Is the 2nd law by definition true?

[pivoxa15]

The 2nd law states that "Any large system in equilibrium will be found in the macrostate with the greatest entropy (aside from fluctuations too small to be measured)."

This is saying any system will tend (over time) to exist in the state with the greatest multiplicity or entropy. In other words it will tend towards states that have increasingly greater probabiliy. Is this not by definition true? Because probabilities are calculated with how likely an event will occur. The highest probability => most likely scenario if given the opportunity for long enough. So by itself the system will always tend toward states that are more probable by definition. This would mean that the 2nd law is not based on empirical observation hence will be true in any universe or system.

 

[christianjb]

No- it relies on the fact that states of equal energy are equally probable- which is not necessarily true in all possible universes.

It is not easy to see if the second law holds in Conway's game of life- for which there is no conserved energy-like quantity for different states.

 

[lpfr]

Physics laws are not True or False (with a big F and T). Physics laws are just hypothesis accepted as true as long as they describe reality within acceptable errors. The question in the poll is inadequate.

 

[russ_watters]

This is saying any system will tend (over time) to exist in the state with the greatest multiplicity or entropy. In other words it will tend towards states that have increasingly greater probabiliy. Is this not by definition true? Because probabilities are calculated with how likely an event will occur.

Why is the state of higher entropy the more likely?

It seems you think that entropy is defined as the most likely state. It isn't. It is defined as the most disordered state. It just so happens that the most disordered is the most likely - which is what the 2nd law states. But if there were a universe where order were more likely than disorder, then the 2nd law would work backwards.

 

[vanesch]

The second law of thermodynamics has something tautological to it, in the same way as "survival of the fittest" has something tautological to it. That doesn't mean it is trivial though.

In classical thermodynamics, the existence of the entropy function is absolutely not evident. In fact, the second law is used to demonstrate that such a function can exist.
However, the statistical mechanics version of it is more tautological. There, entropy is defined as the (logarithm of) the number of microstates that correspond to a given macrostate, and then the second law becomes rather "trivial": we always go from a macrostate with few microstates to a macrostate with many, we never go in the other direction. That's (almost) trivial because only a few special microstates of all the possible ones in the "big" box would be able to evolve into one of those few microstates that correspond to the small box. So if the micro-evolution law ignores everything about our "cutting up" the space of microstates into boxes corresponding to "macrostates", then such a "random" evolution would not be able to "aim back".

However, the non-trivial part is then the following:
1) how come that we have well-defined macrostates which correspond to such vastly different-sized boxes of microstates ?
2) how come we are not already in the biggest box ?

These are highly non-trivial questions. It is only when 1 and 2 are given, that the second law in its statistical version becomes "trivial and tautological". Indeed, in just any computer experiment where we have a state space sliced up in chunks of vastly different size, such that the (deterministic or stochastic) dynamics is unrelated to the choice of the chunks, and where we start in a small chunk, we will inevitably have that the state of our system will evolve from small boxes to larger boxes. THAT part is trivial. What is highly non-trivial is why things are set up that way in the first place.

To 1), a part of the answer can be that macrostates are defined by observables which only depend on low-order correlations. So the distributions of the low-order correlations can define vastly differently-sized boxes, which have nothing to do with the micro-evolution which is determined by the high-order correlations.
To 2), the answer is less clear, and comes down to the question of the "initial state of the universe" in one way or another. It might be answered by something like the anthropic principle (which some people do not consider as an answer).

Nevertheless, 1) and 2) are highly non-trivial, but once given, then the "second law" becomes trivial.

 

[marcusl]

However, the statistical mechanics version of it is more tautological. There, entropy is defined as the (logarithm of) the number of microstates that correspond to a given macrostate, and then the second law becomes rather "trivial": we always go from a macrostate with few microstates to a macrostate with many, we never go in the other direction. That's (almost) trivial because only a few special microstates of all the possible ones in the "big" box would be able to evolve into one of those few microstates that correspond to the small box. So if the micro-evolution law ignores everything about our "cutting up" the space of microstates into boxes corresponding to "macrostates", then such a "random" evolution would not be able to "aim back".

However, the non-trivial part is then the following:
1) how come that we have well-defined macrostates which correspond to such vastly different-sized boxes of microstates ?
2) how come we are not already in the biggest box ?

These are highly non-trivial questions. It is only when 1 and 2 are given, that the second law in its statistical version becomes "trivial and tautological". Indeed, in just any computer experiment where we have a state space sliced up in chunks of vastly different size, such that the (deterministic or stochastic) dynamics is unrelated to the choice of the chunks, and where we start in a small chunk, we will inevitably have that the state of our system will evolve from small boxes to larger boxes. THAT part is trivial. What is highly non-trivial is why things are set up that way in the first place.

