Is the second's law of thermodynamics fundamental?

[fluidistic]

Is the second law of thermodynamics fundamental? Like Newton's laws for example.
I'm having doubts about the increase of entropy of any system. Poincaré stated the recurrence theorem which goes against the increase of entropy of a dynamical system with respect to time. By this theorem, a dynamical system will return at its original state after a finite amount of time, even if it's a really really really big amount of time. Entropy says that irreversible processes are one way and that these processes are really irreversible, while the recurrence theorem implies that there's no irreversible process. (Okay, with several assumptions, like a finite size Universe and some others).
I know that Boltzmann and Zermelo took different positions over the subject of the second's law, but I can't find the letters they wrote to each other.

Now I'm asking a question I really would like to know the answer : Does Quantum Mechanics solves the problem of entropy?
For instance I don't think the recurrence theorem takes into account that a particle can disappear (I think QM does, but I'm unsure since I didn't study it yet), or any other "strange" stuff to Classical Mechanics. And by this the recurrence theorem wouldn't be of a good use to apply to the Universe since it doesn't take into account important realities of the Universe.
Does QM offers a proof that the entropy of any system really increase with respect to time?

I know I'm not qualified at all (currently studying introductory thermodynamics at university), but these questions I ask are important to me, so that I can imagine better all what I'm learning. And understand how things really are.
Thanks very much.

 

[diazona]

Well, technically there's no physical law that strictly forbids running an irreversible process in reverse. The second law is really a probabilistic statement - it says that the probability that an irreversible process will run in reverse is small compared to the probability that it will go forward. But for typical thermodynamic processes, the probabilities involved are incomprehensibly close to either 0 or 1 - you might have something like $1/10^{10^{300}}$ for the probability of an irreversible process running backwards. Something with that probability would almost certainly never have occurred in the entire 13.7-billion-year history of the universe.

 

[Mapes]

The Second Law says that entropy tends to increase, not that it must increase. And the exceptions grow less and less unlikely with increasing system size. For example, it wouldn't be unusual for five gas molecules to group momentarily at the corner of a container. But it would be exceptionally unusual for 10²³ molecules to group together. This much can be seen intuitively without moving beyond classical mechanics.

 

[atyy]

Try the subsection on "Irreversibility" in L9 of http://ocw.mit.edu/OcwWeb/Physics/8-333Fall-2007/LectureNotes/index.htm .

 

[fluidistic]

Thanks to you all!
I quote http://ocw.mit.edu/NR/rdonlyres/Physics/8-333Fall-2007/1E2D4D68-EC43-44C7-91A8-9FC8A7698A78/0/lec9.pdf :

Of course, not all microscopic laws of physics are reversible: weak nuclear interactions violate time reversal symmetry, and the collapse of the quantum wave-function in the act of observation is irreversible.

So I can conclude (or almost...) that Poincaré's recurrence theorem is not applicable to the Universe. (I'm now thinking of Proton's decay). My conclusion could be false if the proton's decay produce another proton a finite amount of time after having decayed. But I don't have any idea about this possibility/impossibility. I guess I shouldn't care that much for now and concentrate on my introductory study of Physics.

 

[atyy]

But Kardar also goes on to say "The former interactions in fact do not play any significant role in everyday observations that lead to the second law. The irreversible collapse of the wave-function may itself be an artifact of treating macroscopic observers and microscopic observables distinctly."

 

[Count Iblis]

Poincaré's theorem is also true in quantum mechanics. In fact, in quantum mechanics the proof is almost trivial. You just write down the general form of the wavefunction of the universe (considered as a closed system) in terms of energy eigenfunctions. Then the time dependent coefficients are exp(- i E_n t/hbar). So, every coefficient of an energy eigenfunction moves along a circle in the complex plane at some rate determined by the energy eigenvalue.

It then follows that if the ratio of the eigenvalues of the eigensates with nonzero amplitude are all rational numbers, then you'll return to the initial state in a finite time, if not then you'll approach it arbritrarily closely.

 

[Count Iblis]

Also note that CP violation and the implied violation of time reversal is not the sort of time irreversibility that is relevant to thermodynamics. This is because unitarity is not violated in the CP violating processes, so no information is lost.

 

[fluidistic]

Poincaré's theorem is also true in quantum mechanics. In fact, in quantum mechanics the proof is almost trivial. You just write down the general form of the wavefunction of the universe (considered as a closed system) in terms of energy eigenfunctions. Then the time dependent coefficients are exp(- i E_n t/hbar). So, every coefficient of an energy eigenfunction moves along a circle in the complex plane at some rate determined by the energy eigenvalue.

It then follows that if the ratio of the eigenvalues of the eigensates with nonzero amplitude are all rational numbers, then you'll return to the initial state in a finite time, if not then you'll approach it arbritrarily closely.

Very interesting. I wish I could understand all this well.
So what is the conclusion about the Second's Law of Thermodynamics? Is it a fundamental law in the sense that it is always valid?

 

[atyy]

So what is the conclusion about the Second's Law of Thermodynamics? Is it a fundamental law in the sense that it is always valid?

