/Cockroaches in a chessboard

Each time you click on the chessboard a new cockroach is created, and it keeps walking in a randomly chosen direction and some fixed speed from the place where you clicked.

According to Poincaré recurrence theorem, the cockroaches will gather inside one single box (or any other region of the chessboard) after some time, and indeed infinitely often, for almost all initial conditions. This ''recurrence time'' depends on the region and on the number of cockroaches, and you may want to estimate how.

Is it reasonable to expect to see such an event when the number of cockroaches is large, say one hundred? And what would you say if it is as large as the number of molecules in a macroscopic amount of gas?

Let's do some estimates. We divide the chessboard into \( n \simeq 100 \) (slightly more than the usual \( 64 \)) equal boxes. We assume the speed is chosen so that any single cockroach spends approximately \( \tau \simeq 1 \) second in each box, and that therefore any single cockroach passes through each one of the boxes approximately once every \( n \cdot \tau \simeq 100 \) seconds (of course, there are countably many periodic orbits, and some of them do not pass through all boxes, but they form a set of probability zero, which we may safely disregard). It is clear that \( 2 \) cockroaches will meet inside a fixed box once every \( n^2 \cdot \tau \) seconds. In general, the time needed to see all the \( N \) cochroaches in a fixed box is of the order $$ n^N \cdot \tau \simeq 10^{2N} $$ seconds. The age of our Universe is estimated around \( 4 \cdot 10^{17} \) seconds, and it is much smaller (a factor \( 1000 \) less) than the time nedded to see just \( 10\) cockroaches inside a fixed single box! A real gas inside a box of the order of a real chessboard contains some \( N \simeq 6 \cdot 10^{23} \) molecules, much more than 10, and you may now understand (as Boltzmann) what the relevance of Poincarè theorem in real thermodynamical systems!