# 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. – Dan Yand Dec 25 '18 at 18:28

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.

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.