Density expansion

In this Section we consider several approaches which differ from the many-point density formalism discussed above. Szabo et al. [45] have introduced a novel method based on the density expansion for the survival probability, uj t). Consider a system containing walkers (particles A) and N traps (quenchers B) in volume V in d-dimensional space. We assume that the particles have a finite size but the traps can be idealized as points and hence are ignorant of each other. When the concentration of the walkers is sufficiently low so that excluded volume interactions between them are negligible, one might focus on a single walker. 

Let r and fj, j = 1. iV be the coordinates of the walker and traps, respectively, in this d-dimensional space at t = 0. The survival probability of the walker for the given initial positions of the traps, = 

Now the boundary conditions are separable, but if k 7 0, the diffusion equation is not separable. The problem is still hard. 

Ultimately we wish to calculate the survival probability in the presence of uniformly distributed traps. This average is denoted by brackets and obtained by an integration over the volume accessible to the traps  

Our starting point is a density analogous to that used in [49] in treating the migration of excitons between randomly distributed sites. This expansion is generalization of the cluster expansion in equilibrium statistical mechanics to dynamical processes. It is formally exact even when the traps interact, but its utility depends on whether the coefficients are well behaved as V and t approach infinity. For the present problem, the survival probability of equation (5.2.19) admits the expansion 

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The same result can also be obtained directly from the virial equation of state given above and the low-density fonn of g(r). B2(T) is called the second virial coefficient and the expansion of P in powers of is known as the virial expansion, of which the leading non-ideal temi is deduced above. The higher-order temis in the virial expansion for P and in the density expansion of g(r) can be obtained using the methods of cluster expansion and cumulant expansion. 

Barzykin A V, Barzykina N S and Fox M A 1992 Electronic excitation transport and trapping in micellar systems— Monte-Carlo simulations and density expansion approximation Chem. Rhys. 163 1-12... 

The reader should note that no restrictions were placed on the form of the density expansion Eq. (3.26) in particular there is no limit on the number of terms. As already noted, therefore Eqs. (3.29) are not conventional Kohn-Sham equations. Rather they are an exact one-particle form of the Hohenberg-Kohn variation procedure and use Hohenberg-Kohn potentials in the definition of the... 

In the opposite limit of a negative second virial coefficient, V2 < 0, corresponding to the bad or poor solvent regime, the polymer coil will be collapsed due to attraction between monomers. In this case, the attraction term in the free energy is balanced by the third-virial term in a low-density expansion (where we assume that V3 > 0),... 

For large values of r, the dominant terms in the density expansions (see Eq. (72)) are those with the smallest exponents. Moreover, because the exponential term is the dominant one, we can approximate F(/(r)) in Eq. (71) by ... 

Molar Volume Change. With decrease in fluid density (expansion) during reaction the increased outflow of molecules from the pores makes it harder for reactants to diffuse into the pore, hence lowering /. On the other hand, volumetric contraction results in a net molar flow into the pore, hence increasing For a first-order reaction Thiele (1939) found that this flow simply shifted the S versus Mj curve as shown in Fig. 18.5. 

Then the radial terms of this expansion can be connected to those of the momentum density expansion of Eq. (5.36) by... 

Furthermore, the functional form of the experimental y(0) can be compared with known analytical expressions based on step density expansions [18,23-26]. The same can in principle be done via an analysis of the particle shape itself. Figure 8b shows a fit of the measured shape according to the expression [43-46]... 

The data show that the ratio of elastic to total pressure is higher, the higher the loading density. Thus, near the C-J state, the LSZK isentrope may be approximated by a polytropic relation with exponent 2.78, in agreement with Deal s experimental value of r= 2.77, at least down to 500 bar. The LSZK equation thus seems to yield not only the proper D— p0 relationship but also the proper isentrope, both near the C-J state and in the large low-pressure low-density expansion limit... 

C third virial coefficient, density expansion cm6/mol2... 

It was recently shown that a formal density expansion of space-time correlation functions of quantum mechanical many-body systems is possible in very general terms [297]. The formalism may be applied to collision-induced absorption to obtain the virial expansions of the dipole... 

Density expansion. The method of cluster expansions has been used to obtain the time-dependent correlation functions for a mixture of atomic gases. The particle dynamics was treated quantum mechanically. Expressions up to third order in density were given explicitly [331]. We have discussed similar work in the previous Section and simply state that one may talk about binary, ternary, etc., dipole autocorrelation functions. 

A. Raczynski. Density expansion of correlation functions. Acta Phys. Polonia, 62 A 303, 1982. 

J. Lebowitz and J. Percus, Kinetic Equations and Density Expansions Exactly Solvable One Dimensional Systems, Phys. Rev., to appear. 

Now a density expansion of the collision operator in Eq. (331) can be performed and the first few terms in this expansion are [57]... 

The inconsistency present in this approach is that in the density expansion of the collision operator it was assumed that [p z) is small compared to Cq (pi z) but finally it has been shown that the former diverges. Thus the assumption made during the density expansion is not correct, and such an expansion cannot be performed. 

It is legitimate to inquire whether sensitization by p re-heating is an intrinsic effect or whether it is due to a decrease in expl density (expansion) as a result of pre-heating. The data of Fig 18, from Ref 32, clearly show that the effect of pre-heating cannot be attributed solely to a decrease in packing density of the preheated expl. Here the line is drawn thru the p50% density data at 25°. For HNS preheated to 110°, the sensitization effect may be solely due to expansion and a decrease in density. However, at 203° and especially at 260° sensitization is clearly greater than that produced by a decrease in density... 

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