Random Walk of KMnO4 in Water: Why Do We Observe Non-Probabilistic Behavior?

[Jigyasa]

I had a question regarding the random walk problem in statistical mechanics. If I drop, say, one molecule of KMnO4 in a beaker of water, what we generally observe (spread of KMnO4 to the ends of the beaker) is different from what we should get from probabilistic assumptions. I must be going wrong somewhere in what I'm thinking but I can't point my finger at it.If I consider one molecule of KMnO4 , then the probability of it taking r steps to the right (or left), out of a total of N steps, is NCr *(1/2)^N (assuming unbiased random walk). For Avogadro's number of molecules, this probability is now raised to the power of Avogadro's number. This is maximum if r = N/2. Physically this means that the probability of the whole solution becoming coloured (KMnO4 traveling to the far ends ) is less than the probability of only a part of the solution becoming coloured (because if KMnO4 moves a total of10 steps, the probability of it moving 5 steps to the right and 5 steps to the left is maximum. In a sense, it oscillating is more probable ) But we almost always see that the whole solution turns purple in due course of time.Is it that the need to overcome concentration gradient dominates so KMNo4 has to reach the ends? If yes, then why do we use probabilities in statistical mechanics when systems may or may not be governed by probabilistic assumptions?

Or maybe I'm wrong in assuming this to be an unbiased random walk

Please help.

 

[Orodruin]

For Avogadro's number of molecules, this probability is now raised to the power of Avogadro's number.

It is not clear what you are trying to do here. What you are actually computing is the probability of all molecules ending up at the same place, r steps away. This would typically not be what you want. What you would typically want is the distribution of the molecules, which will be the same as that of a single molecule.

 

[Jigyasa]

If I only consider a single molecule, even then the probability of the molecule reaching the far ends of the beaker is coming out to be less than the probability of it oscillating somewhere in between (because NCr is max for r = N/2, this will always be the case assuming unbiased random walk of KMNO4)

 

[Orodruin]

Obviously. However, the limiting case as N becomes large is an even distribution.

Unless your beaker is infinite, you also cannot disregard boundary effects.