Distribution function Fourier coefficient

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... 

The important information about the properties of smectic layers can be obtained from the relative intensities of the (OOn) Bragg peaks. The electron density profile along the layer normal is described by a spatial distribution function p(z). The function p(z) may be represented as a convolution of the molecular form factor F(z) and the molecular centre of mass distribution f(z) across the layers [43]. The function F(z) may be calculated on the basis of a certain model for layer organization [37, 48]. The distribution function f(z) is usually expanded into a Fourier series f(z) = cos(nqoz), where the coefficients = (cos(nqoz)) are the de Gennes-McMillan translational order parameters of the smectic A phase. According to the convolution theorem, the intensities of the (OOn) reflections from the smectic layers are simply proportional to the square of the translational order parameters t ... 

As another application of this method of asymptotic integration, we shall consider the problem of the Fourier coefficients pfU(P t) i 1 the limit of long times. As mentioned above, we do not wish to give here a detailed proof of the transport equation for pk] p] t) (see, for instance, Ref. 31). The main result of this analysis is, however, very simple in the limit of long times (t —> oo), the correlations are entirely determined by the velocity distribution function p< p t). One has ... 

The method of separation of variables can be applied in the same manner to other initial distributions of diffusant. The effort lies only in determining the Fourier coefficients, which, for many cases, can be looked up in tables. If the spatial dimension of the system is higher [e.g., c(x, y, z, f)], a separate Fourier series must be obtained for each of the three separate functions in the product X(x)Y(y)Z(z). 

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,... 

Simulations—isoergic and isothermal, by molecular dynamics and Monte Carlo—as well as analytic theory have been used to study this process. The diagnostics that have been used include study of mean nearest interparticle distances, kinetic energy distributions, pair distribution functions, angular distribution functions, mean square displacements and diffusion coefficients, velocity autocorrelation functions and their Fourier transforms, caloric curves, and snapshots. From the simulations it seems that some clusters, such as Ar, 3 and Ar, 9, exhibit the double-valued equation of state and bimodal kinetic energy distributions characteristic of the phase change just described, but others do not. Another kind of behavior seems to occur with Arss, which exhibits a heterogeneous equilibrium, with part of the cluster liquid and part solid. 

Finally, Bertaut imagined the homogeneous domains of coherent diffraction as constituted by columns of elementary cells juxtaposed orthogonally to the diffracting planes, and he defined a size distribution P(n) as the numerical fraction of columns of length n cells. From that size distribution function, the size-only Fourier coefficients can be defined " ... 

Molecular dynamics consists of examining the time-dependent characteristics of a molecule, such as vibrational motion or Brownian motion within a classical mechanical description [13]. Molecular dynamics when applied to solvent/solute systems allow the computation of properties such as diftiision coefficients or radial distribution functions for use in statistical mechanical treatments. In this calculation a number of molecules are given some initial position and velocity. New positions are calculated a short time later based on this movement, and the process is iterated for thousands of steps in order to bring the system to an equilibrium. Next the data are Fourier transformed into the frequency domain. A given peak can be chosen and transformed back to the time domain, to see the motion at that frequency. 

The moment method can be formulated without reference to the Fourier transform of the flux. It can be regarded simply as a technique of constructing flux distribution functions from their (numerically calculated) moments. When information about the singularities of the flux transform is not utilized, the method becomes less well-founded theoretically, but it gains in flexibility and can be applied even when the singularities of the transform are not well understood. This situation arises in the case of the plane oblique source, as well as in connection with the penetration of fast neutrons whose attenuation coefficient may be a rapidly varying function of the energy with many maxima and minima. 

