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Not sure whether this is more of a physics or a chess question.

Looking at the evaluation graph (computer evaluation vs. move number) of a game of chess, I realized that there are some similarities to the physics model of escape from a local potential minimum. Let me explain:

While none of the players make a mistake the evaluation will stay at values around zero corresponding to a draw. This I would associate with the motion of a particle in a local minimum, whose coordinate (=evaluation) might fluctuate a bit around the minimum value.

If a player makes a mistake, the evaluation will increase to let's say +2, corresponding to a won position. Normally (i.e. with a very high probability) white should win this game. Until mate, the evaluation will further increase to +3, +4, +5....

This process is similar to the escape from a local minimum. A blunder can "activate" the particle so that it can leave the local minimum and it will subsequently accelerate to larger coordinates (evaluation increases).

simple model for chess

Have there been any studies on chess using such model?

Does it make sense at all?

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    "Does it make sense at all?" - No! You have shown a continuous function. Chess is discrete not continuous (each move representing a potential step change) and evaluation can jump from +3 to -3 in one move without passing through zero. This question has no relevance to chess. – Brian Towers Mar 21 '17 at 11:40
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    There has been no studies in Chess till now and the concept here is quite abstract . Einstein used to play Chess and he was cleanly associated with a lot of Physics . I hope he could have answered it better . – Seth Projnabrata Mar 21 '17 at 11:53
  • @BrianTowers If you are doing numerics on some (continuous) physics problem you are also dealing with discrete values. I don't see any difference here. Note that the vertical axis, i.e. the potential would be something like "resistance to loss", i.e. describing the (presumed) fact that with perfect play a game ends in draw and that you have some margin (large choice of moves) which all keep the game in draw. Perhaps will ask this question at physics stackexchange. – user1583209 Mar 21 '17 at 11:58
  • @ProjnabrataSeth: Einstein did not have computer evaluation though... – user1583209 Mar 21 '17 at 11:59
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    Chess evaluation can probably be described as a variation on a random walk, but there's no such thing as a local minimum. Given a +0.5 evaluation, it isn't harder for Black to make a bad move than for White. – Glorfindel Mar 21 '17 at 11:59
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I'd argue the process you're talking about can be represented by Markov Chain. It solves the continuous problem raised by @Brian.

I don't know where to find a proper state diagram for your question.The following diagram is for a simple board game. Imagine we model the transition probability from win to lose (or win to draw) as the blunder probability, this is similar to what you are asking.

enter image description here

We can represent "local minimum" described in your question as an absorbing state.

More generally, if you want to assume some latent factors for making a blunder (e.g. bad opening, personal issues), you could generalize to hidden markov model. It's possible to use the EM algorithm to fit a model from engine PV evaluations.

Given a fitted model, you can query the probability of a blunder given a chess game.

Have there been any studies on chess using such model?

No studies that I am aware of.

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