Brownian motion and G.P.Rajaratnam.

G.P.Rajaratnam iyengar (1908-1979) popularly known as Rajaratnam is a well known figure in Kannada literature. He has written many, many books, but he is best-known for his work "Ratnana Padagalu" (ರತ್ನನ ಪದಗಳು) --- life as seen from the eyes of Yendkuduka Ratna (ಹೆಂಡ ಕುಡುಕ ರತ್ನ) (Drunkard Ratna). Ratnana Padagalu has been popularized by well-known Kannada singers, and it is still appreciated and admired. In fact many kannda movie songs are taken from Rajaratnam's "Ratnana Padagalu".

In one of his writings he says there are many things one can learn from a drunkard except drinking. He says a poor drunkard having just enough money to buy a peg of liquor will offer a poorer drunkard standing outside the shop who cannot even afford that to share his peg but a man with all his senses will not share anything in life.

In another of his plays there is a very comical scene where a completely drunk man will be trying to figure his way home. He tells a passer by that "Brother, i have been walking for a very long time towards my home keeping a street light as my reference but after such a long walk the street light is still where it was when i started my walk, the light seem to be going away. Please help me". Comical indeed particularly if you watch this play. But there is an important mathematical concept hidden in this statement. (Long read from here).

Brownian motion is one of the very difficult mathematical concepts to understand. It all began with the english botanist Robert Brown (1773-1858) who in 1827 noticed that wild flower pollen grains suspended in water jiggled about under the lens of the microscope, following a zigZag path. How could these small tiny little things moved about, perpentually and erratically, by itself?

This is very complicated and only statistical mechanics provides an explanation.
Einstein understood that this happened because the suspended particles are being constantly hit by molecules in the sorrounding fluid. These molecules hit from all possible directions and change the path of the particles instantaneously at the moment of impact.

Mathematically , if T is the average time between two collissions, then in time t, the particle suffers N=t/T random collissions. If R is the distance travelled by the particle after N such random collissions , then R=(a){root(N) } [a x square root of N] where a is the average distance covered between two successive collissions. This is famously called the Einstein-Smoluchowski formula.

Atleast for me this is very difficult to understand. Thankfully G.P.Rajaratnam's statement quoted above is for our rescue.

Brownian motion is analogous to an ideal drunkard who has no memory of the previous step.(He has no idea about his relative position with reference to the street light) This lack of memory is the basis for Einstein-smoluchowski formula. Had the steps been corelated, then R=aN after N steps.

Consider each step of length 1 unit. i.e.a=1. If each step is corelated , meaning walker always follows the same direction, then after 10,000 steps, the walker would be 10,000 units from the starting point.
For a random walker, this does not apply. He has no idea of the previous step. He moves randomly. For an ideal drunkard, with a=1 unit, after 10,000 steps, the drunkard is R=1x{root(10000)}= 100 units away.

The drunkard has no destination nor can he return to the starting point. This is the cornerstone of irreversibility of physical system. Matter, or matter in motion exhibits this at every instance by virtue of thermal effects. This is the message that first came from einstein's paper in 1905.

Similarly in the trans-- state, high on alchohol, athma and paramathma (in a crude sense mind and body) collide to produce bodily motion that is random and irreversible with relation to the starting point.

Certainly as G.P.Rajaratnam said, there are many things to learn from the drunkard and one of them is Brownian motion.
Thanks to Rajaratnam, its a little easy for a layman like me to get an idea of Brownian motion.

I have not tried liquor to really experience brownian motion first hand. May be its worth it to give a try to experience brownian motion.