# Is the second law of thermodynamics a “no-go” theorem?

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.

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

• It might fit the context of thermo better to say that after $10^{100}$ flips, you’ll never see an average different from 0.50000000000. – Chemomechanics Dec 25 '18 at 18:00
• @Chemomechanics Yes, I agree. Your version better illustrates why statistical mechanics / thermodynamics work so well. – Chiral Anomaly Dec 25 '18 at 18:28
• So, the second law is more a statement of "improbability" than of "impossibility" ? In which case, one couldn't really view it as a theorem? – N. Steinle Jan 21 at 16:59
• I would say it's a theorem about extreme improbability, as opposed to a theorem about strict impossibility. But the improbability is so extreme that the distinction is not important in practice, at least not when typically-large numbers of molecules are involved. – Chiral Anomaly Jan 22 at 0:58

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 Dan Yand 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.

• Thank you for the answer. Regarding point 3, I wonder if we're getting lost in terminology. For instance, would you consider the "equivalence principle" of GR the same kind of "principle" ?... IN that it encodes failed attempts to find a difference between inertial and gravitational masses? But the EP is used as an axiom in GR, is it not? – N. Steinle Jan 21 at 17:02
• Regarding point 4, that's interesting. May you provide an example or a source where I may read some example systems? – N. Steinle Jan 21 at 17:03
• @N.Steinle About terminology, obviously one can choose different terms, but I think that all 'principles' are used as axioms, but, at variance with axioms of mathematical theories, in addition to the requirement of mutual compatibility, there is a requirement of compatibility with the exprimental evidence. If any experiment would show a difference between inertial and gravitational mass it couldn't be used as axiom in any physical theory. – GiorgioP Jan 21 at 19:00
• @N.Steinle about point 4, I had in mind the case of two-dimensional cristals. On the one hand, Mermin's theorem forbids the existence of a 2D crystal characterized by non vanishing Bragg's peaks . On the other hands very good experimental realizations of 2D crystal are possible (electrons on the surface of liquid He or rare gases adsorbed on graphite). The two things can coexist if one realizes that the vanishing of Bragg's peaks is related to a very slow logarithmic divergence of Debye-Waller factor which in practice becomes important only .. – GiorgioP Jan 21 at 19:11
• for crystals much larger than our Solar System (clearly irrelevant for lab size crystals). By tomorrow I should be able to write a precise reference. – GiorgioP Jan 21 at 19:11

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.