Explicit distribution

The explicit distribution of molecular sizes can now be obtained by solving recursively the Smoluchowski equation at Eq. (8). For ci we get... 

The subscript indicate the differentiation variables and 0 no differentiation before x and y are set to 1.) It is not difficult to solve the sets analytically for the stoichiometric mixture (Ao= Bo = 1/2). The solutions together with explicit distribution functions are collected in Table 3. 

Table 5. Moments and explicit distributions generated by the Smoluchowski coagulation equation (prior to gel point) for models 1 and 2 in Table 4. [For both models Mi=l]...

It is now easy to see that if these Wk fulfill Eq. (80), the explicit distributions Ck in Table 5 necessarily fulfill the relation... 

Again, the constant in the denominator of the fraction can be absorbed in renormalization of / (l1. Although / and f have been expressed as explicit distributions in T. these also represent distributions in Dy, since T = q2D. ... 

Here, process iterand is a thread-safe routine that produces elements of the X vector, and the programmer has explicitly distributed the work among the threads in a round-robin fashion. 

OpenMP provides work-sharing constructs that make it unnecessary for the programmer to provide code that explicitly distributes the work among threads. A work-sharing construct binds to the thread team created by the innermost parallel directive within which the construct is nested. 

Here we have carried out the velocity integral over the required half-space and used the explicit fonn of the Maxwell-Boltzmaim distribution fiinction, given by equation (A3.1.3). 

Fig. 7. Writhe distributions for closed circular DNA as obtained by LI (see Section 4.1) versus explicit integration of the Langevin equations. Data are from [36].

The relation between the dusty gas model and the physical structure of a real porous medium is rather obscure. Since the dusty gas model does not even contain any explicit representation of the void fraction, it certainly cannot be adjusted to reflect features of the pore size distributions of different porous media. For example, porous catalysts often show a strongly bimodal pore size distribution, and their flux relations might be expected to reflect this, but the dusty gas model can respond only to changes in the... 

The semi-empirical methods of HyperChem are quantum mechanical methods that can describe the breaking and formation of chemical bonds, as well as provide information about the distribution of electrons in the system. HyperChem s molecular mechanics techniques, on the other hand, do not explicitly treat the electrons, but instead describe the energetics only as interactions among the nuclei. Since these approximations result in substantial computational savings, the molecular mechanics methods can be applied to much larger systems than the quantum mechanical methods. There are many molecular properties, however, which are not accurately described by these methods. For instance, molecular bonds are neither formed nor broken during HyperChem s molecular mechanics computations the set of fixed bonds is provided as input to the computation. 

An important question is whether one can rigorously express such an average without referring explicitly to the solvent degrees of freedom. In other words. Is it possible to avoid explicit reference to the solvent in the mathematical description of the molecular system and still obtain rigorously correct properties The answer to this question is yes. A reduced probability distribution P(X) that depends only on the solute configuration can be defined as... 

The reduced probability distribution does not depend explicitly on the solvent coordinates Y, although it incorporates the average influence of the solvent on the solute. The operation symbolized by Eq. (4) is commonly described by saying that the solvent coordinates Y have been integrated out. In a system at temperature T, the reduced probability has the form... 

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. 

Essential for MD simulations of nucleic acids is a proper representation of the solvent environment. This typically requires the use of an explicit solvent representation that includes counterions. Examples exist of DNA simulations performed in the absence of counterions [24], but these are rare. In most cases neutralizing salt concentrations, in which only the number of counterions required to create an electrically neutral system are included, are used. In other cases excess salt is used, and both counterions and co-ions are included [30]. Though this approach should allow for systematic smdies of the influence of salt concentration on the properties of oligonucleotides, calculations have indicated that the time required for ion distributions around DNA to properly converge are on the order of 5 ns or more [31]. This requires that preparation of nucleic acid MD simulation systems include careful consideration of both solvent placement and the addition of ions. 

We studied vacancy segregation near interphase and antiphase boundaries using the MFA and PCA approaches described in Sec. 6 below. For the A-B alloy with vacancies, the stationary distribution of mean vacancy occupations =< > can be explicitly... 

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