George wrote:
Thanks Bruno, Bruce, Brent, Liz, John for your responses.
1) Regarding convection currents in a gas column with an adiabatic
temperature profile. There is no convection current even though gas near
the floor is hotter than gas near the ceiling. The reason is that gas
rising in an adiabatic column expands and cools exactly at the same rate
as the adiabatic temperature lapse and therefore the gas is in
equilibrium. All gradients ranging from isothermal to adiabatic cannot
support convection. To get convection one needs a gradient steeper than
the adiabatic gradient. This point is well understood by meteorologists.
It should be made clearer in physics classes.
I think there is a difference between having a column of gas at thermal
equilibrium absent an external field (gravitational or other) and the
stable situation in a gravitational field. I think the stable situation
in the field is such that the temperature is uniform throughout (so
there is no convection as there could be in a transitional state when
some filed is turned on). The thing to remember is that the pressure of
the gas varies with height -- lower levels are at a higher pressure to
support the mass of gas above. Pressure can be increased either by
raising the temperature or by raising the density. I think the stable
situation for the column of gas in a gravity field is for a constant
temperature but a density gradient.
2) Regarding Liz’s comment regarding the Second Law and gravity.
Yes, the Second Law is linked to gravity (and to other forces as well).
See the paper by Erik Verlinde “On the Origin of Gravity and the Laws of
Newton” at http://arxiv.org/abs/1001.0785. The violation that I am
discussing is at the intersection of gravity and QM.
Verlinde's ideas about 'entropic gravity' caused a short-lived stir in
some circles, but the idea didn't really lead anywhere and is now
somewhat out of favour.
3) Regarding why Loschmidt was wrong. Brent is the one who got
closest to the answer.
Loschmidt ignored the fact that the energy of the molecules is
correlated with their vertical direction of movement. For example, those
molecules which are at the top of their trajectories (zero vertical
kinetic energy) must always experience their next collision at a lower
elevation. In general the smaller the kinetic energy of a molecule, the
more likely it is to experience its next collision at a lower elevation.
Gravity operates as an energy separator shifting upward molecules with
higher total energy. This effect exactly counterbalances the effect
Loschmidt was relying on that (i.e., molecules get cooler as they rise
against gravity). The gas column remains isothermal.
This analysis seems a little one-dimensional to me. Although the
potential is in one dimension, at any particular height in the column
collisions in the transverse directions are going to cause rapid
thermalization at that level. In other words, the increase in vertical
momentum for a falling molecule is rapidly distributed over the
transverse directions as well. In the transitional phase, molecules are
going to sink on average because the pressure must be higher at lower
levels. The steady state is, as I have claimed, a density gradient at a
uniform temperature throughout.
Bruce
A more formal approach is to utilize molecular distributions. For a
Maxwell gas subjected to an energy gradient the energy distribution is
If the molecules are subjected to a potential energy , a Boltzmann
factor needs to be added and the above equation becomes
The red curve shows the distribution with V = 0 (ground) and the blue
curve with V>0. Notice the lowering in the density.
The renormalized equation is given by
(notice that renormalization eliminated the V term because it is
expressed in an exponent and can be seen as a constant factor )
The resulting curve at elevation > ground is identical to the original
**
This shows that Loschmidt was wrong. A column of gas following Maxwell’s
distribution cannot spontaneously develop a temperature gradient. It
remains isothermal.
Best
George Levy
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