Reduced configurational distribution functions

Consider the configuration space distribution function P(r ), Eq. (5.6). Mathematically, it is the joint distribution function (see Section 1.5.2) to find the N particles of the system in their respective positions in configuration space, that is, P(r ) /r is the probability to find particle 1 in the range /ri near n, and particle 2 in the range dr2 near r2, and particle 3 in the range dr, near rs, and so on. 

We may also define a reduced distribution (see Section 1.5.2). The probability to find particle 1, say, in the neighborhood Zr i of ri irrespective of the positions of all other particles is P / (ri) /ri, where 

If all the particles in the system are identical then r i and r2 can be the coordinates of any two particles in the system. It is sometimes convenient to use a normalization that will express the fact that, if we look at the corresponding neighborhoods of ri and r2, the probability to find these neighborhoods occupied by any tM o particles increases in a statistically determined way with the number of particles in the system. This is achieved by multiplying the joint distribution function (r i, r2) 

Noting that N(N — 1) is the total number of pairs in the system, represents the density of such pairs per unit volume. This concept can be generalized the reduced joint distribution function for particles 1. n is given by 

As already noted, in a homogeneous fluid does not depend on the 

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The local functions depend upon at most 3N coordinates of the electron configuration space. This number can be reduced to 3k by integration over the positions of N-k electrons. For example, in the case of spinless reduced density distribution functions [40] ... 

Fig. 6.2.2. Left Simulated NMR lineshapes that are averaged by various characteristic segmental motions. In the case of fast rotation, y represents the angle between the rotation axis and the C—bond. For a two-site jump, j3 denotes the angle between the C—bond in the two configurations, and the effective asymmetry parameter becomes 17 7 0. Right Calculated 2D exchange spectra for a two-site jump with /3 = 120° (top), and for continuous diffusion (bottom). The distribution functions P(/3) of the reorientation angle are shown, together with the contour maps of the corresponding spectra. All data are displayed on the reduced frequency scale in units of Cq, and mixing times are set equal to the motional correlation time r,..

A natural goal of simulation would be the computation of the relative probabilities of these various states. A more elementary task is to compute the radial distribution which gives the distribution of distance between atom pairs observed. The radial density function may be approximated from a histogram of all pan-distances observed in a long simulation. (There are 21 at each step, so the amount of data is helpfully increased, reducing the sampling error .) This distribution is displayed in Fig. 3.5. The peaks of the radial distribution function are correlated with the various interatomic distances that appear in the cluster configurations shown in Fig. 3.4. 

The simplest is that proposed by Wall and developed by him and Flory using only the distribution function equation (83). Despite Brownian motion. Wall supposed that the crosslinks did not really move at all, and thus the problem reduces to that of a single chain between two fixed ends, and when the solid deforms all chains deform affinely, i.e. all microscopic dimensions deform the same way as the macroscopic dimensions. Under these circumstances the number of configurations Q is given by A Cl R)... 

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