the arrow of time

LITThe microscopic laws barely care which way time runs, yet gas spreads and never unspreads. Boltzmann tied the arrow to entropy: there are vastly more mixed states than ordered ones. Loschmidt objected that reversing every velocity should un-mix the gas, and it does, in principle. In practice, any tiny error grows chaotically and the reversal fails.

COMPUTEDHard discs start in the left half. The coarse-grained entropy S = −Σ p ln p over a 10×6 grid is computed live. "Reverse" flips every velocity. On an exact integer lattice gas, reversal returns the gas perfectly and S falls back. Add a "perturb" of one part in a million and the return is ruined.

reverseperturb 1e−6

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