Entropic Repulsion

When I finished my masters courses in applied maths I had to write a dissertation to obtain the degree (you may read it here). After talking to several of the probability professors and their different projects I decided to do the research on an application of an entropic repulsion result of Brownian motion (if you want to know about Brownian motion I can recommend Peter Mörters's book which is free here) that translates into one of Galton-Watson trees (to know about these, I recommend this, also free, book here).

I don't want to go through the maths in here, but rather the concepts. So to keep things in perspective think of a snake moving through the desert.

I guess it's not the most reader friendly example, to make it a bit friendlier you may think is one of those elephant eaters from Le Petit Prince (also free here) although I'm not sure how that helps... if it does.

The point is that we would like to imagine it is happily walking through the desert (no, snakes don't walk, they slither, but I'm trying to make a soft transition from reality to the mathematical model, so bear with me) and if we see the path it's taken we'll may find something that looks like this:

Which roughly seems like the snake wants to go to the right although it moves with certain random variations going up and down... but in general to the right.

The question is now, what would the snake do if it encounters a big rock in its way? Entropic repulsion means that in order to preserve the variations (the movements up and down) intrinsic to the system (the path of the snake) the system will force itself to overachieve the obstacle rather than satisfy the minimum condition (that would be stay moving close to the rock). Something like this:

I don't know if I made myself clear enough. Another example of entropic repulsion would be the amount of time studied for the final maths exam by a high school student who usually does very bad and desperately needs a passing mark. It's easier to aim towards a very good mark rather than aiming for the minimum passing grade (he doesn't want to risk the failing mark). Was that clear enough?

Well, here's the punch line of this whole story. A boy in love with a girl can be seen as a system with a clear path (he wants to see and be with her) with some random variations (some times he'll try reaching her on the phone, go for dinner, to the cinema... who knows... many dimensions in here rather than just movements up and down). Now, what if there's a big obstacle in he's way? Well, in light of the entropic repulsion phenomenon we should expect he tries to overachieve his goal (make her be in love with him too). So maybe instead of writing her one casual text he writes her poems, who knows... what sort of obstacles could we give as an example? Well, say he suddenly finds himself on the other side of the world without the budget to make the trip back... who knows.

The bad news is that in contrast with the snake or the the student, the love system has a goal that's also moving in the other side and has it's own variations so it's not necessarily a direct analogy.

Anyway, I'm just writing this because I'm sort of proud I found a mathematical way of analyse an otherwise psychological case. Although I'm not sure real applications can actually be done, I think it is a nice thought that some boys are willing to climb really big rocks.

By the way, last Friday was Pi Day... and although Vihart is a pro-Tau person, I still love her videos. Enjoy!