>But this does not by itself imply entropy of the system increased.
doesn't matter what way it got there, entropy of the gravitational system oil-water-Earth is lower and the gravitational potential is higher when oil bubbles are inside the water when compared with the same system in the state when all oil is on top of the water. Thus system naturally moves from the former state to the latter, from lower entropy to the higher, from higher gravitational potential to the lower.
Of course, gravitational potential energy of the system is higher when the oil is scattered temporarily in the water, because the water is higher than it would be in equilibrium state. However, I do not see how this has anything to do with the concept of entropy. The non-equilibrium state cannot be easily assigned thermodynamic entropy, because there is no obvious thermodynamic state variable that could describe the state of mixed water and oil. If you know how to introduce such a variable and corresponding entropy, then please explain it.
>However, I do not see how this has anything to do with the concept of entropy.
in short - entropy of the system with lower gravitational potential is higher (https://en.wikipedia.org/wiki/Entropic_gravity). When oil is mixed into water, the thermodynamic entropy "St,mix" of oil/water system is higher than "St,unmixed" while the entropy "Sg,mix" of the gravitational system of oil/water/Earth is lower when mixed than the "Sg,unmixed" of the unmixed case. In case of bad solubility, like oil in water, we have ["St,mix" - "St,unmixed" < "Sg,unmixed" - "Sg,mix"] and thus system on its own will evolve toward unmixed state.
Entropic explanation of gravity force is not an accepted physics theory. Regardless of whether it is right or not, the concept of entropy in common thermodynamics I was referring to above is different from that in entropic gravity. In thermodynamics, in contrast with theories of entropic forces, entropy is assigned only to states of thermodynamic equilibrium that can be achieved by exchanging heat reversibly. State that is not a state of mechanical equilibrium (such as state of accelerating bodies, unmixing water+oil) is not such a state.
the thermodynamics (and specifically the notion of "heat") is just statistical/aggregate description of the interaction between the forces which are subject to the same entropic description like the entropic gravity and obey the same 2nd law. It is not surprising once you see that "minimal action principle" is just the 2nd law ("entropy evolves along the gradient") expressed using the language of Lagrangian in mechanics. Thus all the known forces are subject to it, ie. in addition to gravity electromagnetic forces is subject to the same, and the Schrodinger too. That allows to calculate and use entropy for the whole system along its whole evolution path.
Your statements are incompatible with classical physics developed for centuries and enjoying high level of acceptance. The concept of thermodynamic entropy brings nothing to understanding of mechanics. There is no time in thermodynamic theory, thermodynamic entropy is not a function of time so there is no possibility of deriving equations of motion of mechanics from the laws of thermodynamics. You may need to study mechanics and thermodynamics and their history from several books before you realize that.
well, i have MS in Math with ~3.85 GPA (all A except for ROTC program and English) from one of the best math school in Russia which also included graduate level courses in mechanics, mathematical physics, etc.
What about you?
>there is no possibility of deriving equations of motion of mechanics from the laws of thermodynamics
i didn't say that. It actually works the other way - thermodynamics is derivable as an aggregation from the laws of motion. And thermodynamical entropy is just a facet of the entropy (which in general can be defined using relative volume in the space of states of possible configurations of the system for a given values of generic coordinates)
I have a hard time to understand what you mean by "aggregation" and "facet of the entropy which in general can be defined using relative volume in the space of states of possible configurations of the system for a given values of generic coordinates". Please post a link to some exposition of your theory, I'd like to take a look.
doesn't matter what way it got there, entropy of the gravitational system oil-water-Earth is lower and the gravitational potential is higher when oil bubbles are inside the water when compared with the same system in the state when all oil is on top of the water. Thus system naturally moves from the former state to the latter, from lower entropy to the higher, from higher gravitational potential to the lower.