Ulamized variables

This Ulamized variable y has potential as a random number generator because the numbers produced are uniformly distributed on the interval (0, 1) (see Figure 31.5). 

Note that although the Ulamized variables are distributed uniformly on the interval (0,1), unfortunately the variables do have correlations. This is also true for the logistic variables, as evidenced by the noise-spheres in Figures 31.6 and 31.7. Therefore, users of this approach will have to sample the generated variables at some suitably large interval to eliminate correlations. I recommend users skip iterates, taking every tenth iterate, at which point no correlations are visually apparent (Figure 31.8). 

If readers are interested in using the preceding for 3-D extraterrestrial terrain generation, convert the z distribution to a radial one in polar coordinates, and use the Ulamized variables y as angles ... 

Another powerful tool is the cellular automata method invented by John (or Janos) von Neumann and Stanislaw Marcin Ulam (under the name of cellular spaces ). The cellular automata are mathematical models in which space and time both have a granular structure (similar to Monte Carlo simulations on lattices, in MD only time has such a structure). A cellular automaton consists of a periodic lattice of cells (nodes in space). In order to describe the system locally, we assume that every cell has its state representing a vector of N components. Each component is a Boolean variable i.e., a variable having a logical value (e.g., 0 for false and 1 for true ). 

The first use of the Monte-Carlo method in a form similar to that used today was by Ulam, von Neumann and Metropolis. The goal is to calculate the expectation of some observable g(X) of a random variable X (assumed to be defined by a complicated process) having a given probability density f x). By definition... 

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