What About Dimension

What happens in two dimensions? After all, we have two dimensions in our smallest hemocoel. Step randomly to the north, south, east or west each time. Once the walker returns to the origin, it starts over again, and then given enough time, there is a second return to the origin, and then a third return, and so on. Two-dimensional random walks, then, visit all points of the surface if walkers have sufficient time to complete their walks.

Now what happens to walkers in three dimensions? Here things are different. Very different. Walkers can go up and down as well as in the four compass directions. Use a standard six-sided die to determine your particle's movements. Now however, even if our walker takes infinite time and infinitely many steps, its probability of getting back to the origin or any other specified place in the hemocoel is only about a third or point three four. There is so much space available to get lost in, that unless our particle happens to make it back to the origin in just a few steps, it is most likely to get lost forever. There are so many ways for aimlessly wandering molecules to get lost. It is for this reason that three-dimensional hemocoels do not transfer particles from one position to another unless convection adds direction to diffusion (Ref: Walking Particles).

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