Statistical Mechanics

Statistical mechanics seeks to describe systems with large numbers of particles, even when the behavior of individual particles cannot be described precisely.

The fundamental quantity in statistical mechanics is the entropy , where is the number of possible configurations of the particles. For instance, one can try to count the number of ways there are to arrange a system with a given total energy, , by counting the number of ways one can distribute the energy among the particles. This number is and the entropy is its logarithm. The fundamental assumption is that each configuration happens with equal probability. This seems like the most natural assumption one could make, yet it can lead to surprising predictions.

Puzzle

I have picked a number between and . You can guess a number, and I will tell you if it is equal, above or below the selected number. Try to find the number in the least possible number of guesses.

Solution

You may be familiar with a “binary” search, which involves always trying to guess the middle of the possible range. This is the best approach because you are effectively minimizing the entropy of the system. ¹ Basically, you want to put yourself in the position of gaining as much information about the situation as possible from each guess. So dividing the range equally is the optimal strategy since it allows you to eliminate half the possibilities on any one question, which is the best you can do. Any other splitting may land you with the bigger than half of the possibilities left over. So one tries to optimize the worst case scenario.