Sunday, November 06, 2005

What is temperature.....

Reading the several versions of Essex and McKitrick anyone familiar with thermodynamics (heat engines, blackbodies, chemical reactions, etc.) will start to scratch their heads. One peculiar statement after another appears dealing with temperature and other basic stuff. It turns out that Essex is using a rather special definition of temperature for a non-equilibrium radiation field. If you want to read about it look up "How hot is radiation", C. Essex, D.C. Kennedy and R.S. Berry, Am. J. Phys. 71 (2003) 969 . It really is a nice paper for a field known for impenetrability and crypticisms of all sorts.

As is standard in equilibrium thermodynamics, temperature is defined as (dU/dS)p (that should be the partial derivative of the system internal energy with respect to entropy at constant pressure, but hey, Blogger has a lousy equation editor). The energy and entropy of the ensemble are then obtained from the Hermetian density operator and the Hamiltonian of the radiation field.

In the equilibrium limit, the radiation emitted by a black body has the same temperature as the body, and the non-equilibrium temperature collapses to that from a black body. However, as Essex, Kennedy and Berry point out, while their definition is "relatively" simple for a radiation field, it is not so simple to define a non-equilibrium temperature for (radiation field + matter). They say:

"The Gibbs-Duhem relation for radiation, SdT-Vdp = 0 implies that the two intensive thermodynamic parameters, pressure, P (conjugate to volume) and temperature T (conjugate to energy), reduce to one independent intensive parameter, which is usually identified as T. This feature of radiation thermodynamics, like the photon's zero mass and lack of rest frame, makes radiation thermodynamics much simpler than that of matter, which has conserved particle numbers and nonzero chemical potentials. It also makes generalizing intensive thermodynamic parameters out of equilibrium much easier. Thus radiation is a natural context in which to introduce non-equilibrium temperature."

So, pretty clearly Essex is talking about non-equilibrium thermodynamics, and probably playing telephone with McKitrick. BUT, then they are clearly treating the atmosphere as a non-equilibrium system, and that reminds me of the jokes about for all practical purposes:

A mathematician and a physicist agree to a psychological experiment. The (hungry) mathematician is put in a chair in a large empty room and his favorite meal, perfectly prepared, is placed at the other end of the room. The psychologist explains, "You are to remain in your chair. Every minute, I will move your chair to a position halfway between its current location and the meal." The mathematician looks at the psychologist in disgust. "What? I'm not going to go through this. You know I'll never reach the food!" And he gets up and storms out. The psychologist ushers the physicist in. He explains the situation, and the physicist's eyes light up and he starts drooling. The psychologist is a bit confused. "Don't you realize that you'll never reach the food?" The physicist smiles and replies: "Of course! But I'll get close enough for all practical purposes!
You may prefer these versions: Third one down


Anonymous said...

Surely that should be T = (dU/dS)_V, constant volume rather than constant pressure, no ?

I think Essex fails to distinguish between "nonequilibrium" and "far from equilibrium" systems. Of course the atmosphere is not in equilibrium. It is, however, sufficiently near to local equilibrium that one can define a local temperature in a straightforward way. The temperature changes in space and time, no big deal.

The laser, on the other hand, is very far from equilibrium so a completely different approach is required. In general, one doesn't even try to define a temperature for such a system. In order for there to be a temperature, there has to be some regularity in the system that makes it possible to describe the distribution of population over states in terms of a single parameter. You can always do that for a two-level systems (hence such esoterica as the"negative temperatures" used to describe population inversions in NMR and two-level atoms) but for an arbitrary set of states there is not going to be any such thing.


EliRabett said...

You would think so, but they define everything for constant pressure. This is an important ambiguity in their argument. (See the AJP reference Eq. 7 ) OTOH, energy and enthalpy are the same if the change in PV is zero.

I am not sure how they would handle radiation pressure in this scheme, but, again they are explicitly excluding matter and radiation-matter interactions here.

I fully agree with your second point, and it was one that I was trying to make, although you expressed it more clearly.

Anonymous said...

I think you misinterpreted their eq. 7 - the subscript "p" does not stand for pressure, it's some sort of component index (an index that runs over the subsystems that make up their composite system - this is based on a very superficial skim of the paper.)

EliRabett said...