EhrenfestUrnModel

EhrenfestUrnModel - Stat 333 Ehrenfest Urn Model of...

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Stat 333 Ehrenfest Urn Model of Molecular Diffusion The simplest statement describing the process of diffusion is to say that molecules tend to move from regions of high concentration to regions of low concentration. Imagine a drop of black ink being placed in a glass of water. Initially the ink is concentrated in the small black drop and the surrounding water is clear. However, as time passes, the ink molecules spread throughout the glass, ultimately turning the water a murky grey colour. Sometimes this process is described as the result of a force or gradient pulling the ink molecules outward from the drop, but in fact this behaviour can be described in simple probabilistic terms. Moreover, this probabilistic description shows that no special energy or active force is required to accomplish diffusion (that is, this is a passive form of transport) at any given time far in the future there is a microscopic (but non-zero) probability that all the ink molecules will be crowded back together in the original drop. (This is counter-intuitive and is a kind of reverse diffusion . More about this alarming assertion later!) the physical setup: Suppose we have two containers, A and B , of equal size, separated by a membrane. Further suppose we have K molecules (e.g. ink molecules) which are scattered randomly and uniformly between and within the two containers. We assume that the membrane is permeable to the molecules i.e. if a molecule strikes the membrane then it will pass through and cross to the other container. We further assume that it is not possible for two or more molecules to cross the membrane at exactly the same time. Some remarks: 1. The assumption that the containers are of equal size is just to make things simple when we set up the mathematical model (see below). In this case any random molecule in container A has the same chance of crossing the membrane as does any random molecule in container B (because the shape and volume of space they occupy is the same). Thus all molecules “behave the same” under the equality assumption. On the other hand, if the containers are not of equal size, the crossing probabilities for molecules are not the same for the two containers. Imagine a very small box interfacing with a very large box. On average, the distance between the membrane and a molecule in the large box will be much greater than the distance between the membrane and a molecule in the small box; hence the molecule in the large box will be less likely to hit the membrane than will the molecule in the small box. However , having said that, the same basic form of the results hold (with probabilities adjusted for size) even when the containers are of unequal sizes.
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EhrenfestUrnModel - Stat 333 Ehrenfest Urn Model of...

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