Approximate Distribution Functions

Recently, we have developed a new method of data analysis which permits us to obtain an approximate distribution function in terms of translational diffusion coefficients or sizes (14,15). 

Figure 6.2 Approximate distribution functions of relaxation (A/i) and retardation (L ) times for polyisobutylene. (Reproduced with permission from Marvin, Proceedings of the 2nd International Congress of Rheology, Butterworths, London 1954)...

The function so calculated is approximated by a positive and normalized function parameterized with some parameters (e.g., log-normal, Raleigh, or Pearson IV distributions) behaving approximately in the same manner as the condensation approximation distribution function in the experimental region. 

Integral equation approximations for the distribution functions of simple atomic fluids are discussed in the following. 

In the limit of zero ion size, i.e. as o —> 0, the distribution functions and themiodynamic fiinctions in the MS approximation become identical to the Debye-Htickel limiting law. 

The state of the surface is now best considered in terms of distribution of site energies, each of the minima of the kind indicated in Fig. 1.7 being regarded as an adsorption site. The distribution function is defined as the number of sites for which the interaction potential lies between and (rpo + d o)> various forms of this function have been proposed from time to time. One might expect the form ofto fio derivable from measurements of the change in the heat of adsorption with the amount adsorbed. In practice the situation is complicated by the interaction of the adsorbed molecules with each other to an extent depending on their mean distance of separation, and also by the fact that the exact proportion of the different crystal faces exposed is usually unknown. It is rarely possible, therefore, to formulate the distribution function for a given solid except very approximately. 

The histogram is a graphical device which is both attainable in practice and also an approximation to a theoretical distribution function. 

It should be observed that Eq. (3.102) may be viewed as a distribution function for relaxation times. In fact, if N,. is large enougli, integer increments in p may be approximated as continuous p values. This makes Tp continuous also. The significance of this is that Eq.(3.90) can be written as an integral in analogy with (3.62) if p is continuous ... 

It is possible to go beyond the SASA/PB approximation and develop better approximations to current implicit solvent representations with sophisticated statistical mechanical models based on distribution functions or integral equations (see Section V.A). An alternative intermediate approach consists in including a small number of explicit solvent molecules near the solute while the influence of the remain bulk solvent molecules is taken into account implicitly (see Section V.B). On the other hand, in some cases it is necessary to use a treatment that is markedly simpler than SASA/PB to carry out extensive conformational searches. In such situations, it possible to use empirical models that describe the entire solvation free energy on the basis of the SASA (see Section V.C). An even simpler class of approximations consists in using infonnation-based potentials constructed to mimic and reproduce the statistical trends observed in macromolecular structures (see Section V.D). Although the microscopic basis of these approximations is not yet formally linked to a statistical mechanical formulation of implicit solvent, full SASA models and empirical information-based potentials may be very effective for particular problems. 

As we have said, the key to the analysis of asystemlike this one is tohave a function that approximates to the actual residence time distribution. The tracer experiment is used to find that distribution function,butwewillworkfroman assumed function to the tracer concentration-timecurvetoseewhattheexperimentaloutcomemightlooklike. 

The phase diagrams of the 2D binary alloys are shown in Fig.5. In Fig.6, we show the point distribution functions f and fg of the binary alloys. The dashed curve in Fig.5 shows the phase separation determined by the conventional CVM with the pair approximation The parameter is taken such that 4e = 2e g - ( aa bb)- The solid curve is calculated using the present continuous CVM, with the... 

Euler s equation (equation 9.7) may be recovered from Boltzman s equation as a consequence of the conservation of momentum, but only in the zeroth-order approximation to the full distribution function. Setting k — mvi in equation 9.52 gives, in component form. 

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

The Burnett Expansion.—The Chapman-Enskog solution of the Boltzmann equation can be most easily developed through an expansion procedure due to Burnett.15 For the distribution function of a system that is close to equilibrium, we may use as a zeroth approximation a local equilibrium distribution function given by the maxwellian form ... 

Coefficient Equations.—To determine the coefficients of the expansion, the distribution function, Eq. (1-72), is used in the Boltzmann equation the equation is then multiplied by any one of the polynomials, and integrated over velocity. This gives rise to an infinite set of coupled equations for the coefficients. Only a few of the coefficients appear on the left of each equation in general, however, all coefficients (and products) appear on the right side due to the nonlinearity of the collision integral. Methods of solving these equations approximately will be discussed in later sections. 

It was shown, in Eqs. (1-73), (1-74), (1-75), that a = 1, afy r> => 0, a = 0. As the zeroth approximation we shall assume that A mid /a are zero (their effects are negligibly small) if Eqs. (1-86) and (1-87) are multiplied by /a and A, respectively, we obtain the condition that og0 and -oSi are zero higher order equations would show that all the coefficients are zero. Thus, the coefficients are proportional to some power of /a (or A). The zero-order approximation to the distribution function is just the local maxwellian distribution... 

Eqs. (1-76) show that these values of the coefficients produce the Navier-Stokes approximations to pzz and qz [see Eq. (1-63)] the other components of p and q may be found from coefficient equations similar to Eq. (1-86) and (1-87) (or, by a rotation of coordinate axes). The first approximation to the distribution function (for this case of Maxwell molecules) is ... 

To see the type of differences that arises between an iterative solution and a simultaneous solution of the coefficient equations, we may proceed as follows. Bor the thirteen moment approximation, we shall allow the distribution function to have only thirteen nonzero moments, namely n, v, T, p, q [p has only five independent moments, since it is symmetric, and obeys Eq. (1-56)]. For the coefficients, we therefore keep o, a, a 1, k2), o 11 the first five of these... 

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. 

Although this lowest order approximation is used in determining the first order corrections to the distribution function, it is necessary to go to a higher order of approximation in determining the collision integral of Eq. (1-140). If we keep terms to first order in the small quantity m/M, the collision integral may be evaluated to give 28... 

The next step in the argument will be to make it plausible (a strict proof involves some limit arguments that would take us too far afield) that the (one-dimensional) distribution function of the random variable 7(f) is gaussian. To do so, we argue that the integral in Eq. (3-273) can be approximated as closely as desired by a sum of the form... 

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