Ideal 2D gas, no inter-particle collisions (Maxwell original limit). The demon measures each particle approaching the door and opens it only for fast particles going left to right or slow ones going right to left.
T = / k_B (2D: = k_B T)
S_side = N k_B [ln V + ln T] (c_v = k_B per particle)
dS_gas = S_now - S_start (negative: gas gets more ordered)
Landauer: erasing 1 bit costs >= k_B T ln 2 of heat -> +k_B ln 2 entropy
net = dS_gas + bits ln 2 >= 0
Entropy is in units of k_B. With erasure counted, the second law is restored.