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Are there any analytical proofs for the 2nd law of thermodynamics?

Or is it based entirely on empirical evidence?

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In the presence of gravity there is no proof (now and never will). The 2nd law was stated outside of gravity environment. – Helder Velez Jun 2 '11 at 11:17
never proved, always loved, it is not a fundamental physical law, is actually a computer science law. – Wolphram jonny Jul 13 at 23:31
up vote 17 down vote accepted

It's simple to "roughly prove" the second law in the context of statistical physics. The evolution $A\to B$ of macrostate $A$, containing $\exp(S_A)$ microstates, to macrostate $B$, containing $\exp(S_B)$ microstates, is easily shown by the formula for the probability "summing over final outcomes, averaging over initial states", to be $\exp(S_B-S_A)$ higher than the probability of the inverse process (with velocities reversed). Because $S_B-S_A$ is supposed to be macroscopic, such as $10^{26}$ for a kilogram of matter, the probability in the wrong direction is the exponential of minus this large difference and is zero for all practical purposes.

The more rigorous versions of this proof are always variations of the 1872 proof of the so-called H-theorem by Ludwig Boltzmann:


This proof may be adjusted to particular or general physical systems, both classical ones and quantum ones. Please ignore the invasive comments on the Wikipedia about Loschmidt's paradoxes and similar stuff which is based on a misunderstanding. The H-theorem is a proof that the thermodynamic arrow of time - the direction of time in which the entropy increases - is inevitably aligned with the logical arrow of time - the direction in which one is allowed to make assumptions (the past) in order to evolve or predict other phenomena (in the future).

Every Universe of our type has to have a globally well-defined logical arrow of time: it has to know that the future is being directly evolving (although probabilistically, but with objectively calculable probabilities) from the past. So any universe has to distinguish the future and the past logically, it has to have a logical arrow of time, which is also imprinted to our asymmetric reasoning about the past and the future. Given these qualitative assumptions that are totally vital for the usage of logic in any setup that works with a time coordinate, the H-theorem shows that a particular quantity can't be decreasing, at least not by macroscopic amounts, for a closed system.

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Isn't there an underlying assumption that the microstate probability density doesn't vary or do any fishy business for the derivations in statistical thermodynamics to work. I mean, as you write it is easy to visualize entropy when looking at microstates in configuration space and macrostates, but that moves the "problem" one level deeper: crudely put, the reason it works is because particles are "dumb". Am I opening another box of worms here? :) – BjornW Jun 2 '11 at 9:10
Dear Bjorn, it's an OK can of worms. In the particular proof Boltzmann chose, he assumed molecular chaos, a particular assumption about the lack of correlations in the initial state. There can be correlations but it's still true that it's exponentially unlikely that they will modify the qualitative answer - they would just make the quantitative calculation more ambiguous. Moreover, there may also be an assumption that particles are dumb but this assumption can be proved, cannot it? ;-) Things that are clever, like eggs and fridges, may lower their entropy but they increase $S_{environment}$. – Luboš Motl Jun 2 '11 at 12:36
Yes it can be and is demonstrated but even if it's trivial, it's still an underlying assumption that the reductionalist in me is left with after consuming the other derivations :) After all, it is the whole business with Maxwell's demons in another guise, right.. but I guess that a "clever" particle (for example a particle which starts with but also maintains and changes correlations) would undermine and destroy the derivation in other ways anyway! :) – BjornW Jun 2 '11 at 22:31
@Lubos: Lorschmidt's paradox is not as silly as you make it sound--- if you actually reverse the spins of a relaxing nuclear spin system, it will decrease in apparent entropy with time. This is a physical realization of Lorschmidt's paradox. The absence of entropy decreasing correlations in the motion of molecules is really a separate assumption, which is equivalent to the statement that the entropy is low only in the past, and the state is generic in the future. – Ron Maimon Nov 15 '11 at 20:17
@LubošMotl "Every Universe of our type has to have a globally well-defined logical arrow of time". Is that something you can derive from ST? If so then, yes, I can understand your reasoning - you're saying that the arrow must be one way or the other and experimentally we know which way it is. But does that invalidate the Loschmidt argument? I mean, it certainly tells us why the LP is not a problem, but it seems to me that this is done by bringing other info (your statement) and the experimental fact to the table. LP then says, "we can't decide without further info" and you say "here it is!". – WetSavannaAnimal aka Rod Vance Oct 22 '13 at 10:17

If we assume time evolution is unitary and hence reversible, and the total size of the phase space subject to constraints based upon the total energy and other conserved quantities is finite, then the only conclusion is Poincare recurrences cycling ergodically through the entire phase space. Boltzmann fluctuations to states of lower entropy might occur with exponentially suppressed probabilities, but the entropy would increase both toward its past and future. This is so not the second law as Boltzmann's critics never tire of pointing out.

