Is quantum uncertainty principle related to thermodynamics?

Question

Would like to ask a question, but first i would like to say Hello Everybody in a way that plays the system, since some geniouses decided that one should not be able to say hello in a question.

The uncertainty principle in quantum mechanics is well known and considered one of most basic properties of natural reality. The 2nd Law of thermodynamics is also well known and also considered one of the most basic processes of natural reality.

The uncertainty principle uses and is related to Planck's constant. Planck's constant has the dimensions of action and in a statistical mechanics approach, also relates nicely with the partitioning of the phase-space providing the basic measure for the entropy functional (this answer provides a nice outline of this).

Apart from that, there are relatively recent papers which relate the Heisenberg Uncertainty Principle in quantum mechanics directly and intuitively to the 2nd Law of Thermodynamics.

Is this relation correct? And if so can we derive one from the other?

Thank you

PS. One can also check this question, which although not the same, is related in an interesting way.

UPDATE:

anna's answer is accepted since by mentioning the derivation of (part of) the 2nd law from unitary dynamics, answers the question at least in one way. Please consider this as still open so you can add another answer. There are more alternatives (and one of which is my stance, ie thermodynamics -> uncertainty)

Answer 1

That is a very nice question and the article (which I did not know) is interesting as it states that thermodynamics can't be true and the microscopic uncertainty relations false at the same time. Regardless of whether it's true or not, it is an interesting claim as it departs from the traditional wisdom of the full bottom-up approach.

I haven't checked whether the argument they put forward is sound or not but the way they do it is definitely reasonable for different reasons:

• If uncertainty relations have to do with thermodynamics, the link is likely to be at the information level. That is because, on the one hand, uncertainty relations imply that there exist such things as incompatible events which in turn says that the maximum amount of information we can have on a quantum system is a set of eigenvalues of a complete set of commuting observables. This information is seemingly always less than or equal to the classical information we could have on such a system. On the other hand, Landauer has proposed that the minimum amount of energy one has to give to change one bit of information is $k_BT \ln 2$ in order not to violate the second principle of thermodynamics. This limit is very important has it was a first reasonable solution to the Maxwell demon problem.

• At first glance, the authors of the paper seem to propose a thought experiment akin to the Maxwell's demon one but accounting for the quantum properties of the particles. From their reasoning they conclude that one can in principle obtain an infinite energy source (i.e. we would be back to the original problem of the Maxwell's demon) unless the uncertainty relations are true.

I still need to study more thoroughly the paper, but my take on it is that they have showed that one can seemingly violate the Landauer bound (and hence the second principle of thermodynamics) over one cycle if uncertainty relations are not enforced.

It has been shown recently with a classical system that indeed one could violate the original Landauer bound in an erasure procedure of information. However, this does not violate a more elaborate version of the Landauer bound that takes into account the rate of success of the erasure protocol and I wonder if the result from Hanggi et al. could be interpreted in this broader context.

Unfortunately, I have no definite opinion neither about the work you point out not about the actual claim that thermodynamics implies uncertainty relations. But I do think that if it were to be true, then assessing the cost of information erasure would be the right direction to look into and also, in my view, how fluctuation relations behave in the quantum world needs to be better understood to provide more definite arguments on this issue.

Answer 2

You say your self :

The uncertainty principle in quantum mechanics is well known and considered one of most basic properties of natural reality.

In fact quantum mechanics and its postulates and laws are the underlying framework on which any classical theory is built.

The "laws" of classical theories emerge from the underlying quantum mechanical framework. In the paper you quote they claim that :

More precisely, we show that violating the uncertainty relations in quantum mechanics leads to a thermodynamic cycle with positive net work gain, which is very unlikely to exist in nature.

As an experimentalist I am in no position to check whether their conclusion is correct, this is the work of peer review in journals, and it has been accepted in Nature and , I hope, peer reviewed. Well done if it is correct, because it is one more validation of the underlying quantum mechanical framework.

I do not know whether it is related to the statement in the wiki article :

In statistical thermodynamics, the second law is a consequence of unitarity in quantum mechanics

It seems from the references to be connected to the many worlds interpretation , so this new derivation might be a more mainstream connection of the quantum mechanical framework to the second law.