In a previous post "A new attempt" I sketched a model which could possibly explain the atmospheric lapse rate as the solution of a relatively simple heat equation, which of course in the end must also take into account convection. However, the solution had an arbitrary constant which in normal cases is taken care of through boundary conditions. I am now speculating whether this constant should instead be fixed using a somewhat different reasoning. Let Fa be the total amount of heat absorbed by the atmosphere.
The temperature could then be fixed using an energy argument in combination with the gas law:
Total energy = Fa = 3/2Nk_bTa,
where Ta is some suitably averaged temperature of the atmosphere. A note of caution however! It could be that we should instead put:
Total energy = Fa = 5/2Nk_bTa,
since according to statistical physics the heat capacity of the gas increases in a gravitational field. But this requires further thinking.