My understanding is that it is not fundamental. It has to do with the initial conditions of the universe, and the fact that we coarse grain when looking at "thermodynamic" variables.

 

[D H]

Suppose you remove all but a couple of gas molecules from each of two 1 liter vessels. Now open a valve that connects these two vessels, let the system be for a while, close the valve, and count the molecules in each vessel. All of the molecules might well be in the same 1 liter vessel. If they aren't repeat the experiment until you find that they are all in the same vessel. It won't take too long before you succeed. With four molecules, the probability all four will be in the same vessel is 1/8.

If on the other hand each of the vessels initially had half a mole of gas molecules in it, the probability all of the molecules would be in the same vessel after waiting a while is 1 out of $$2^{6*10^{23}}$$. This event is technically possible but effectively never happens.

All of the molecules moving to one of the two 1 liter vessels represents a violation of the second law of thermodynamics. The second law of thermodynamics is essentially a statistical law. This law can be violated; it is just that the odds are against it. On a microscopic scale, the odds that the second law will be violated are not that bad; violations do happen. On a macroscopic scale, the odds that the second law will be violated are incredibly high, so extremely high that violations never happen.

However, let's go back to the original post:

Is the second law of thermodynamics fundamental? Like Newton's laws for example.

Newton's laws are not fundamental, either.

 

[Naty1]

As described above, the second law of thermodynamics (SLOT)is statistical in nature and given that limitation is still one of the most widely regarded foundations in all of modern physics.

I don't know what "fundamental law" means, but I believe SLOT is almost universally regarded by scientists as a valid test for all sorts of new ideas. It IS a cornerstone of modern physics.

I'm having doubts about the increase of entropy of any system.

That would be a mistake, I believe, subject to the the explanations already posted by others. Try wikipipedia at
http://en.wikipedia.org/wiki/Second_law_of_thermodynamics for a decent discussion of the different forms of SLOT.

The MIT notes reflect: "Reconciling the reversibility of laws of physics governing the microscopic domain with the observed irreversibility of macroscopic phenomena is a fundamental problem." which I think sums it up.

Part of the issue might be, and I am no expert here, trying to apply classical "logic" and "reason" to quantum mechanics.

For an interesting discussion on a reversible process I just happened to read one last evening in THE BLACK HOLE WAR (2008) by Leonard Susskind:

He says this: run a photon through a slit... then run the photon in reverse. Does it reappear at it's original location?

He says it WILL return to it's original position if not observed; but will NOT return if observed. "Quantum mechanics, despite its unpredictability, nevertheless respects the conservation of information.

So maybe any apparent paradox with QM might be more clearly defined and resolved via information theory rather than traditional entropy arguments. I've posted about entropy being a subset of information elsewhere.

Susskind goes on to say:

Running the laws of physics backward is perfectly feasible-mathematically speaking. But really doing it? I very much doubt anyone will be able to reverse any but the simplist system...The mathematical reversibility of quantum mechanics (called unitarity) is critical to its consistency...Without it quantum logic would not hold together.

 

[fluidistic]

Suppose you remove all but a couple of gas molecules from each of two 1 liter vessels. Now open a valve that connects these two vessels, let the system be for a while, close the valve, and count the molecules in each vessel. All of the molecules might well be in the same 1 liter vessel. If they aren't repeat the experiment until you find that they are all in the same vessel. It won't take too long before you succeed. With four molecules, the probability all four will be in the same vessel is 1/8.

If on the other hand each of the vessels initially had half a mole of gas molecules in it, the probability all of the molecules would be in the same vessel after waiting a while is 1 out of LaTeX Code: 2^{6*10^{23}} . This event is technically possible but effectively never happens.

I agree but it is not impossible for the half mole molecules to be in one vessel as you pointed out. If you had the required time to wait, you would see that the Second's Law can be violated even in a big system. On the other hand if the gas is hydrogen (or any other gas) and one proton decays, the system will never reach its original state (correct me if I'm wrong) no matter how long one waits.

Newton's laws are not fundamental, either.

I know they assume that the speed of light is infinite and maybe they are not of a perfect accuracy, but I don't think that anything can violate say the Law of Gravitation. I might be wrong of course.

I think I'm now convinced that some processes are really irreversible (the proton decay. And about

He says this: run a photon through a slit... then run the photon in reverse. Does it reappear at it's original location?
He says it WILL return to it's original position if not observed; but will NOT return if observed. "Quantum mechanics, despite its unpredictability, nevertheless respects the conservation of information.

But did he observe the photon passing through the slit at first? If yes, then it would be an irreversible process I believe.) despite the conservation of energy.
I'm now imagining a stone falling off a mountain : it will never lift up to be where it was. So it's a one-way process... but I'm maybe mixing things/concepts up.

 

[Naty1]

But did he observe the photon passing through the slit at first? If yes, then it would be an irreversible process I believe.) despite the conservation of energy.

No observation...reversable process...That is what Susskind says.

 

[RonL]

I'm now imagining a stone falling off a mountain : it will never lift up to be where it was. So it's a one-way process... but I'm maybe mixing things/concepts up.