The H-theorem depends upon the stosszahlansatz assumption that separate events in the past are uncorrelated, but that is statistically exceedingly improbable assuming a uniform probability distribution.

If the total size of the phase space is infinite, Carroll and Chen proposed that in eternal inflation there can be some state with finite entropy with entropy increasing in both time directions.

To me, the most likely scenario is to drop the assumption of unitarity and replace that with time evolution using Krauss operators acting upon the density matrix.

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It was first found empirically, and later dervied from various more theoretical assumptions.

There is a proof in Section 7.2 of Chapter 7: Phenomenological Thermodynamics of Classical and Quantum Mechanics via Lie algebras, based on a few axioms for thermodynamics, and a proof in Chapter 9 that these laws follow from the standard assumptions in statistical mechanics.

The reversibility objections (Loschmidt's paradox) are unjustified since the Poincare recurrence theorem assumes that the system in question is bounded, which is (most likely) not the case for the real universe.

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note this is a similar "proof" statistically (much like Boltzmann's H-theorem) and not a full proof (of course whether this is relevant or not, we will have to see) – Nikos M. Jul 3 '14 at 23:35

I never saw a proof and don't know where to find one. Infinitely stable substances only exist according to a simplified quantum mechanical theory where gravity and nuclear chemistry don't exist. Whether or not there has been a published proof that proves it from that simplified quantum mechanical theory, I don't know. Maybe it has never been proven that the second law of thermodynamics is true. Maybe it has never been proven that no solid has 3 forms at a given temperature and pressure such that it has a tendency to go in circles nucleating one form from another form. If such a solid exists, we might be able to get perpetual motion. Acording to, a weaker statement than the second law of thermodynamics is true, that is, the zeroth law of thermodynamics that two substances that are in thermal equilibrium with a third substance are in thermal equilibrium with each other. That's probably pretty easy to confirm with observations because if 3 substances are the slightest bit off from following that law, we probably could have detected it. Another similar law that two substances that are in solubility equilibrium with a third substance they're immiscible with are in solubility equilibrium with each other if they're immiscible with each other, is probably also true because otherwise, it easily could have been detected that it weren't. On the other hand, it's pretty hard to find a substance with 3 forms with a tendency to nucleate from one form to another in circles even if there is one, so we don't know that such a substance doesn't exist. There does however exist another implication of the second law of thermodynamics that's much easier to test, that the more soluble form of a substance is always the more stable form. From that, it follows that the temperature at which both forms of sulfur are equally soluble in water is exactly the same as the temperature at which they're equally stable. Solubility is determined by surface interactions so I don't see a reason those temperatures should be exactly the same. If those temperaures differ by even a 20th of a degree celsius, it should be possible to set up an experiment to detect the difference, or if the difference is too small to detect, do the math to figure out that there is a difference according to the simplified quantum mechanical theory. Actually, solubility can at any temperature can only be defined exactly for a form of a substance that's the most stable and least soluble. All we have to do is create a closed system regulated in between those temperatures and then put a single crystal of each type of sulfur in the same body of water at that temperatures and the crystal of the less stable form at that temperature will be the one that grows because it's less soluble. The crystal of the less stable form could then be reconverted to the more stable form while maintained at that temperautre, and we might be able to use that to get perpetual motion.

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The problem when you include gravity or other long range forces, is that thermodynamics becomes non extensive. For instance, the energy of the union of two systems is not the sum of the energies of the individual systems.

To handle those cases, generalized entropies have been proposed. By generalized it means that these formalisms allow for long range forces and non-extensivity, for certain parameters of the definition of entropy, but reduces to the classical extensive entropy for certain value of the parameter. One of such extended entropies is TSallis entropy. It depends on a parameter $q$, and for $q=1$ it reduces to the standard classical entropy. It has been shown that this entropy works well in some gravitational systems, where it predicts the correct distribution of temperatures and densities, for instance in a polytropic model of a self-gravitating system. It has also been shown that this entropy satisfies the second law for any parameters $q$ in the classical case, and at least for $q\in(0,2]$ in the quantum case.

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