Positional distribution function

Positional Distribution Function and Order Parameter. In addition to orientational order, some Hquid crystals possess positional order in that a snapshot at any time reveals that there are parallel planes which possess a higher density of molecular centers than the spaces between these planes. If the normal to these planes is defined as the -axis, then a positional distribution function, can be defined, where is proportional to the... 

In some Hquid crystal phases with the positional order just described, there is additional positional order in the two directions parallel to the planes. A snapshot of the molecules at any one time reveals that the molecular centers have a higher density around points which form a two-dimensional lattice, and that these positions are the same from layer to layer. The symmetry of this lattice can be either triangular or rectangular, and again a positional distribution function, can be defined. This function can be expanded in a two-dimensional Fourier series, with the coefficients in front of the two... 

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Positional Distribution Function and Order Parameter. In addition to orientational order, some liquid crystals possess positional order in that a snapshot at any time reveals that there arc parallel planes which possess a higher density of molecular centers than the spaces between these planes II the normal to these- planes is defined as the -axis, then a positional distribution function. g( ). can be defined, where gOd is proportional to the fraction of molecular centers between r and + Since yO is periodic, it can he represented as a Fourier scries (a sum uf a sinusoidal function with a periodicity equal to the distance between ihe planes and its harmonics). To represent the amount ol positional order, the coefficient in front of the fundamental term is used as the order parameter. The more Ihe molecules lend to form layers, the greater the coefficient in front of ihe fundaiucnlal sinusoidal lerm and [he greaicr the order parameter for positional order,... 

For the equilibrium properties of an ideal gas it is thus the distribution function of the velocities which is required. For nonideal gases or liquids, a position-distribution function is needed for a system not at Equilibrium but changing in time, distribution functions of velocity and position which were the proper functions of time would similarly serve to establish the properties of the system. 

Fig. 2. Simulations of a one-dimensional harmonic oscillator coupled to Nose-Hoover chains of length M = 1 (a-c), M = 3 (d-f), and M = 4 (g-i). (a),(d),(g) The Poincare sections for these oscillators. (b),(e),(h) The momentum distribution functions. (c),(f),(i) The position distribution functions...

The modified position distribution function is incorporated directly into the rate constant calculation based on the voxel method, with the penetrant positioned at the voxel center. Each single-voxel configuration integral [Zj in Eq. (32)], referenced to the energy of the local minimum origin state, contains an integral over polymer conformation fluctuations sampled via the parameter. 

There have been a number of reports discussing where a dopant sits in the Ge2Sb2Tc5 host network [8], It is therefore interesting to see how these dopants (Cu, Ag, Au) adapt to the local structure. The atomic structure of our models is studied through a set of pair correlation functions. A pair correlation function is a position distribution function based on the probability of finding atoms at some distance r from a central atom. Following [9], we tersely develop the expressions for correlation functions and present this below. A general expression for the pair distribution function [9] is ... 

For a paracrystal system, the stmcture factor Sp(q), which is also known as the interference function or lattice factor, can be determined from the Fourier transform of a complete set of lattice points. In this sense, Sp(q) is the same as that defined in the previous subsection. However, the paracrystal model assumes to have the second kind of crystal imperfection, so its lattice points are no longer fixed at certain positions but instead are described by a positional distribution function. In the simple case where the autocorrdation function of a crystal lattice is given by the convolution product of the distributions of the lattice points along three axes and the distribution fimaion is a Gaussian, Sp(q) can be expressed by the following equation ... 

Instead of plotting tire electron distribution function in tire energy band diagram, it is convenient to indicate tire position of tire Fenni level. In a semiconductor of high purity, tire Fenni level is close to mid-gap. In p type (n type) semiconductors, it lies near tire VB (CB). In very heavily doped semiconductors tire Fenni level can move into eitlier tire CB or VB, depending on tire doping type. 

Because the correlation of atomic positions decreases as r — co, = 1. The function 47T p (, the radial distribution function (RDF), may also be... 

For a removal attempt a molecule is selected irrespective of its orientation. To enhance the efficiency of addition attempts in cases where the system possesses a high degree of orientational order, the orientation of the molecule to be added is selected in a biased way from a distribution function. For a system of linear molecules this distribution, say, g u n ), depends on the unit vector u parallel to the molecule s symmetry axis (the so-called microscopic director [70,71]) and on the macroscopic director h which is a measure of the average orientation in the entire sample [72]. The distribution g can be chosen in various ways, depending on the physical nature of the fluid (see below). However, g u n ) must be normalized to one [73,74]. In other words, an addition is attempted with a preferred orientation of the molecule determined by the macroscopic director n of the entire simulation cell. The position of the center of mass of the molecule is again chosen randomly. According to the principle of detailed balance the probability for a realization of an addition attempt is given by [73]... 

The probability given by Eq. (2) is a function of an enormous number of variables. We can neither compute nor display such a function. The most with which we can deal are functions of the coordinates of one, two, three, or, at the outside, four molecules. It takes six variables to specify the positions of four molecules. Therefore, it is helpful to integrate over the positions of most of the molecules. The h molecule distribution function is given by... 