I can think of several ways the stone can return to the point from which it fell, the least likely, a freak event of nature where a tornado might happen to be at that exact spot and pick it up, and drop it out where it started.

Mixing things/concepts up, is to me one of the most valuable tools we possess, provided we always keep track of what and why things have been mixed.

I do understand the intent of your example
I also think I understand your questions of entropy increase and decrease.

Ron

 

[D H]

I agree but it is not impossible for the half mole molecules to be in one vessel as you pointed out. If you had the required time to wait, you would see that the Second's Law can be violated even in a big system.

Be prepared to wait a *long* time. The mean time between collisions for molecular oxygen at a density of 0.5 mol/liter and temperature of 300 K is 1.29×10^-11 seconds. Suppose the configuration changes once every 10^-11 seconds, and each configuration has a 1 out of 2^6e23 chance of having all of the molecules in one vessel. The amount of time need to have at least a 50/50 chance of seeing all of the molecules congregate in one vessel, even for a brief 10^-11 seconds, is amazingly long: 4×10²³ times the current age of the universe. It effectively never happens.

I know they assume that the speed of light is infinite and maybe they are not of a perfect accuracy, but I don't think that anything can violate say the Law of Gravitation. I might be wrong of course.

I assumed by fundamental you meant "not derivable from some deeper model". You have something else in mind. Furthermore, Newton's law of gravitation is not exact. Einstein developed general relativity for the explicit purpose of finding a better description of gravity.

 

[Mute]

The Fluctuation-Theorem gives a mathematically precise version of the second law. It states that ratio of the probability that the time averaged entropy production will attain some value A to the probablility that it will have value -A is exponential in At. This implies the "second law inequality", that if we have an ensemble of experiments the ensemble-averaged time-averaged entropy production is strictly positive for all time.

These statements pertain to non-equilibrium systems, so I believe the standard second law of thermodynamics follows as a special case.

Note, however, that there is still a problem with time reversibility: because the laws of mechanics are time-reversible, if you look at the system at a given time in some state, the fluctuation theorem predicts that entropy will increase in the future, but also that the entropy of the system was larger in the past.

http://en.wikipedia.org/wiki/Fluctuation_theorem

 

[sokrates]

Very good discussion. I have one question:

Is there any other way to justify the emergence of irreversiblity from reversible laws other than running 10^5 particle simulations?

I mean, could it be proved theoretically (because as far as I know thermodynamics developed independently) ?

It could be exciting to see a theoretical bottom-up view of thermodynamics.

Edit: I think what Mute is saying is relevant to what I ask, unfortunately we posted within the same minute :)

 

[fluidistic]

Be prepared to wait a *long* time. The mean time between collisions for molecular oxygen at a density of 0.5 mol/liter and temperature of 300 K is 1.29×10^-11 seconds. Suppose the configuration changes once every 10^-11 seconds, and each configuration has a 1 out of 2^6e23 chance of having all of the molecules in one vessel. The amount of time need to have at least a 50/50 chance of seeing all of the molecules congregate in one vessel, even for a brief 10^-11 seconds, is amazingly long: 4×10²³ times the current age of the universe. It effectively never happens.

It would happen only if the molecules of the gas doesn't decay or react chemically with the walls of the vessels which seems to me impossible due to the enormous amount of time. So in reality even if you had that much time to wait, you wouldn't even see the original configuration, meaning that the process is irreversible.

I assumed by fundamental you meant "not derivable from some deeper model". You have something else in mind. Furthermore, Newton's law of gravitation is not exact. Einstein developed general relativity for the explicit purpose of finding a better description of gravity.

I have a more precise definition about what I wanted to mean : an intrinsic property of the Universe so that it can never be violated. As far as I know 2 masses m will always attract each other according to a gravitational force (maybe not Newtonian. I don't know in Relativity. It would be : 2 masses m always curves space) and you won't see a case in which it is false, even if you wait $$10^{23}$$ times the age of the current Universe (I know this is so gigantic I can't even imagine).


So for me the SLOT wouldn't be a fundamental law if matter wouldn't evolve within an enormous period of time ( but I don't think it happens). So you would think I consider it as a fundamental law, but I do not : Imagine I had the chance in my lifetime to fall over a macroscopic violation of the SLOT like in the gas example...

The Fluctuation-Theorem gives a mathematically precise version of the second law. It states that ratio of the probability that the time averaged entropy production will attain some value A to the probablility that it will have value -A is exponential in At. This implies the "second law inequality", that if we have an ensemble of experiments the ensemble-averaged time-averaged entropy production is strictly positive for all time.

These statements pertain to non-equilibrium systems, so I believe the standard second law of thermodynamics follows as a special case.

Note, however, that there is still a problem with time reversibility: because the laws of mechanics are time-reversible, if you look at the system at a given time in some state, the fluctuation theorem predicts that entropy will increase in the future, but also that the entropy of the system was larger in the past.

http://en.wikipedia.org/wiki/Fluctuation_theorem

That's really interesting. Thanks for the information.



P.S.: I don't believe in a Big Crunch theory and the following question has certainly be asked before, but does the theory of a Big Crunch implies that the entropy of the Universe decreases to a level as it was in the early Universe?