A finite universe would have low entropy at the beginning with continuing increases marked by divergence (expansion). As you can see from the link, I prefer the infinite universe model, as it explains the development of ordered systems through convergence as well as their destruction through divergence.

So if the Earth were emitting the same amount of energy as it receives and thus remaining somewhat inert, how would you describe global warming?

Mind-boggling. Here’s another little excersise: given the rate of energy that the earth gets from the sun, how many seconds of solar input would be needed to raise the temperature of the seas/atmosphere by the amount that global warming predicts?

Here’s a hint, Chris: every morning the sun manages to raise the temperature of the earth by several degrees. Even in a global warming scenario, the energy in and out is almost exactly balanced. Or we’d all be either toast or icicles.

In a sense, it can be called a much stronger version of the 2nd law but this relationship is not really universal as far as I know. It depends on the details of the system like the overall scale of the kinetic constants and is usually studied in a limited context such as the Boltzman master equation etc (i.e a Markov process)

But it would be interesting nonetheless..

But there is a more interesting comparison to make — by making the distinction between preparing a low entropy state (what Sean was really talking about) and maintaining a low entropy state.

The 2nd law applies to the former. The latter is quite different and the calculation of entropy production per UNIT TIME using T_{sun} and T_{earth} is really related to the latter.

As Sean notes, the existence of life (say at a steady state) is a low entropy state — lower than thermal equilibrium by some amount Delta S. To prepare such a state, of course, some entropy must have been produced elsewhere.

To maintain a system in a non-equilibrium steady state such as this, it turns out that one needs a constant RATE of entropy production elsewhere. The T_{Sun} and T_{earth} argument really answer this question of rate.

Crucially, the RATE of entropy production to maintain a system in a low entropy state is NOT simply directly related to the low entropy of that system — it depends on the kinetic constants between states of that system in a sense. For example, that required RATE of entropy production can be made as high as one wishes by scaling all kinetic constants without affecting the low entropy state one is able to maintain.

It would be a very interesting and perhaps difficult generalization of Sean’s post here to estimate what RATE of entropy production is needed to maintain life of given entropy on earth and compare that RATE to the Q (1/T_{sun} – 1/T{earth}) quantity..

Just look up definitions of entropy, the math is very simple. The definitions involving information (i.e from the information perspective) are a little easier to read than the physical ones.

For photons, imagine the following: a photon can exist in a fixed number of configurations, or ’states’. If you have more photons, the state space expands. How it expands depends on how you look at things. This is because for every possible state in the first photon, there are a similar number of possibilities for the next photon and so on, so the increase is exponential if you look carefully. If you look through a dusty lens, you don’t see that, you just see more photons (as many as you added). This number – the volume of the state space, is what entropy tries to measure.

The author said ‘20 times’ because that is the coarse grained view that physicists see. If you pack 20 times the particles you get 20 times the density. The fine grained view is that you get 2^20 times the entropy, if each particle can be in 2 states. Happy Mothers Day!

blackbody entropy = (4b/3)T^3 = 3.6 Nk , where N = number of photons

http://www.csupomona.edu/~hsleff/PhotonGasAJP.pdf
equation (8)

Xerxes– Trust me, if they’ve read this far in the book, they’ve learned a lot about coarse-graining!

“It turns out, after a bit of math, that twenty times as many photons directly translates into twenty times the entropy.”

I think I can handle a bit of math- can anyone point me to a reference (online or otherwise) that contains this derivation?