# What is the real second law of thermodynamics?

1. Feb 18, 2011

### moonman239

What is the "real" second law of thermodynamics?

Title says it all.

2. Feb 18, 2011

### G01

Re: What is the "real" second law of thermodynamics?

The second law of thermodynamics says that the entropy of a closed system will maximize its entropy.

3. Feb 18, 2011

### Curl

Re: What is the "real" second law of thermodynamics?

Multiplicity increases, or equivalently, information about a system decreases. As much as it can...

4. Feb 18, 2011

### Andrew Mason

Re: What is the "real" second law of thermodynamics?

The easiest form of the second law to understand is the Clausius statement of the second law:

"No process is possible whose sole result is the transfer of heat from a body of lower temperature to a body of higher temperature."​

The Kelvin statement can be shown to be equivalent:

"No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work."​

AM

5. Feb 19, 2011

### RedX

Re: What is the "real" second law of thermodynamics?

I'd have to go with entropy in a closed system increases over time.

However, just to give you some food for thought, is it possible for the entropy of a closed system to decrease?

Say you have an insulated box of gas, but the gas is only occupying half of the box. If you wait awhile the gas will eventually occupy all of the box, and the entropy will have increased.

But maybe if we wait awhile longer, the gas will return to just occupying half the box?

Keep in mind that all the molecules of the gas obey Newton's laws, and in Newton's laws there is no uncertainty: in principle if we have a strong enough computer, we can calculate where all the particles will be in the box exactly.

In fact, there is even a theorem in classical mechanics, called the Poincare Recurrence relation, that states that for anything that obeys Newton's laws, if you wait long enough, your system will return arbitrarily close to the initial state. This is a 100% certainty.

So there is a 100% certainty that the gas will be in only half the box again at some really long time.

http://en.wikipedia.org/wiki/Poincare_recurrence#Recurrence_theorem_and_entropy

6. Feb 19, 2011

### Fra

Re: What is the "real" second law of thermodynamics?

This is why I personally think the second law is best phrased not as a forcing law, but as an expectation and rational constraint on decisions.

Given no further information besides the entropy and the macroscopic constraints, the entropy of a system is by construction more likely to increase than to decrease during any change.

This certainly does not mean that it WILL or MUST increase. Strictly speaking I'd say that form of the second law is not quite correct. That fact that in the FAPP sense, it is true is just in the sense that there is a saying that "events with sufficiently low probability, simply never happens".

So the second law is one of the rare few things in physics which is clear. The only not to trivial things about probabilit theory and entropic reasoning, is the role of the CHOICE of the microstructure (and thus prior). Because it's no denial if you look close enough that the measure of disorder (entropy) is actually relative in the sense that it depends on an ergodic hypothesis. And genereally, at first analysis level, there is no prevention from different observers making DIFFERENT ergodic hypothesis, yielding them inequivalence entropic expectations. This latter thing is the only thing that is not so trivial in there. Essential this means that each observe has their own distored direction suggested by second law.

But in CLASSICAL nonrelativistiv physics, this latter issue is not there, and entropy is objective. But both in relativity, and in observer dependent theories, these things is far more intricate. In QM one can define different entropy mesures, like the von neumann entropy. But this is in fact NOT the same measure, though both are called entropy. Here are again lots more food for thought too.

/Fredrik

7. Feb 19, 2011

### Staff: Mentor

Re: What is the "real" second law of thermodynamics?

Yes, where "awhile" means much much [...] much much longer than the age of the universe. As I recall, estimating this time is an exercise in the book that I used the last time I taught thermodynamics.

This reflects the difference between classical thermodynamics and statistical thermodynamics, in which the Second Law is a statistical statement, not an absolute one.

8. Feb 19, 2011

### RedX

Re: What is the "real" second law of thermodynamics?

All this is a little over my head. I always found it amazing that if you set up a system of differential equations together with boundary conditions, that if you let time go to infinity, you'll get a Boltzmann distribution as your asymptotic behavior, no matter what your differential equations are. But I've also seen the proof of the Poincare Relation before, so I'm a little confused since the Poincare Relation seems to disprove that asymptotic solutions of a system of differential equations even exists.

Now of course a Boltzmann distribution is the same as equal distribution for states with the same energy (i.e. canonical and microcanonical are equivalent). This is the result that's derived in every class, that the fluctuations in energy of the Boltzmann distribution goes as the specific heat which is negligible.

