Box dimension

To deal with fractals that are not self-similar, we need to generalize our notion of dimension still further. Various definitions have been proposed see Falconer (1990) fora lucid discussion. All the definitions share the idea of measurement at a scale e —roughly speaking, we measure the set in a way that ignores irregularities of size less than , and then study how the measurements vary as — 0. 

This power law also holds for most fractal sets S, except that d is no longer an integer. By analogy with the classical case, we interpret d as a dimension, usually called the capacity or box dimension of, S. An equivalent definition is 

This solution illustrates a helpful trick. We used a discrete sequence = (1/3) that tends to zero as n — , even though the definition of box dimension says that we should let — 0 continuously. If e (1/3) , the covering will be slightly wasteful— some boxes hang over the edge of the set—but the limiting value of d is the same. 

A fractal that is not self-similar is constructed as follows. A square region is divided into nine equal squares, and then one of the small squares is selected at random and discarded. Then the process is repeated on each of the eight remaining small squares, and so on. What is the box dimension of the limiting set  

Pick the unit of length to equal the side of the original square. Then S, is covered (with no wastage) by N = 8 squares of side e = 7. Similarly, is covered by V = 8 squares of side = (7). In general, TV = 8 when e = (7) . Hence 

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Also in this case, Tp corresponds to a relaxation time which determines the coupling of the modulated variable to the external bath. The pressure scaling can be applied isotropically, whidi means that the factor is the same in all three spatial directions. More realistic is an anisotropic pressure scaling, because the box dimensions also change independently during the course of the simulation. 

For a rectangular btix, the riorihotuled cutoff should be less than onc-lialf the smallest box dimension. 

A switched function extends over the range of inner (Ron) to outer (Roff) radius and a shifted function from zero to outer (Roff) radius. Beyond the outer radius, HyperChem does not calculate non-bonded interactions. The suggested outer radius is approximately 14 Angstroms or, in the case of periodic boundary conditions, less than half the smallest box dimension. The inner radius should be approximately 4 Angstroms less than the outer radius. An inner radius less than 2 Angstroms may introduce artifacts to the structure. 

The collisions per second with the wall, on the other hand, depend upon the container dimension and the velocity (as the molecule bounces back and forth between the walls). We can assume that one-third of the molecules bounce back and forth in a given direction between two opposite walls. Then if there are n molecules in the container, there are n/3 hitting these two walls. One of these walls receives a collision each time one of the molecules travels the box dimension d and back, a distance of 2d. 

It is the local dimension that describes the irregularity of the self-affine fractal. The local dimension can be determined by such methods as the box-counting method1,61,62,65 and the dividerwalking method.61,66 The box dimension dEB is defined by the... 

Determining g(r) from a simulation involves a procedure quite similar to that described above for determining the continuous distribution of a scalar property. For each snapshot of an MD or MC simulation, all A-B distances are computed, and each occurrence is added to the appropriate bin of a histogram running from r = 0 to the maximum radius for the system (e.g., one half the narrowest box dimension under periodic boundary conditions, vide infra). Normalization now requires taking account not only of the total number of atoms A and B, but also the number of snapshots, i.e.,... 

Choose a tube length between 30 and 60 ft or so, so as to make the box dimensions roughly comparable. The exposed length of the tube, and the inside length of the furnace shell, is 1.5 ft shorter than the actual length... 

Initially, the spheres are positioned randomly in the box, periodic boundary conditions are used in the x- and /-direction and no-slip on the -direction. The spheres move according to the flow field and the viscosity is calculated for several time steps, and for each configuration an average suspension viscosity is obtained. The box is divided into 216 elements with 650 nodes, and each sphere into 96 elements with 290 nodes. The computational time depends, as for any particulate simulation, on the number of spheres. Two different sphere radii were used in the simulations 0.05 length units and 0.07 length units. In the same way, the box dimensions were set to lxlxl (length units)3 and 0.8 x 0.8 x 0.8 (length units)3. Each case was simulated with 10,20, 30 and 40 spheres. 

Another method used to examine microstructure is the box counting or grid dimension analysis. This dimension is referred to as a grid dimension because for mathematical convenience, the boxes are usually part of a grid that is laid over the image (65). The box dimension is dehned as the exponent D in the relationship ... 

Fig. 22. (a) Snapshot of an interface between two coexisting phases in a binary polymer blend in the bond fluctuation model (invariant polymerization index // = 91, incompatibihty 17, linear box dimension L 7.5iJe, or number of effective segments N = 32, interaction e/ksT = 0.1, monomer number density po = 1/16.0). (b) Cartoon of the configuration illustrating loops of a chain into the domain of opposite type, fluctuations of the local interface position (capillary waves) and composition fluctuations in the bulk and the shrinking of the chains in the minority phase. Prom Miiller [109]... 

When computing the box dimension, it is not always easy to find a minimal cover. There s an equivalent way to compute the box dimension that avoids this problem. We cover the set with a square mesh of boxes of side e, count the number of occupied boxes TV(e), and then compute d as before. 

Even with this improvement, the box dimension is rarely used in practice. Its computation requires too much storage space and computer time, compared to other... 

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