To 1), a part of the answer can be that macrostates are defined by observables which only depend on low-order correlations. So the distributions of the low-order correlations can define vastly differently-sized boxes, which have nothing to do with the micro-evolution which is determined by the high-order correlations.
To 2), the answer is less clear, and comes down to the question of the "initial state of the universe" in one way or another. It might be answered by something like the anthropic principle (which some people do not consider as an answer).

Nevertheless, 1) and 2) are highly non-trivial, but once given, then the "second law" becomes trivial.

In practical cases, the answer to 2) is that typically we put a system into a small box or bit of phase space and then ask how it will evolve. When you open a bottle of perfume or drop a bit of ink into a jar, you prepare the system in a non-equilibrium state. Stat Mech then answers how the perfume molecules or ink will spread through room or jar until equilibrium is reached.

I think an explanation for 1) is that macrostates are characterized by a very few average values (parameters such as volume or pressure) that obliterate the innumerable fine details of the actual system. Since we are ignorant of those details, we shouldn't be surprised if different values of a gross parameter occur with vastly different probabilities.

Finally, the original posted question rather misses the mark because the Second "Law" is not a law like Newton's are, but rather a statement of overwhelming probabilities. Perfume molecules are allowed to diffuse out of the room and back into the bottle--the laws of physics (they are laws) don't prevent it--but the probability of it happening is vanishingly small.

Vanesch talks of the difficulty of a system "aiming back", and this is the key to the second "law". It is put into perspective when we realize how enormous the numbers are when dealing with a statistical mechanical ensemble. E.T. Jaynes points out that a macrostate having entropy just one microcalorie lower than that of the maximum entropy or equilibrium state occupies a region of phase space that is a factor of exp(10^16) smaller. The maximum entropy state is so overwhelmingly more probable (we can't even write the ratio using numbers) that it takes on the force of a law even though it is instead a statement of odds. Maxwell may have realized this distinction when he wrote
"The second law of thermodynamics has as much truth as that, if you poured a glass of water into the ocean, it would not be possible to get the same glass of water back again."

EDIT: So I vote "no" because it's true in fact and by statistical preponderance, not by definition.

 

[marcusl]

No- it relies on the fact that states of equal energy are equally probable- which is not necessarily true in all possible universes.

Good point, pivoxa is using Boltzmann's definition of entropy in terms of multiplicities of equally probable microstates

$$S=-k\log{W} .$$

Gibbs's definition

$$S=-k\sum_i{p_i \log{p_i}}$$

is more general since it deals directly in probabilities.

 

[vanesch]

In practical cases, the answer to 2) is that typically we put a system into a small box or bit of phase space and then ask how it will evolve.

Yes, but my point was: even to consider this as a possibility (namely, even considering that it is possible to put a system in a small box in a very peculiar far-from-equilibrium state), we need to have an entire universe out of equilibrium; otherwise we would not be able to put a small system in such a peculiar state (and we wouldn't be here, either). THIS is what is not understood. It has to do with the "initial conditions" of the universe. Of course, this has something anthropic to it, in the sense that we couldn't be in a universe which is in equilibrium (which is in a state which has by far the highest odds). We are in a very very peculiar universe where we are NOT in such a (initial) state. There is a priori absolutely no reason why this should be so.

Your point about the "few gross parameters" is indeed what I was alluding to by saying that macrostates are defined mostly by using low-order correlation functions.

 

[marcusl]

The alternative--an maximum entropy universe--would be a very drab and dull place indeed!

 

[pivoxa15]

No- it relies on the fact that states of equal energy are equally probable- which is not necessarily true in all possible universes.

It is not easy to see if the second law holds in Conway's game of life- for which there is no conserved energy-like quantity for different states.

I should also have added that the actual calculation of probabilties is empirical and depends on the physical laws so if in a system states with equal energies do have different probabilities of occurring then that is fine but the 2nd law is still true in that over time the state with the higher probability will tend to occur.

Physics laws are not True or False (with a big F and T). Physics laws are just hypothesis accepted as true as long as they describe reality within acceptable errors. The question in the poll is inadequate.

The 2nd law is certainly different to the other laws like Newton's laws or even conservation of energy. They may be violated. The former has already and the latter people have thought about changing like Hisenberg when analying Neutrinos and recently Hawking but both have been proved wrong. Has anybody thought about disproving the 2nd law as I have stated?

 

[vanesch]

Has anybody thought about disproving the 2nd law as I have stated?

This is why I voted "yes" (but...)
The second law, in one form or another, that states that a system will probably evolve from a less probable macrostate to a more probable one, is IMO, not "violable". It can be of no use, it can not be applicable, but it will, IMO, never be proven wrong. At most, in such a potential case, we would have the probabilities all wrong, but not the fact from probably doing something probable.

 

[pivoxa15]

Why is the state of higher entropy the more likely?

It seems you think that entropy is defined as the most likely state. It isn't. It is defined as the most disordered state. It just so happens that the most disordered is the most likely - which is what the 2nd law states. But if there were a universe where order were more likely than disorder, then the 2nd law would work backwards.

