Entropy increase,coarse-grained vs. Landauer's principle justification

[Evaristo]

First off, just clarify that I have a very, very superficial knowledge of Physics, so my apologies if my question is based on an obvious misunderstanding of the basic principles underlying the second law of thermodynamics or if it has a rather simple answer.

The doubt that I have is related to the justification behind the entropy increase, in particular, the fact that I have read two different approaches that in my view appear to arrive at completely different conclusions.

One of them is related to the idea of the complexity required to selectively control the dynamics of the different particles of the system in order to decrease the entropy, i.e. the Maxwell's demon thought experiment, and the fact that this would require to keep record of an increasingly long set of parameters regarding the location of each particle in the phase space and that in the long run bit of information of this long list would have to be deleted in order to make room for new data and that according to the Landauer's principle the bit erasure produces an entropy increase of the total system including the storage space for keeping these parameters. According to this justification, the apparent decrease of the entropy would be only an illusion, because actually there is an overall increase of entropy of the whole system, which must necessarily include the demon interacting and modifying the dynamics of the particles and the storage space used for keeping track of the dynamics of the system.

The second justification is completely different. It is argued that the reversibility of the Laws of Physics and in particular the Liouville's Theorem prevents that the entropy increases at all, but due to the complexity character of the dynamics of most physical systems, each point gets separated according to the Lyapunov exponent in a fractal like manner and it gets progressively more difficult to track the exact position of the different particles in the phase space, and therefore in a real situation it becomes impossible, after a certain time has elapsed, to have accurate enough information of the system as to reverse the process and decrease the entropy. A coarse grained entropy is for this reason defined with a summation over the phase-space of all possible states with a unit size corresponding to the maximum resolution of the measurements and this is the entropy that is increased in all cases, whereas the fine grained entropy is left unvaried due to Liouville's Theorem.

Therefore, according to the first justification the entropy is increased whereas according to the second, at least the fine grained entropy is kept unchanged. Any idea regarding how to make this two justification with this apparent contradiction consistent with each other?

Thank you very much in advance for your help!

 

[atyy]

I can't provide a rigourous argument, but heuristically, entropy increases in both cases when information is erased or lost. In the first case, Landauer argued that if the demon did not erase information, entropy would not increase. In the second case, if one never loses information at the fine grained level, entropy does not decrease. However, the coarse grained description inherently loses information compared to the fine grained case.

 

[Evaristo]

Thanks, Atvy! Your answered has helped me. I understand now that in both cases there is a theoretical manner to achieve a "no increase" in entropy, but that real-life limitations make it impossible to be realized. I did not want anyway a rigorous argument. It is actually my getting lost in the details of the argumentation of the Landauer's principle that made me see this principle in a different light that the coarse grained argumentation, when actually there is a clear analogy between loss of information and erasure. Now I have a better image of the bigger picture, which is what really mattered to me!

 

[atyy]

You're welcome. I'm glad you got the idea, even though there's a terrible typo in my post!

 

[sardar]

I can offer some clarification on these two different justifications for entropy increase. The first explanation, based on Landauer's principle and the Maxwell's demon thought experiment, is focused on the idea that in order to decrease the entropy of a system, one would need to selectively control the dynamics of each individual particle. This would require keeping track of a large amount of information, which would eventually need to be erased, leading to an overall increase in entropy. This explanation is based on the idea that in order to decrease entropy, one would need to go against the natural tendency of systems to move towards disorder and randomness.

On the other hand, the second explanation is based on the concept of complexity and the idea that it becomes increasingly difficult to track the exact position of each particle in a system as time goes on. This leads to a coarse-graining of the system, where the entropy is calculated based on larger units of measurement rather than individual particles. This explanation is based on the idea that even though entropy may appear to decrease on a fine-grained level, it is actually increasing on a larger, coarse-grained level.

Both of these explanations are valid and do not necessarily contradict each other. They are simply different ways of looking at entropy increase. The first explanation focuses on the physical processes involved in decreasing entropy, while the second explanation looks at the mathematical concept of entropy and how it is calculated. Both explanations ultimately lead to the same conclusion - that entropy increases over time in closed systems.

It is important to note that these explanations are based on theoretical models and thought experiments, and may not fully capture the complexity of real-world systems. As a scientist, it is important to continue researching and exploring these concepts in order to gain a better understanding of entropy and its role in our universe.