So are there asymptotic solutions to (phase-volume preserving) differential equations with boundary conditions, and is the asymptotic solution that all states with the same energy are equally probable?

As for adding relativity to thermodynamics, this is really confusing to me. You don't expect the partition function to be relativistically invariant. However, the path integral is relativistically invariant, and the partition function is merely the path integral (using imaginary time), except you sum over all possible states for the endpoints. If the partition function is Lorentz invariant, then so should be the entropy, since temperature is a scalar (partition function invariant -> helmholtz energy invariant -> entropy invariant).

As always I suspect that its because the boundary conditions change when converting the partition function to the path integral, and this is the cause of the breaking of Lorentz invariance. But I thought Lorentz invariance can be restored by introducing as a 4-vector the velocity of the heat bath.

Anyways, in actual calculation of entropy, you do sum over all microstates, even microstates of the box that are only half-occupied. It changes the final result by so little it can be ignored.

9. Feb 19, 2011

### Andy Resnick

Re: What is the "real" second law of thermodynamics?

Going along what Andrew Mason said, the second law is a fundamental statement about what kinds of physical processes can occur in nature. It does this by defining the entropy of a process. Most other statements regarding the second law involve different ways of defining the entropy.

10. Feb 19, 2011

### Andy Resnick

Re: What is the "real" second law of thermodynamics?

According to the fluctuation-dissipation theorem, it regularly does:

http://prl.aps.org/abstract/PRL/v89/i5/e050601

11. Feb 20, 2011

### SW VandeCarr

Re: What is the "real" second law of thermodynamics?

If you took Avogadro's number($$N_{A}$$) of fair coins and tossed them all at once, the overwhelming probability is that the results would be proportionately very close to 0.5 heads (the point of maximal entropy). However, there exists a finite probability that they all could be heads, specifically $$(1/2)^{N_{A}}$$. With much smaller numbers (or scales), the probability of larger deviations from the expected values (maximal entropy) increases.

Last edited: Feb 20, 2011
12. Feb 20, 2011

### Andrew Mason

Re: What is the "real" second law of thermodynamics?

You can say that there is a finite probability. But the chances are still infinitessimally small that it has ever occurred anywhere in the history of the universe or that it ever will. That is about the probability in quantum mechanics that an apple will not obey Newton's law of gravity. So we call it a law.

AM

Last edited: Feb 20, 2011
13. Feb 20, 2011

### Pythagorean

Re: What is the "real" second law of thermodynamics?

One of my physics professor conveyed a definition of thermodynamics to me that has stuck. I believe it was originally presented by Susskind (who has lectures available online).

The second law is only really justifiable theoretically with quantum mechanics (the classical mechanics description is limited by the Planck constant and is more of an experimental fact).

But let's start with the classical view, using phase space (a plot of the position vs. the momentum of a particle or set of particles). You pick a point and that represents a particle and you trace it through phase space. Since they're deterministic equations in the classical view, you can trace them back to their origin with no problem, even chaotic systems.

Now, if we consider quantum mechanics, we suddenly have an issue when we trace the particle back to it's origin on the phase plot. Namely, that it could have come from any arbitrary point within a circle the size of planck's constant (which is an area on the phaseplot).

That is, due to indistinguishability and Heisenburg uncertainty, we have an inherent loss of information in the universe about the state of the particles whose motion (characterized by position and momentum) is directly related to energy and this loss of information is entropy.

14. Feb 20, 2011

### Andrew Mason

Re: What is the "real" second law of thermodynamics?

I don't think that entropy is really that esoteric. It does not depend at all on quantum effects. When I break 15 balls on a pool table (one that has no friction, has perfectly elastic collisions between balls and cushions and no pockets), the energy of the cue ball will disperse into the other 15 and no matter how long I wait, the motion of the cue ball prior to impact will never be recovered.

Energy tends to disperse from more concentrated forms to less concentrated forms. That is the principle behind the second law of thermodynamics.

AM

15. Feb 20, 2011

### Andy Resnick

Re: What is the "real" second law of thermodynamics?

I'm not sure how that pertains to the reference: a *small* system, with *short* time scales.

16. Feb 20, 2011

### Pythagorean

Re: What is the "real" second law of thermodynamics?

The point, I think, is how you qualified "tendency". Classically, thermodynamics must be defined this way (statistically) because the equations are deterministic. We observe entropy classically, but there's no way to predict it theoreically without QM.

