Distribution function calculation average value

Here, p(r) is the average density of atoms found in a thin shell at a radius r from an arbitrary atom in the material, and p is the average density of the entire material. For very small values of r, g(r) —> 0, since atoms cannot overlap one another. For large values of r, on the other hand, g(r) —> 1, because atoms that are separated from one another by large distances in a disordered material are not influenced by one another. The distribution functions calculated by Lewis et al. for liquid and amorphous InP are shown in Fig. 9.4. As might be expected, the amorphous material has considerably more structure than the liquid. One important feature in the amorphous material is the peak in the P-P distribution near r 2.2 A. This peak shows the existence of P-P bonds in the amorphous material, a kind of bond that does not exist in the crystalline solid. 

E. Collision Frequency between Maxwellian Molecules. Finally, we can calculate the average number of collisions made by a molecule going through a Maxwellian gas if the molecule does not have a fixed velocity V, but has instead a velocity distribution which is itself Maxwellian. This may be done by multiplying Zc [Eq. (VII.8D.4)] by the Maxwellian distribution function and averaging over all values of Vx ... 

In order to compute average properties from a microscopic description of a real system, one must evaluate integrals over phase space. For an A -particle system in an ensemble with distribution function P( ), the experimental value of a property A( ) may be calculated from... 

The potential of mean force is a useful analytical tool that results in an effective potential that reflects the average effect of all the other degrees of freedom on the dynamic variable of interest. Equation (2) indicates that given a potential function it is possible to calculate the probabihty for all states of the system (the Boltzmann relationship). The potential of mean force procedure works in the reverse direction. Given an observed distribution of values (from the trajectory), the corresponding effective potential function can be derived. The first step in this procedure is to organize the observed values of the dynamic variable, A, into a distribution function p(A). From this distribution the effective potential or potential of mean force, W(A), is calculated from the Boltzmann relation ... 

The important point we wish to re-emphasize here is that a random process is specified or defined by giving the values of certain averages such as a distribution function. This is completely different from the way in which a time function is specified i.e., by giving the value the time function assumes at various instants or by giving a differential equation and boundary conditions the time function must satisfy, etc. The theory of random processes enables us to calculate certain averages in terms of other averages (known from measurements or by some indirect means), just as, for example, network theory enables us to calculate the output of a network as a function of time from a knowledge of its input as a function of time. In either case some information external to the theory must be known or at least assumed to exist before the theory can be put to use. 

It will be assumed for the moment that the non-bonded atoms will pass each other at the distance Tg (equal to that found in a Westheimer-Mayer calculation) if the carbon-hydrogen oscillator happens to be in its average position and otherwise at the distance r = Vg + where is a mass-sensitive displacement governed by the probability distribution function (1). The potential-energy threshold felt is assumed to have the value E 0) when = 0 and otherwise to be a function E(Xja) which depends on the variation of the non-bonded potential V with... 

The radial distribution ftinctions may be used to calculate expectation values of functions of the radial variable r. For example, the average distance of the electron from the nucleus for the Is state is given by... 

Table 4.2 Two-dimensional displacement distribution function W on W 110 heating periods were 60 s each experimental values were averaged over equivalent directions. Theoretical values calculated from the experimental mean square displacments are listed in... 

The moments of the distribution function are of practical value, since the average molecular weights can be calculated from their ratios ... 

The workhorse for the calculation of cross sections in full collisions is the so-called Monte Carlo technique (Schreider 1966 Porter and Raff 1976 Pattengill 1979). The application to photodissociation proceeds in an identical fashion. Within the Monte Carlo method an integral over a function f(x) is approximated by the average of the function over N values Xk randomly selected from a uniform distribution,... 

The MFA [1] introduces the perturbation due to the solvent effect in an averaged way. Specifically, the quantity that is introduced into the solute molecular Hamiltonian is the averaged value of the potential generated by the solvent in the volume occupied by the solute. In the past, this approximation has mainly been used with very simplified descriptions of the solvent, such as those provided by the dielectric continuum [2] or Langevin dipole models [3], A more detailed description of the solvent has been used by Ten-no et al. [4], who describe the solvent through atom-atom radial distribution functions obtained via an extended version of the interaction site method. Less attention has been paid, however, to the use of the MFA in conjunction with simulation calculations of liquids, although its theoretical bases are well known [5]. In this respect, we would refer to the papers of Sese and co-workers [6], where the solvent radial distribution functions obtained from MD [7] calculations and its perturbation are introduced a posteriori into the molecular Hamiltonian. 

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