the partition function to compute an average. 

We assume that A is a symmetric and positive semi-definite matrix. The case of interest is when the largest eigenvalue of A is significantly larger than the norm of the derivative of the nonlinear force f. A may be a constant matrix, or else A = A(y) is assumed to be slowly changing along solution trajectories, in which case A will be evaluated at the current averaged position in the numerical schemes below. In the standard Verlet scheme, which yields approximations y to y nAt) via... 

The numerical value of the exponent k determines which moment we are defining, and we speak of these as moments about the value chosen for M. Thus the mean is the first moment of the distribution about the origin (M = 0) and is the second moment about the mean (M = M). The statistical definition of moment is analogous to the definition of this quantity in physics. When Mj = 0, Eq. (1.11) defines the average value of M this result was already used in writing Eq. (1.6) with k = 2. 

Equation (2.39) is the weight average molecular weight as defined in Sec. 1.8. It is important to note that this result. My = M y, applies only in the case of nonentangled chains where 17 is directly proportional to M. A more general definition of My for the case where 17 a is... 

The number average degree of polymerization for these mixtures is easily obtained by recalling the definition of the average from Sec. 1.8. It is given by the sum of all possible n values, with each multiplied by its appropriate weighting factor, provided by Eq. (5.24) ... 

If we multiply the time elapsed per monomer added to a radical by the number of monomers in the average chain, then we obtain the time during which the radical exists. This is the definition of the radical lifetime. The number of monomers in a polymer chain is, of course, the degree of polymerization. Therefore we write... 

Now we consider how the averaging implied by the overbar is carried out. What this involves is multiplying cos(srj, cos 7) by P(7) d7-the probability that a particular angle is between 7 and 7 + d7-and then integrating the result over all values of 7 in keeping with the customary definition of an average quantity. 

Fig. 13. Definition of effective average slopes of equilibrium line (45).

---