Cross sections of the tube at random positions

Two different models of the biased random walk were envisaged. In model 

2 direction, the probability toward the output end (z+) is now (I/2) + e, while the corresponding probability toward the input end (z ) is (1/z) — e (where 2 is the coordination number of the underlying space, e.g., z = 6 in a 3-dimensional space). This model has the characteristic that diffusion is equally probable in all possible directions, the species spending equal times in all of them, but due to the factor, when the 2 direction is chosen a positive flow drives the solution to the output end. 

In a second model II, more emphasis is given to the motion toward the output and less to the other directions. The probabilities for motion in the different directions are now defined differently. While in the simple random walk the probability for motion in a specific direction is 1/z, here the probability for motion in the output direction is (1/z) + e, while the probability in any of the other five directions is 

When the diffusing species come in contact with a closed site (such as the villi sites of the model) they have two options. In the first option, the particle does not feel the presence of the closed site, and it may attempt, unsuccessfully, to go to it. This model is called the blind ant model. In the second model, the particle feels the presence of the closed site, and thus it never attempts to land on it. This is called the myopic ant model. The difference between these two models is that the blind ant consumes long times in unsuccessful attempts, and thus its motion is slower than the myopic ant case. 

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Figure 6.10 Cross sections of the tube at random positions for various concentrations of villi, TVvini = 50,100,150,200. Reprinted from [216] with permission...

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