This is why I voted "yes" (but...)
The second law, in one form or another, that states that a system will probably evolve from a less probable macrostate to a more probable one, is IMO, not "violable". It can be of no use, it can not be applicable, but it will, IMO, never be proven wrong. At most, in such a potential case, we would have the probabilities all wrong, but not the fact from probably doing something probable.

I haven't looked into the mathematics of the 2nd law specifically so I might have created this thread a little prematurely. Vanesch what do you think about Russ Watter's post?

I like your comparison of the 2nd law with the survival of the fittest hence evolution. Evolution to me was self evident to the point that it seemed to me (incorrectly) to be more philosophical than empirical but a theory I 'like' very much because it can explain so many things and is the foundations of biology. I need to study stat mech more but the 2nd law may turn out to be as 'good' as evolution which at the moment is my 'favouriate' theory in science if I may say so.

 

[vanesch]

I haven't looked into the mathematics of the 2nd law specifically so I might have created this thread a little prematurely. Vanesch what do you think about Russ Watter's post?

I think the answer to that depends heavily on how exactly one defines the quantity "entropy". As I said, macroscopically (as in classical thermodynamics), the existence of a state function called "entropy" is highly non-trivial. If you want to assign it to the meaning of "disorder", then it is absolutely not evident that entropy should increase. It is when one looks at the microscopic definition of entropy (which happens to coincide with the older, macroscopic function!) that things turn out to be more evident. So I think that Russ looks upon entropy as in classical thermodynamics, while I looked upon it in its statistical mechanics definition (where indeed entropy is nothing else but a measure of the probability to be in the macro state at hand, assuming that all allowed-for micro states are "equally probable" - or even, as was pointed out by marcusl, any other reasonable probability distribution of micro states.

 

[Andrew Mason]

Why is the state of higher entropy the more likely?

It seems you think that entropy is defined as the most likely state. It isn't. It is defined as the most disordered state.

I am not sure what you mean here. A thermodynamic system will naturally move toward a state of equilibrium, which results in an overall increase in entropy. In thermodynamics, this means that the quantity: $\int dq/T$ for the system and the surroundings will not be negative. Essentially this means that heat will flow from hot to cold. Whether this results in a more disordered stated depends on how you define disorder. In terms of energy, it means that energy will become more disperse - concentrated energy will spread out.

It just so happens that the most disordered is the most likely - which is what the 2nd law states. But if there were a universe where order were more likely than disorder, then the 2nd law would work backwards.

This is an interesting comment. The second law does not depend on the physical properties of the system, so I am not sure that there could be a "universe" in which entropy would decrease. If the laws of physics that arose out of the "big bang" were completely different than our laws of physics, the big bang would still be an event in which energy was more concentrated before than after.

AM

 

[pivoxa15]

This is an interesting comment. The second law does not depend on the physical properties of the system, so I am not sure that there could be a "universe" in which entropy would decrease. If the laws of physics that arose out of the "big bang" were completely different than our laws of physics, the big bang would still be an event in which energy was more concentrated before than after.

AM

What about the hypothesised big crunch where after a point, the universe will collapse on itself to a singularity? If that were to happen, would that mean the 2nd law is violated and infact reversed? Or is it the case that it's more probable for the universe to 'shrink' as hypothesised by the theory. So the 2nd law still holds true but if and when it happens energy will flow from low concentrated to more concentrated form which will violate the 2nd law?

 

[vanesch]

What about the hypothesised big crunch where after a point, the universe will collapse on itself to a singularity? If that were to happen, would that mean the 2nd law is violated and infact reversed? Or is it the case that it's more probable for the universe to 'shrink' as hypothesised by the theory. So the 2nd law still holds true but if and when it happens energy will flow from low concentrated to more concentrated form which will violate the 2nd law?

Be careful. When gravity plays a role, more lumped matter means Higher entropy. A black hole has higher entropy than the equivalent amount of mass, spread out uniformly. These are very tricky considerations.

 

[Andrew Mason]

What about the hypothesised big crunch where after a point, the universe will collapse on itself to a singularity? If that were to happen, would that mean the 2nd law is violated and infact reversed? Or is it the case that it's more probable for the universe to 'shrink' as hypothesised by the theory. So the 2nd law still holds true but if and when it happens energy will flow from low concentrated to more concentrated form which will violate the 2nd law?

Interesting question.

It is not a violation of the second law for heat to flow from a less concentrated form to a more concentrated form provided energy is added - a refrigerator is a good example. As you know, when you convert matter to energy, you release an enormous amount of energy.

If all of the matter in the universe were to be pulled back into a single point in space-time, it would require a great deal of energy. So, the big crunch could occur, from and energy and entropy point of view, provided it had a smaller mass than exists in the current universe.

AM