# 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.

• The second law of thermodynamics cannot be proved from statistical physics considerations. In fact the second law is incompatible with mechanics. This is the so-named "arrow of time problem" and the main reason why there is no general and fully accepted nonequilbrium statistical mechanics formulation. Commented May 10, 2019 at 5:43

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. Commented Dec 25, 2018 at 18:00
• @Chemomechanics Yes, I agree. Your version better illustrates why statistical mechanics / thermodynamics work so well. Commented Dec 25, 2018 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? Commented Jan 21, 2019 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. Commented Jan 22, 2019 at 0:58
• @ChiralAnomaly Then we have kept it law because we still have hope to disprove the statement? As the law is valid for temperatures greater than 0k but not equal to :) Commented Dec 5, 2020 at 3:11

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.

• 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? Commented Jan 21, 2019 at 17:02
• Regarding point 4, that's interesting. May you provide an example or a source where I may read some example systems? Commented Jan 21, 2019 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. Commented Jan 21, 2019 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 .. Commented Jan 21, 2019 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. Commented Jan 21, 2019 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.

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.

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.

• What is the definition of "heat bath" in stochastic thermodynamics? Commented Jan 15, 2021 at 13:06

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)

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.