Physics, Nature and Society by Joaquín Marro

Author:Joaquín Marro
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

When the aggregate has reached a certain size, a new particle—always generated in a place at random—is easily trapped by its outer parts, then it rarely manages to penetrate inside the form. The zones that grow quickly screen others, which therefore become less accessible. Consequently, ramified aggregates form that emanate from the origin, such as those in Fig. 5.15. If the probability p of becoming trapped reduces, the particle ricochets more often, in any random direction, so that such local random movement then tends to fatten the branches and produce somewhat more compact aggregates. The result is suspiciously reminiscent of the third form shown in Fig. 5.14. Forms similar to the other in this figure can also be obtained if starting with a line of fixed particles.

The simulation that we have described is perhaps suitable as a metaphor of the microscopic dynamics during electro-deposition of ions, even for the development of a coral, but surprisingly DLA forms also occur in other, very different scenarios. The fact that similar forms are observed under varied conditions without any apparent relation with the details of the simulation suggests that the DLA process is somewhat general. In effect, similar forms show up, although softened, when a fluid penetrates a cavity occupied by another fluid with which it does not mix. This is the case, for example, when injecting water into porous rocks to extract the petroleum stored inside. A significant fact is that, to thus obtain DLA forms, it is necessary for the fluids to have the property of responding with a speed proportional to the pressure gradient that they undergo. This requirement can consequently be seen as the macroscopic reflection, that is, a mean global result of the DLA microscopic process.

It is also a symptom of generality that the DLA forms are fractal objects. As we later discuss, this means that those aggregates show a potential relation, N = r D , between the number of particles, N, and a measure, r, of their size. The parameter D, called fractal dimension, is D ≃ 1.7 for a planar aggregate (d = 2) and D ≃ 2.5 for an aggregate in the three-dimensional space (d = 3). Furthermore, by making DLA simulations in spaces whose dimension surpasses the case d = 3, it is observed that D tends towards d − 1, so that the fractal dimension D is always under the dimension d of the space that contains the object.

When studying growth, organisms and populations have also been observed that develop according to a random multiplication process. This means that growth at each stage or time is a certain percentage of the size at that time and that, due to unknown factors, that percentage is different and unforeseeable each time. To specify this idea, let’s assume that the object starts from a size and that at each stage n = 1, 2, 3…, its size is multiplied by a random variable, A, so that . Given that the result depends on the distribution

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