Does entropy always involve heat?

[techmologist]

Are there other ways to increase the entropy of the universe than dispersing energy as heat? For example, say I have a jar filled with 100 white marbles, and I shake it up. I have increased the entropy of the universe by dispersing heat energy from a high temperature (burning the fuel in my body) to a low temperature (heating up the marbles a little). Now suppose I have an identical jar with 50 red marbles neatly packed on top of 50 yellow marbles, and I shake the jar the same way I did the other one. Is the increase in entropy the same as with the white marbles? Or is it more because I have mixed up two different color marbles, in addition to heating them up?

 

[torquil]

Are there other ways to increase the entropy of the universe than dispersing energy as heat? For example, say I have a jar filled with 100 white marbles, and I shake it up. I have increased the entropy of the universe by dispersing heat energy from a high temperature (burning the fuel in my body) to a low temperature (heating up the marbles a little). Now suppose I have an identical jar with 50 red marbles neatly packed on top of 50 yellow marbles, and I shake the jar the same way I did the other one. Is the increase in entropy the same as with the white marbles? Or is it more because I have mixed up two different color marbles, in addition to heating them up?

I think that if your marbles are considered classical objects, two yellow marbles are just as distinguishable from each other as a white and a red one. So the two cases should result in the same amount of entropy increase, I think.

But I guess your point could be made using two different species of quantum particles as well. But in that case you could probably not arrange so that the heat production in both cases (glass of one type, glass of both types) is exactly the same during the shaking, since the different properties of the quantum particles will result in different internactions among them.

And if you arrange to obtain the same heat increase in each case, then maybe the entropy increase will be the same also, even though you end up with a different amount of mixing in each case.

Torquil

 

[techmologist]

I think that if your marbles are considered classical objects, two yellow marbles are just as distinguishable from each other as a white and a red one. So the two cases should result in the same amount of entropy increase, I think.

But I guess your point could be made using two different species of quantum particles as well. But in that case you could probably not arrange so that the heat production in both cases (glass of one type, glass of both types) is exactly the same during the shaking, since the different properties of the quantum particles will result in different internactions among them.

And if you arrange to obtain the same heat increase in each case, then maybe the entropy increase will be the same also, even though you end up with a different amount of mixing in each case.

Torquil

I think you are right. I wouldn't think the entropy increase should be so subjective as to depend on whether a human observer can easily distinguish one classical object from the next. In other words, what does the universe care whether a marble is red or yellow or white? Still, I am trying to figure out whether (delta Q)/T is the most general formula for entropy. Maybe there are more general kinds of increase in disorder that have to be accounted for in calculating the change in physical entropy, beyond the dispersal of energy as heat.

 

[torquil]

This doesn't have much to do with this particular problem, but I've lately read Penrose's "The road to reality", and he tries to explain the origin of the law of entropy increase. As a basis for his argument, he uses a "course-graining" of the phase space of the universe, and the volumes of each of these subsets. The choice of course-graning in phase space expresses which states that are considered equal from a thermodynamic standpoint. It was a very instructive read, although I don't agree with everything he says regarding entropy.

It is a book I recommend highly to anyone with at least some university math background. Mostly for its introduction to lots of different subjects within math and physics. I would be surprised if a better "advanced popular science" book exists. It is not to be ignored that it is far more difficult than ordinary popular science books, so it's not for everyone.

Torquil

 

[techmologist]

Ah, The Road to Reality! When that book first came out ~5 yrs ago, I salivated over it at the bookstore. But as I thumbed through it and read little snippets, I realized that I would be biting off more than I could chew to try it now. It would simply take up all my time, and there are so many other things I want to read. Still, I think it would be fun to take a couple years and go through it slowly. It certainly is tantalizing to think, "Here it is, all in one book. Now if I could just get through it, I would understand everything." My guess is that the people who get the most out of that book are the ones who are already familiar with much of what's in it. Perhaps the part on entropy is self-contained enough that I could read it... Congrats on finishing it!

Another example of increase in disorder that is does not exactly involve heat, but is suggestive of it, is the following: Say you had a frictionless billiard table (no pockets) with elastic cushions and balls. 15 balls are initially placed randomly on the table, and you pick one and shoot it in some random direction. Most of the time, after a while, some large fraction of the kinetic energy of that ball will have been dispersed to the other balls. Their energies will follow something like a Maxwell distribution. If you played a video of this process backwards, it would look very strange. Of course, given the initial velocity of the first ball, everything after that is deterministic and perfectly reversible. I don't know if random motion of large (6 oz) billiard balls can be considered thermal motion, with a characteristic temperature, but this macroscopic example still seems to illustrate the arrow of time--that the universe proceeds from less probable to more probable.