Fractals exact solution

Giona, M., Schwalm, W.A., Schawalm, M.K., and Adrover, A. Exact solution of linear transport equations in fractal media. Renormalization analysis and general theory, Chem. Eng. Sci., 51, 4717,1996a. 

Given the paucity of exact solutions in this area, it seems reasonable to look for some artificially constructed graphs, e.g. fractals, for which an exact solution can be found. This solution then, can be considered as an approximation to the original problem. The advantage of this approach, over other ad-hoc approximations like the Flory approximation, is that one is assured of weU-behavior requirements like the convexity of the free energy, and avoids problems like getting two different values for a quantity ( e.g. the pressure for hard-sphere systems), if one calculates it in two different ways within the same approximation. 

Of course there are many unsolved problems, and possible directions for further research in this area. The most interesting problem would be to try to extend these exact solutions to some fractals with infinite ramification index. There are some studies of statistical physics models of interacting degrees of freedom on Sierpinski carpets, using Monte Carlo simulations, or approximate renormalization group using bond-moving, or other ad-hoc approximations. An exactly soluble case would be very instructive here. 

Thus, the results stated above demonstrated that Flory-Huggins interaction parameters described exactly enough interactions system for mac-romolecular coil in solution, controlling its fractal dimension value. The main problem at the Eq. (58) using with the purpose of prediction is the absence of the empirical parameter calculation technique [67],... 

The fact, that macromolecular coil in diluted solution is a fractal object, allows to use the mathematical calculus of fractional differentiation and integration for its parameters description [72-74]. Within the framework of this formalism there is the possibility for exact accounting of such nonlinear phenomena as, for example, spatial correlations [74]. In the last years the methods of ftactional differentiation and integration are applied successfully for pol5mier properties description as well [75-77]. The authors [78-81] used this approach for average distance between polymer chain ends calculation of polycarbonate (PC) in two different solvents. 

Within the frameworks of this formalism to account for consistently nonlinear phenomenon complex nature is a success, such as memory effects and spatial correlations. In addition the earlier known solutions are not only reproduced, but their nontrivial generalization is given. Another important feature is connected with fractal structures self-similarity using. Unlike the traditional methods of system description on the basis of averaging different procedures, when microscopic level erasing occurs, in fractal conception medium self-affine structure and thus within the frameworks of this conception system micro and macroscopic description levels are united. Exactly such method is important for complex multicomponent systems, discovered far from thermodynamic equilibrium state [35], which are polymers [12], The authors of Refs. [31, 32] are attempted two indicated trends combination. 

A wide range of complex structures were obtained in in situ silica-filled poly(dimethylsiloxaiie) networks prepai ed by various synthetic protocols [4,33]. Nevertheless, the typical fractal patterns and morphologies described in the case of polymerization of silica in solution are not exactly those observed when the polymerization is carried out in PDMS [4]. This is most probably due to the constraints provided by the polymer environment. 

---