The case A = 2 is of greatest interest. Since the force is central, it is not necessary to use rj and ri as variables. The single variable r 2 is sufficient since the position of the center of mass is irrelevant. Thus, we have the radial distribution function (RDF), g r 12). 

Differentiation of Eq. (8) with respect to the position of a molecule gives a hierarchy of integro-differential equations, each of which relates a distribution function to the next higher order distribution function. Specifically,... 

Figure 14 shows the displacement of the distribution function towards high / , i.e. the uncoiling of molecules under the influence of stretching for polyethylene (A = 3 x 10-9 m, N = 100 and T = 420 K). This displacement will be characterized by the position of the maximum of the distribution curve, the most probable value of / , i.e. j3m, as a function of x (Fig. 15). Figure 15 also shows the values of stresses a that should be applied to the melt to attain the corresponding values of x (o = xkT/SL, where S is the transverse cross-section of the molecule). 

Discussion of the Equation.—The Boltzmann equation describes the manner in which the distribution function for a system of particles, /x = /(r,vx,f), varies in terms of its independent variables r, the position of observation vx, the velocity of the particles considered and the time, t. The variation of the distribution function due to the external forces acting on the particles and the action of collisions are both considered. In the integral expression on the right of Eq. (1-39), the Eqs. (1-21) are used to express the velocities after collision in terms of the velocities before collision the dynamics of the collision process are taken into account in the expression for x(6,e), from Eqs. (1-11) and (1-12), which enters into the k of Eqs. (1-21). Alternatively, as will be shown to be useful later, the velocities before and after collision may be expressed, by Eq. (1-20), in terms of G,g, and g the dynamics of the collision comes into the relation between g and g of Eq. (1-19). 

Boltzmann s H-Theorem. —One of the most striking features of transport theory is seen from the result that, although collisions are completely reversible phenomena (since they are based upon the reversible laws of mechanics), the solutions of the Boltzmann equation depict irreversible phenomena. This effect is most clearly seen from a consideration of Boltzmann s IZ-function, which will be discussed here for a gas in a uniform state (no dependence of the distribution function on position and no external forces) for simplicity. 

This binary collision approximation thus gives rise to a two-particle distribution function whose velocities change, due to the two-body force F12 in the time interval s, according to Newton s law, and whose positions change by the appropriate increments due to the particles velocities. 

If the distribution function does not change rapidly with position, i.e., the macroscopic distances involved are large compared with the distance over which a collision takes place, we may neglect the A and Ar2. Since VVl = —, and Av2 = — AVj, the integrand of Eq. 

If the interval r is large compared with the time for a collision to be completed (but small compared with macroscopic times), then the arguments of the distribution functions are those appropriate to the positions and velocities before and after a binary collision. The integration over r2 may be replaced by one over the relative distance variable r2 — rx as noted in Section 1.7, collisions taking place during the time r occur in the volume g rbdbde, where g is the relative velocity, and (6,e) are the relative collision coordinates. Incomplete collisions, or motions involving periodic orbits take place in a volume independent of r when Avx(r) and Av2(r) refer to motion for which a collision does not take place (or to the force-field free portion of the... 

Anderson (A2) has derived a formula relating the bubble-radius probability density function (B3) to the contact-time density function on the assumption that the bubble-rise velocity is independent of position. Bankoff (B3) has developed bubble-radius distribution functions that relate the contacttime density function to the radial and axial positions of bubbles as obtained from resistivity-probe measurements. Soo (S10) has recently considered a particle-size distribution function for solid particles in a free stream ... 

Applying the TABS model to the stress distribution function f(x), the probability of bond scission was calculated as a function of position along the chain, giving a Gaussian-like distribution function with a standard deviation a 6% for a perfectly extended chain. From the parabolic distribution of stress (Eq. 83), it was inferred that fH < fB near the chain extremities, and therefore, the polymer should remain coiled at its ends. When this fact is included into the calculations of f( [/) (Eq. 70), it was found that a is an increasing function of temperature whereas e( increases with chain flexibility [100],... 

In the discussion of hypoelectronic metals in ref. 4, the number of ways of distributing Nv/2 bonds among NL/2 positions in a crystal containing N atoms with valence v and ligancy L was evaluated. The number per atom is the Nth root of this quantity. Structures for which the number of bonds on any atom is other than v-l,v, orv + l were then eliminated with use of the binomial distribution function [only the charge states M+, M°, and M are allowed by the electroneutrality principle (5)]. In this way the following expression for rhypo, the number of resonance structures per atom for a hypoelectronic metal, was obtained ... 

Fig. 5. The electron distribution function for a Dirac 2s electron in atoms with the indicated atomic numbers. The vertical broken line shows the position of r for...

The simplest way computationally of obtaining a sedimentation coefficient distribution is from time derivative analysis of the evolving concentration distribution profile across the cell [40,41]. The time derivative at each radial position r is d c r,t)/co /dt)r where cq is the initial loading concentration. Assuming that a sufficiently small time integral of scans are chosen so that Ac r t)/At= dc r t)ldt the apparent weight fraction distribution function g (s) n.b. sometimes written as (s ) can be calculated... 

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... 

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