Mass fractal definition

The proportionality constant Nf in Eq. (21) is a generalized Flory-Number of order one (Np=l) that considers a possible interpenetrating of neighboring clusters [22]. For an estimation of cluster size in dependence of filler concentration we take into account that the solid fraction of fractal CCA-clusters fulfils a scaling law similar to Eq. (14). It follow directly from the definition of the mass fractal dimension df given by NA=( /d)df, which implies... 

However, the same difficulty that observers meet in defining a cluster exist for theorists to define clusters in a numerical simulation typical numerical simulations handled several millions dark matter particle and a similar number of gas particle when hydro-dynamical processes are taken into account the actual distribution of dark matter, at least on non linear scales is very much like a fractal, for which the definition of an object is somewhat conventional Different algorithms are commonly used to define clusters. Friend of friend is commonly used because of its simplicity, however its relevance to observations is very questionable, especially for low mass systems. On the analytical side... 

Fractals can be defined in two different ways. The first definition considers a fractal to be a structure that is self-similar at any scale. The second definition considers a fractal to be a stmcture with a noninteger Hausdorff dimension. Let M be the fractal mass, i.e., the number of points of the fractal. If the mass density is constant, the mass is proportional to the fractal volume. The latter is proportional to, where L is the fractal length and the Hausdorff, or fractal, dimension. Both definitions... 

This definition deviates from the one some authors use for compact aggregates, where the fluid captured in the voids are considered as part of the aggregate. However, in the case of fractal aggregates it is difficult to define a meaningful aggregate surface, which is necessary for calculating a total volume or mass. 

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