Susskind gives the classical description in his entropy lecture (available online) in which he uses chaos (fractalization of phase-space) to recover this inconsistancy between centropy and determinism by partitioning the initial distribution. But this isn't valid if your partitions have an area less than plancks constant.

it doesn't appear so esoteric to me, but I admittedly don't know the quantum formalism of entropy and have taken my interpretation of my professors words on good faith. It makes sense to me qualitatively, that HUP would contribute to entropy.

Perhaps you've heard the basis of the arguments before and already rejected them. I'm not sure, I know there's been several discussion motivated from information theory before here on physicsforums:

http://lcni.uoregon.edu/~mark/Stat_mech/thermodynamic_entropy_and_information.html

Last edited: Feb 20, 2011
17. Feb 21, 2011

### Andrew Mason

Re: What is the "real" second law of thermodynamics?

I am not sure what you mean by predicting entropy. We simply predict that the entropy of a closed system will always increase. A closed system tends to equilibrium. That is what is observed. That is the second law.

I think it would be better if Susskind started at the beginning of the history of the concept of entropy. Instead he starts at the end.

I don't see an inconsistency between determinism and entropy at all. Entropy is a macroscopic concept that has a statistical explanation. The cue ball does not regain all its lost energy due to some quantum concept. It does not regain it because there are an infinite number of states that the balls can have and only one will result in the balls regaining their initial state.

At the quantum level, there is a different kind of statistics operating. Entropy at that level is conceptually different and perhaps should not be called entropy.

AM

18. Feb 21, 2011

### Andy Resnick

Re: What is the "real" second law of thermodynamics?

I'm not sure I agree with this- calorimeters directly measure changes in the enthalpy and/or the Gibbs free energy, these are related to changes in the entropy: ΔG = ΔH – TΔS.

http://www.microcal.com/technology/dsc.asp

19. Feb 21, 2011

### Pythagorean

Re: What is the "real" second law of thermodynamics?

What I mean, and now it's a question set, since I'm not sure anymore is:

can you predict the second law without the observation (from other "first principles") in classical mechanics? What about QM?

But now that you've said they're completely different things, then the idea that QM explains the mechanisms behind classical entropy is faulty? What's the classical mechanism, then? I've only seen it defined by observation (what systems tend to do)?

How do you mean? I'd be interested in a different pedagogical approach. I wasn't interested in thermodynamics when I went through my undergrad, i just did what I had to for good marks, and now I'm going back and watching lectures online with a renewed interest.

Perhaps it was purely pedagogical and there was is only an intuition block between determinism and entropy (did you watch the lecture, by chance?). He discusses it from a phase space diagram (and it's for systems in which the energy is conserved, so the area of phase space stays constant, but still spreads out, becoming more porous)

Newtonian determinism (I thought) says the process is reversible, so there's nothing that disallows all the particles from condensing back to a volume in phase-space and if you wait long enough, they just might.

But if you fractilize the phas-space as fine as you can, you actually measure a volume difference after the system has reached equilibrium (a volume difference on the phase map, meaning an energy gain/loss, which violates conservation). But you can fix this problem by, as I said, partitioning the initial volume (when it's in a low entropy state) so that none of the fractalized branches have volumes smaller than the planck area (phasemap area).

Was that more coherent? If there's anything incorrect about what I've said here (in the last paragraph, the previous was more the interpretation) please let me know.

Well, I'm not sure either anymore. I hear different things about entropy from every professional I ask.

But there's a problem with the website's language that you quoted. It says "inferred". Could it be argued that your counterpoint is an "inference"? Possibly, but then so can the whole empirical method, so it's kind of difficult to understand what is meant without a rigorous definition of inference.

It's occurred to me that I might be harboring silent assumptions from the field of quantum chaos, too, which studies how chaos can arise from quantum systems to emerge in classical systems. I guess I've had a feeling that HUP is significant when looking at perturbations in a chaotic system, but please kill this belief now if it's faulty.

Last edited: Feb 21, 2011
20. Feb 22, 2011

### Andy Resnick

Re: What is the "real" second law of thermodynamics?

I'm not trying to be dense, but I didn't see the word 'inferred' on the page- can you be a little more specific?

In any case, do you perhaps mean something analogous to "measuring" a spring constant by hanging weights off a spring and measuring the change in length is, in fact, not directly measuring the spring constant?