Is the second law of thermodynamics a "no-go" theorem?

Question

As defined here, there are several no-go theorems in theoretical physics. These theorems are statements of impossibility.

The second law of thermodynamics may be stated in several ways, some of which describe the impossibility of certain situations.

The question is: if we view the second law of thermodynamics as a theorem (that is, a proposition that can be either proved to be true or untrue), then is it a no-go theorem?

I understand that the second law of thermodynamics is a physical "law" in the sense that it is axiomatic in thermodynamics (i.e. we don't prove Newton's laws in classical mechanics), however, one can "prove" the second law of thermodynamics from statistical physics considerations. So, if you'd rather not call the second law of thermodynamics a "theorem," then perhaps it is a "no-go law"?

Perhaps I'm missing a key or subtle point here, all input is very much appreciated. It may be just a matter of terminology, but I'm curious either way.

Answer 1

From the perspective of statistical mechanics, the second law is neither an axiom nor a strict no-go theorem. It's a practical no-go theorem in the same sense that getting $10^{100}$ heads when flipping a fair coin $10^{100}$ times will never happen. It's not strictly impossible (in contrast to the strict impossibility of solving $x^3+y^3=z^3$ with positives integers $x,y,z$), but you can rest assured that it will never happen. A more complete answer is given here:

Explain the second principle of thermodynamics without the notion of entropy

Answer 2

The reciprocal status of Thermodynamics and Statistical Mechanics is not a basic law of Physics and there is space for different point of views. However there are a few facts which should be borne in mind.

1. A strict correspondence between Statistical Mechanics results and Thermodynamics emerges only after taking the so-called thermodynamic limit, i.e. extrapolating finite size results to the limit of an infinite system. In this limit, the practical impossibility Chiral Anomaly is referring in his answer becomes a real impossibility (exactly zero probability). However, we have to notice that the proof depends on the specific interaction law.
2. Statistical Mechanics is an almost (see point 4) successful attempt to derive Thermodynamics laws from basic Mechanics and from models of the interaction laws between individual elementary degrees of freedom. However, the basic laws of Thermodynamics depend neither on Statistical Mechanics assumptions, nor on any modeling of interactions.
3. Thermodynamics laws (like the principles of Mechanics or other basic laws in physics) do not have the same role as axioms in mathematical theories. The fundamental difference is in the fact that they embody a huge number of experimental results. So for example, and referring to the specific question, the second principle can be seen as a "no-go" principle, i.e. it encodes in a short sentences (whose exact formulation may vary) all the failed experimental attempts to build a perpetual motion machine of the second kind.
4. There are systems whose average behavior is better described by a finite number of degrees of freedom (no thermodynamic limit). For such systems it is not possible to prove something fully equivalent to the second principle.

In conclusion, taking into account only the previous points 1 and 2, and confining the analysis only to situations where no problem is expected from thermodynamic limit, one could see the 2nd law as a theorem. But keeping separate Thermodynamic laws from Statistical Mechanics allows to use the second law even in cases where the Statistical Mechanics machinery in not fully under control.

Answer 3

This is really just a matter of semantics. The folk meaning of 'no-go' theorem is along the lines of, "given that X is a nice property you would like your models of reality to have, or a nice mathematical simplification you would like to make, it turns out that X is impossible, so don't even try". That's the gist of all the theorems listed here.

So is the second law of thermodynamics a no-go theorem? I suppose that depends on your position. If you're working within pure thermodynamics, it's an axiom; you can barely do anything without it. If you're working within statistical mechanics, the microscopic theory underlying thermodynamics, it's an emergent property. But if you're trying to build a perpetual motion machine, I suppose it could count as a no-go theorem, though we don't usually think of them in such applied contexts.

Answer 4

I was writing this post for a question that was closed as duplicate before I finished, so I put it here because the answers here do not contain my point.

One has to understand what "laws" are in physics theories. Physics theories are mathematical models. Mathematical theories have a large space of parameters and solutions, In order to pick up those solutions that can model physical results one uses the so called "laws" of physics. They are extra axioms that come from observations and measurements and are imposed on the mathematical solutions so that the units refer to measured or observed values. That is how the solutions used by the physics theory are predictive of new values, and the theory can be validated.

Thermodynamics is a physics theory that developed this way, and is rigorous mathematically and predictive . If a thermodynamics law could be derived within thermodynamic variables , then a different axiom should take the place of it in order to have the observed behavior in units and data. ( this is true for pure mathematical theories too, if an axiom becomes a theorem a theorem should become an axiom to keep the consistency).

As the answers discuss, thermodynamics can be seen as an emergent theory from statistical mechanics, but that does not invalidate the law within the theory of thermodynamics.

Answer 5

Consider for concreteness the Kelvin-Planck statement that 'you cannot extract net average work in a closed cycle from a single heat-bath'. This certainly has a flavour of a no-go statement. To call it a theorem we normally demand that it is derived (non-trivially) from some other definitions/axioms. One can indeed derive Kelvin-Planck statement after defining work, heat-bath and closed cycle mathematically (using stochastic thermodynamics). So it seems fair to call it a no-go theorem. We should bear in mind that the domain of validity is very specific, e.g. many systems around us are not heat-baths as defined in stochastic thermodynamics.

Answer 6

There is an argument that can be made that adding the additional but intrinsic to science restriction of limiting physical phenomenon to those that can be empirically attested to renders the obeying of the Second Law of Thermodynamics tautological rather than just "very high likelihood statistically." Relying on the link between entropy and information, the argument basically goes that any process that decreases the entropy of a system has to inherently destroy the correlations within the system's state necessary for the loss of entropy to be determinable afterwards without correlations to outside degrees of freedom. If the entropy decrease of a system could be attested to by another system (like a human outside a box of gas), then the correlations in the combined system of the two can't prove the combined system decreased in entropy, thus endlessly moving the goalpost for a decrease in global entropy to have an empirically distinguishable influence.

(This was originally intended to be an answer to this closed question, which was more precisely asking if the Second Law of Thermodynamics represents just very low likelihood of entropy decrease rather than outright physical impossibility)

Answer 7

From the point of view of Statistical Mechanics, the second law is the consequence of a Statistical observation.

By definition, maximizing the combined entropy of subsystems that make up an isolated system (for instance the Universe) is the same problem as finding the state which has the maximum probability of occurrence.

Most situations-practically all observable systems-display a binomial distribution around the maxima of the probability distribution. And it is the property of the binomial distribution, that the greater the total number over which it is distributed, the smaller the relative spread around the maxima.

This means for a system that has many possible distributions (in the order of Avagadro's number and higher) the distribution of states around the state of maximum probability-and therefore maximum entropy-is almost negligible. So, when we look at macroscopic systems that are isolated its entropy increases is absolute. Because the chance of the entropy decreasing is too small to be feasible.