Heat Death

A box of gas, hot on one side, cold on the other. Stand a tiny demon at the door and let it pass fast molecules one way, slow ones the other. It seems to un-mix the box for free, beating the second law. It cannot. Finding out why took a century, and the answer is about information.

The concept

Left alone, the box mixes: hot and cold blur to lukewarm, and never spontaneously separate. That arrow is the second law of thermodynamics, and it is statistical, not mechanical. There are vastly more lukewarm arrangements than sorted ones, so the system drifts toward the many. Maxwell's demon seems to cheat by sorting without work.

The math

Entropy counts arrangements (Boltzmann):

$S = k_B \ln \Omega$

where $\Omega$ is the number of microstates. The demon does lower the gas's entropy. But to sort, it must measure and remember which molecules are which, and erasing one bit of that memory to reset for the next molecule costs at least (Landauer's bound)

$\Delta E \geq k_B T \ln 2$

dumped as heat into the surroundings. Account for the demon's own information and total entropy rises after all. The bookkeeping closes exactly.

Why it stays strange

The resolution rewired physics: information is physical, erasing it has a thermodynamic price, and the boundary between "knowing" and "heating" is not a metaphor. The demon does not fail because sorting is impossible. It fails because forgetting is not free.

Further reading

• Landauer (1961), "Irreversibility and Heat Generation in the Computing Process."
• Bennett (1982), "The Thermodynamics of Computation."

@article{landauer1961,
  author  = {Landauer, Rolf},
  title   = {Irreversibility and Heat Generation in the Computing Process},
  journal = {IBM Journal of Research and Development},
  volume  = {5},
  number  = {3},
  pages   = {183--191},
  year    = {1961}
}