Sunday, March 21, 2010

Eureka

Continuing the thoughts from my previous post...

I finally found a quite simple and straightforward explanation as to why it seems to be generally accepted that the entropy of blackbody radiation (such as sunlight) expanding away from its source remains unchanged. If this is true, then the thermodynamic potential of sunlight to do work (on a per joule basis -- the fraction of solar energy that can be converted to electricity or mechanical work) will remain unchanged regardless of the distance from the source. The effective yield of "useable" energy from the sunlight will depend only on the difference in temperature of the source and the temperature of the environment in which the radiation is being put to work. The explanation is straightforward, and wrong. It violates one of the other basic laws of physics (the Uncertainty Principle).

The basis of this argument is that any reversible process cannot involve any change in entropy. This is straightforward enough: if a process results in a change in entropy, then according to the Second Law it can only run one way, the way that causes total entropy to increase. A reversible process is one that can be reversed without the use of any energy or the working of any work. Reversible processes happen all the time at the molecular and subatomic levels; in the macro world where we live they are only ideal theoretical constructs, not things that can actually be built, but they serve as thought experiments to derive principles and calculate values.

Given this, the thought experiment runs as follows:

Take that theoretical blackbody cavity, filled with blackbody radiation at thermodynamic equilibrium, and poke a pinhole in it. Some of the radiation now leaks through this pinhole and radiates out in to space. A real emitter of thermal radiation can be seen as an arbitrarily large collection of these arbitratily small pinholes, so what applies to this pinhole emitter should apply to a real emitter. If you were to place a perfectly spherical, perfectly reflecting mirror perfectly centered around this pinhole, it would reflect the radiation exactly back to the pinhole, reversing the expansion of the light without doing any work or adding any energy. Hence, the expansion of the light away from the source is thermodynamically reversible, and thus the entropy of the light never changes. QED?

No, QE-non-D. Even in this theoretical world of perfect concepts, this thought experiment fails. In order for it to succeed you have to be able to exactly specify the position and momentum of a photon at three different points -- when it leaves the pinhole, when it interacts with the mirror, and when it returns to and reenters the pinhole. Any tiny deviation at any of these points will cause at least some of the photons to miss the pinhole on their return, meaning that the expansion was not actually reversed. However, you cannot simultaneously specify the exact position and momentum of a photon (or in fact anything else) ever, not even once, certainly not three times. Even a theoretical perfect mirror perfectly positioned cannot perfectly focus light. As your arbitrarily small pinhole becomes arbitrarily smaller, more and more of the photons will miss the pinhole. The same applies if the diameter of the perfectly spherical perfectly centered perfect mirror increases.

In another similar thought experiment, you can (somehow) arrange to release your blackbody radiation into the center of the perfectly spherical perfect mirror. In the classical description, the photons will continue to pulse in and out at the speed of light, expanding and contracting perfectly forever, with no change in entropy. But given the realities of uncertainty, with each pulse they will get a bit more spread out, so they will eventually be bouncing randomly around in the perfect sphere, perfectly dispersed. Remember that this is not dependent on imperfections in the mirror. An abslutely perfect mirror will yield this result.

So, the expansion of sunlight out in to the cosmos is not a thermodynamically reversible process (even if the sun were the only object in the universe, which of course it is not). The entropy of sunlight increases with distance, and its capacity to do "useful" work (in the thermodynamic sense) decreases. I don't believe I am equipped to calculate HOW MUCH it has decreased, however, so I can't tell you if it has lost a miniscule and insignificant fraction, or a large portion, of the usable energy it had when it left the sun. But I can say that the statement that it does not change is incorrect and a violation of the fundamental laws of physics.

4 Comments:

At 2:00 PM, Blogger Frank said...

The cat's out of the box now...

Obscure physics joke...

 
At 2:07 PM, Blogger Bill Pulliam said...

Mrs. Schroedinger: "What did you do to the cat? He looks half-dead!"

 
At 8:33 PM, Blogger DIYer said...

That cat's gotta be dead. It's been in that box for, like, 85 years and nobody has ever fed it.

 
At 8:50 PM, Blogger DIYer said...

I'm not real keen on trying to calculate the entropy of cosmoses. Cosmi? It's pretty certain that they are not reversible, though.

But photons have been known to stay the same for billions of years. They lose a bit of energy to red shift, but just cruise along until they hit something. At 8.3 light minutes, the ones from our local star still have a lot of spunk:
http://www.youtube.com/watch?v=8tt7RG3UR4c
(actually I watched the whole show this came from a while back, and these guys make some silly wild claims about what they can accomplish with their mirror, but that little demo was interesting anyhow)

Of course it isn't really worthwhile trying to collect them from any other star. It would probably take a dish mirror the size of Pluto's orbit to do so.

 

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