Chess and Statistical Mechanics

If you want to understand the working of any complex system then you would first try to break everything down to its fundamentals. Then you would understand the fundamentals very clearly and then use that understanding of the fundamentals to completely understand the system. This is the very basis of Physics, which is one of the most successful endeavours of humankind. We have been able to take this approach whereby we break down systems to fundamentals and then use its knowledge to have the most detailed and clear understanding of a lot of things.

There is an important detail however, which is that, when you break down the system to fundamentals, you might end up with plethora of fundamentals, that, even if you understand them in the most precise form, you would not be able to combine that information to make and understanding of the system. “Why not?” you would ask. The simple answer is that, you might not be able to handle the sheer number of fundamentals simultaneously, especially if they are not independent and are interacting.

My stubborn brain insists that I always use the fundamental principles to understand the unit and then combine them, but as you will quickly come to the reality of handling the independent units, you have no choice but to give up that idea. It sort of feels very uncomfortable, at least to my brain, that I have the complete understanding of the parts, and somehow I can’t combine them to understand a system. But the reality is that the computation/space requirement of the calculation needed to use the fundamentals to the interacting system is a very daunting and almost impossible task. Impossible certainly to human brain but surprisingly to a very fast computer too. We are then left with searching for alternatives.

Imagine an atom. Hydrogen for example. A proton in a relatively confined electron cloud. If you think that you would be able to use Schrödinger’s equation to have complete understanding of each of the constituents, you would relatively easily be able to solve those and have interaction term to have a good enough understanding of the whole system. Now imagine going to deuterium or tritium. Then jumping to helium and further would be a very daunting task. We can’t then imagine Uranium, or even Carbon or Nitrogen.

What is the solution? Come up with some other “fundamental” law that would govern a cornucopia of such particles. One such solution in nature, which has been astonishingly successful to explain the complex behaviour of system is “Statistical Mechanics”. I’ve written about my amazement of such a discipline, but the fact that we can express a “fundamental” principle that would govern such system and use of which can be used to describe a system is completely bonkers. What is that “fundamental” principle you ask. It’s that “every interacting system progresses in such a direction which would extremize a function”. The function is called ’entropy’ which is function of uniquely identifiable parameters of such interacting system.

How is this related to chess? The first similarity is the fact that both are formed by a collection of interacting entities of which we know the fundamental guiding principle. While its harder to have a fully agreed upon and completely understood model of a particle, we have, on the other hand, chess, whose principles are not only fully understood, but by definition are the only guiding principles behind the interaction and evolution of the chess pieces.

As a physicist you should exclaim that you should be able to solve the system completely. Given any position in chess you would be able to give me the best move. This is in fact true. While computer are very powerful compared to human brain, and can calculate orders and orders of magnitude faster and on similarly larger phase space, human brain has a limitation on both time and phase space it can handle.

So, in chess, we come up with “fundamental” rules like, “develop your minor pieces early”, “castle early”, “Knights before Bishops”, “double pawn are bad”, “knights are bad against rook pawns” etc. These are “fundamental rules” so long as the other player follows the same “fundamental” rules. We have a rule in statistical mechanics which is that “system progress towards a configuration which minimizes the entropy” akin to all the rules stated about the chess rules. But for the largest majority of problems, these rules work just as the “fundamental chess rules” do for the largest majority of chess players.

Its astonishing how “most” of the chess moves you would end up playing, following the real chess rules (“Knight moves L” etc) can also be approximated by following the “fundamental rules” (“castle early” etc). But also very fascinating to me that if you are pushing boundary, you “break” the rules. Basically “Magnus can play this move but not you guys at home” akin to “well may be string theory, more dimensions etc, and not just Schrodinger’s euqation”.