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Jason

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[Mar. 23rd, 2010|08:37 pm]
Jason
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Entropic forces are really weird.

Imagine you have a gadget --- like a polymer, kind of --- that admits as a configuration of itself any fixed-length "walk" to and fro along the real line, taking N steps of length Δx, each one either to the left or the right. Supposing that this walk starts at the origin, the total number of configurations that end up at point n looks like a Gaussian distribution: there's a hell of a lot more walks that end up close to where they started. So the polymer "wants" to be in a scrambled, balled up configuration, more so than it "wants" to be stretched out, because there are many more ways for it to be scrunched up than stretched out.

If you do a little thermodyamics back-of-the-napkin calculation, you find that apparently it does actually behave like a honest-to-goodness Hooke's-law-satisfying spring. Its "spring constant" turns out to be 2kBT/(Δx2N) where kB is Boltzmann's constant and T is the temperature in Kelvin. Or if you forget about the particulars of N and Δx and just think in terms of σ the standard deviation of the Gaussian distribution of possible displacements from one end of the polymer to the other, the spring constant is the beautifully simple kBT/σ2

So Boltzmann's constant is kind of a constant of proportionality between entropic forces and real forces, I guess?
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Comments:
[User Picture]From: physics_dude
2010-03-24 04:12 am (UTC)
What is the basis of your gadget's state changes? Spontaneous bit flipping due to quantum noise? An entropic force would have to be emergent as a statistical property of some underlying mechanism.
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[User Picture]From: jcreed
2010-03-24 12:18 pm (UTC)
I fully admit I don't know how to answer this question very well!

But the "theorem", such as it is, is that any physical thing that admits such and such many configurations (such as a flexible chain of molecules whose actual configurations are determined by molecular bonds) tends to behave as if there were a force trying to restore it to macrostates where there are more microstates.
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[User Picture]From: physics_dude
2010-03-24 07:55 pm (UTC)
Okay, that is sort of how the wikipedia article presented it, which I think is a little deficient. It is a useful rule in many cases but it glosses over the fact that it is purely an emergent phenomenon, and there is no actual "second law" force, for example, that causes buildings to tend toward a more entropic collapsed state. Sure, there are earthquakes and such, but if a building happened to survive the exhaustion of energy input to the earth from the sun and the earth's core, then it is entirely conceivable said building would survive the heat death of the universe.
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From: redglasses
2010-03-24 06:10 pm (UTC)
Random oscillations of the component molecules; when there are enough molecules that we treat the system statistically, we call this temperature.
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[User Picture]From: physics_dude
2010-03-24 08:07 pm (UTC)
Temperature is explainable by deterministic mechanics. In deterministic mechanics there cannot really be a second law, only a second tendency. I think you need randomness to invoke any probability theory "proof" of the second law.

Also, temperature only works in more than one dimension. Are you saying that the oscillations of a polymer in one-dimensional space are spontaneous flips from one orientation to another? No classical force allows this. You must be invoking quantum.
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From: redglasses
2010-03-25 12:20 am (UTC)
Stat mech is what you use when you can't treat a system deterministically because there are too many molecules. Without getting into an argument over what is really 'random', stat mech systems are treated as random because they are too big to do any other way, and the gestalt system is where concepts like entropy and temperature arise.

A polymer in one dimensional space does not actually exist, it's a thought experiment. There isn't any real force that corresponds to the example, but you certainly can take the derivative of the system's energy with respect to its entropy and find a temperature.

What jcreed is describing is an even-more-simplified version of an ideal chain, if you want a more-physical view of the same general idea. The article even has a section on entropic elasticity.
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[User Picture]From: easwaran
2010-03-24 08:06 am (UTC)
What?! Why does temperature matter? I would have thought that if all these arrangements have the same energy then temperature should make no difference to where it wants to be or how long it takes to get there (except that things tend to couple better at higher temperature because there are more collisions when there's lots of translational kinetic energy around).
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From: neelk
2010-03-24 09:21 am (UTC)
At a higher temperature, the polymer will take steps in its random walk faster, which should create a higher apparent force. E.g., imagine you've got a fully stretched-out polymer. At zero temperature, there will be no transitions at all, and so the entropic force will be zero.
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[User Picture]From: platypuslord
2010-03-24 08:25 am (UTC)
I am not sure I am satisfied with your definition of "entropic force". You seem to be basically arguing that an "entropic force" is something which causes things which are likely to happen randomly.

So, for example, I could state: "Most weeks, I eat a chicken sandwich four days out of seven, because of the entropic force."

I would be more comfortable with a definition in which a force is something which can be measured in newtons.
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[User Picture]From: platypuslord
2010-03-24 08:28 am (UTC)
Also, I agree with Dave that you should define your gadget's state changes. Using a naive model in which it just flips bits randomly, I can see that the gadget at point X has velocity proportional to -X. But a spring should have that -X as its acceleration rather than its velocity. If your gadget does not have this feature, then it can't have pendulum motion, which makes me feel that it's not really very spring-like.
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[User Picture]From: jcreed
2010-03-24 12:24 pm (UTC)
Well first of all it is not my definition, it is an existing concept in physics, apparently confirmed by experiment. Frankly, I think I find it almost as weird as you do.

The expression kBT/σ2 does in fact have units of Newtons per meter. Boltzmann's constant is Joules per Kelvin. Temperature is Kelvin so that leavs Joules. σ is in meters, so we get Joules per meter squared, which is the same thing as Newtons per meter. This tells us how much force we feel for each meter we stretch the polymer; it's linear, just like an ideal spring.

This isn't a definition of anything, by the way; it's a heuristic derivation, which makes the not-quite-true assumption that the number of configurations is a Gaussian. Actually it's only a finite binomial distribution, and the approximation breaks down if you try to stretch the chain longer than it can possibly go.
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[User Picture]From: jcreed
2010-03-24 02:49 pm (UTC)
Also maybe this will help:

Arguably for a system to "have a well-defined temperature" is by definition for there to be some tradeoff between energy and state-likelihood. When a system is at temperature T, there are e1/kBT times as many states with one less joule of energy in them.
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From: redglasses
2010-03-24 06:15 pm (UTC)
Temperature is just another kind of energy; (without taking the time to dig out my stat mech book and look up the math) I am pretty sure your polymer is a very funny-looking heat engine.
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