The general equations of change given in the previous chapter show that the property flux vectors P, q, and s depend on the nonequi-lihrium behavior of the lower-order distribution functions g(r, R, t), f2(r, rf, p, p, t), and fi(r, P, t). These functions are, in turn, obtained from solutions to the reduced Liouville equation (RLE) given in Chap. 3. Unfortunately, this equation is difficult to solve without a significant number of approximations. On the other hand, these approximate solutions have led to the theoretical basis of the so-called phenomenological laws, such as Newton s law of viscosity, Fourier s law of heat conduction, and Boltzmann s entropy generation, and have consequently provided a firm molecular, theoretical basis for such well-known equations as the Navier-Stokes equation in fluid mechanics, Laplace s equation in heat transfer, and the second law of thermodynamics, respectively. Furthermore, theoretical expressions to quantitatively predict fluid transport properties, such as the coefficient of viscosity and thermal... 

We still need an expression for the cross-response coefficients. The Debye-structure functions 5d(-Ra ) and 5D(i B ) are the Fourier-transforms of the pair distribution functions QAAi f ) and gBBi f") for the A- and B-monomers within their blocks. Considering this definition, it is clear how the coefficients for the cross responses and should be calculated. Obviously, they must correspond to the Fourier-transforms of the pair distribution functions gAB r) and gBAi f ) which describe the probability of finding a B- or A-monomer at a distance r from a A- or B-monomer respectively. Actually, both are identical... 

How can this formal treatment of the distribution function (and resulting order parameters) be generalized to include the smectic-A structure We find the clue in Kirkwood s treatment of the melting of crystalline solids. In a crystal the density distribution function (the translational molecular distribution function) is periodic in three dimensions and can be expanded in a three-dimensional Fourier series. Kirkwood does this and then identifies the order parameters of the crystalline phase as the coefficients in the Fourier series. For simplicity let us consider a one-dimensionally periodic structure (such as the smectic-A but with the orientational order suppressed for the moment). The distribution function, which describes the tendency of the centers of mass of molecules to lie in layers perpendicular to the z-direction, can be expanded in a Fourier series ... 

Kirkwood has pointed out that the density distribution function of a crystalline solid (the translational molecular distribution function) can be expanded in a three-dimensional Fourier series. The coefficients in this series are then identified as the order parameters of the crystalline phase. All these order parameters vanish discontin-uously at the first-order melting point. Empirically, there are no second-order melting transitions, nor do there seem to be any solid-liquid critical points. Though not a proven fact (as far as I am aware), it seems reasonable that crystal melting is always first order because all of the order parameters cannot vanish simultaneously and continuously before the free energy of the solid phase exceeds that of the liquid phase. 

In the smectic-A phase, the single-molecule distribution function, Eqs. [7] and [10] can likewise be represented as a (one-dimensional) Fourier series in which all the coefficients may be considered order parameters. The disappearance of smectic-A order requires the simultaneous vanishing of all the order parameters. That they all can vanish simultaneously and continuously before the free energy of the smectic-A phase exceeds that of the nematic phase seems just as unlikely here as in the (empirically verified) case of the crystalline solids. It seems clear to me that the reason the various theoretical treatments mentioned above can exhibit second-order phase changes is that an insufficient number of order parameters is included. In all the treatments, either the potential, the potential of mean force, or the distribution function are expressed in terms of highly truncated Fourier series. Such truncation automatically limits the number of order parameters. Small numbers of order parameters can then vanish simultaneously and continuously under certain conditions providing the spurious second-order phase transitions. 

Obviously, the theory outhned above can be applied to two- and three-dimensional systems. In the case of a two-dimensional system the Fourier transforms of the two-particle function coefficients are carried out by using an algorithm, developed by Lado [85], that preserves orthogonality. A monolayer of adsorbed colloidal particles, having a continuous distribution of diameters, has been investigated by Lado. Specific calculations have been carried out for the system with the Schulz distribution [86]... 

As suggested by Patterson in 1934, the complex coefficients in the forward Fourier transformation (Eqs. 2.129 and 2.132) may be substituted by the squares of structure amplitudes, which are real, and therefore, no information about phase angles is required to calculate the distribution of the following density function in the unit cell ... 

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