The second and third (optional) arguments are the row and column indices for a particular cell in the grid. We will use a recursive depth first search. Now we need to check for flow through the lattice. > grid.arrange(id(g1), id(g2), ncol = 2)Ī graphical representation makes things a lot clearer. In the text representation of the lattice the occuppied sites are represented by a 1, while a 0 indicates a vacant site. Next a function to generate and populate a lattice, which takes as arguments the width of the lattice and the occupation probability. Creating the Latticeįirst we will load a few handy libraries and set up some constants. The percolation problem has been studied on a menagerie of lattice types but we will focus on the simplest, the square lattice. As we will see there is a well defined range of p (centred on a threshold value p c) for which the probability of percolation decreases rapidly from one to zero. The related bond percolation problem can be posed in terms of whether or not the edges between neighbouring sites are open or closed. This is known as the site percolation problem. This situation can be modelled on a regular lattice where each of the lattice sites is either occupied (with probability p) or vacant (with probability 1-p). Evidently there is a critical porosity threshold which divides these two regimes. If, on the other hand, the porosity is low then such a path may not exist. If it is extremely porous then it is very likely that there will be an open path of pores connecting the top to the bottom and the liquid will flow freely. Whether or not this happens is determined by the connectivity of the pores within the substance. If the substance is porous then it is possible for the liquid to seep through the pores and make it all the way to the bottom of the block. Percolation has numerous practical applications, the most interesting of which (from my perspective) is the flow of hot water through ground coffee! The problem of percolation can be posed as follows: suppose that a liquid is poured onto a solid block of some substance. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. Manfred Schroeder touches on the topic of percolation a number of times in his encyclopaedic book on fractals (Schroeder, M.
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