I don’t think it makes sense to identify an entropic force with a frictional force coming from a Rayleigh function, because a frictional force is almost always velocity-dependent while an entropic force is often not.

Furthermore, the entropic force

involves a partial derivative with respect to while the frictional force

involves a partial derivative with respect to .

Furthermore, the entropic force is proportional to temperature, , while the frictional force is not.

They seem very different.

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so

and then

Therefore, if we identify the entropic part with the dissipative term related to the Rayleigh function, we have

Does this last equation make sense?

]]>The entropic part is proportional to temperature, so it gets bigger when it’s hot. The energetic part doesn’t change.

Before proceeding, note that my is opposite to your $F$ — when the band pulls is *negative* (pressure is negative here, unlike the gas piston). So, to my point. Actually both parts depend on temperature and they both can change. So from your formula one should carefully find

So both derivations are identical indeed (despite the notion difference ).

But I’d like to emphasize again what really matters — the sign

$\frac{\partial S}{\partial L} < 0 $

For a metal rod or a piston (and I guess for a spring) it is positive. Stretching these systems increases the phase space allowed for the system so the entropy increases. Meanwhile if you heat the systems mentioned they expand. Well, just like we were taught at school "when a substance is heated it expands".

The story is opposite for a rubber band. If it is stretched, its entropy decreases. If it is heated it contracts. So the whole thing was to demonstrate how these two "anomalies" are interconnected.

]]>Hi! I don’t think there’s any ‘trick’ to remembering the sign of work. I agree that it’s an annoying issue. But it just means I need to spend a minute deciding whether I’m talking about *the work the system is doing on the environment* or *the work the environment is doing on the system*, which has the opposite sign.

I find it much more annoying when people tell me set my watch “forward” one hour when Daylight Saving Time starts in the spring. Do they mean to set my watch to an earlier time, or a later time? The word “forward” is confusing. The “forward” of a book is near the front, but as you read “forwards” through the book you move toward the back. Similarly, *ancient history* is the study of the time when everything was a lot *younger* than it is now!

I had to learn category theory to really understand this stuff.

Of course, one can try to choose a convention and stick with it. President Kennedy famously said “ask not what your country can do for you—ask what you can do for your country!” So he preferred to always think about the work the system (you) did on its environment (your country).

Thus if you heat the rubber band it will pull harder, it shrinks. I was just curious whether I could prove it :-)

I think your argument is correct, and it’s nice! My argument would be to use the formula I gave:

The force of a rubber band or stretched spring has an entropic part (the first term) and an energetic part (the second term). The entropic part is proportional to temperature, so it gets bigger when it’s hot. The energetic part doesn’t change.

]]>One reason people don’t like thermodynamics is that they don’t understand partial derivatives…

Well, I do love thermodynamics, but the most difficult thing for me is to decide what is the sign near the work term . And what work is it — done by the system or by the environment. May be there is some trick to remember?

Anyway, I hope what follows will be right. So consider a rubber band of length — let it be the only geometric parameter describing the band. Let be the force that *pulls your hand*when you are stretching the band. So if it pulls, it is positive. Then:

or

Hence :

Thus if you heat the rubber band it will pull harder, it shrinks. I was just curious whether I could prove it :-) Maybe I failed but the fact still holds. One need to know this property of rubber in order to explain the rotating sense of a rubber band heat engine.

]]>Hi! No, I haven’t made them available as PDFs. You can get these series of posts on my website:

I think they look better there than here—just click the box on top to get the jsmath set up and the box will go away.

I not put my posts on quantropy or ‘thermodynamics versus classical mechanics’ onto my website yet, but I will, and I’ll let people know when I do. It takes a bit of work. I’ll probably put them into a single series, because they belong together. (In fact all this stuff fits together into a big story, but that’s going to take a while for me to flesh out!)

I’m writing a paper based on the Network Theory series, and I plan to write a paper on quantropy too. They’ll be more polished than these blog posts…

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