>If you mix oil and water, you will see the result does unmix itself again. Does this mean entropy decreased and time direction got reversed? Nah.

Moving an object to higher gravitational potential decreases the entropy of the system (and thus requires work, unsurprisingly).

Mixing oil into water means you displace some volume of water which were closer to Earth with oil bubbles (which are lighter than equivalent volume of water) while you force the displaced water volume higher/further from Earth. Thus by mixing lighter oil with heavier water you decrease the entropy of the gravitational system consisting of the oil, water and Earth. Left to its own devices, the system evolves along the entropy increasing gradient - heavy water goes down displacing lighter oil up, thus decreasing the gravitational potential of the system (the friction during that "unmixing" is the work of the gravitational force that ultimately results in the heat that is lost away into the environment).

When the fluids are mixed by a spoon, surely work has been done by the stirring spoon on the system and energy of the system increased. But this does not by itself imply entropy of the system increased. The reason is the system got to a non-equilibrium state which has no macroscopic thermodynamic description and no unique way to assign entropy and detect its changes.

Entropy always increases over time within a closed system, except in idealized interactions that leave entropy unchanged, which are not practically possible. From what we know, the only thing that might actually be a true closed system is the universe. If you've identified some system that has decreased in entropy, then you've failed to notice something outside of it that has increased in entropy by a greater amount.

Is it possible that our understanding of entropy is flawed? Yes. But unless you have some novel physics experiment or mathematical model to share with us, I'll stick with the second law of thermodynamics.

>But this does not by itself imply entropy of the system increased. The reason is the system got to a non-equilibrium state which has no macroscopic thermodynamic description and no unique way to assign entropy and detect its changes.

Entropy is the tendency to equilibrium. Your earlier statement, that entropy is only a measure of systems that are in equilibrium, is false. If you were right, entropy would be a totally meaningless notion because there are no closed systems and true equilibrium (as opposed to the near-equilibriums that we work with in practical terms) can only exist in one.

My response here is actually overly-simplistic and somewhat wrong. effie is right about classical thermodynamics only defining a measure of entropy (and various other properties of a system) for idealized states of equilibrium. There's been some attempts to define entropy for non-equilibrium systems, but our understanding is incomplete. It's partially a philosophical debate too.

Admittedly, the concept of entropy density has been used somewhat successfully for description of continuum close to equilibrium. But there are problems with extending this to general non-equilibrium case. I think a mix of oil with water is rather awkward to describe thermodynamically, because it is unstable and it is hard to introduce thermodynamic description for a fluid consisting of many drops of oil of different sizes scattered in water. Second law of thermodynamics can be phrased in terms of entropy, but it does not really involve time at all. All it says is after process involving no heat transfer is finished, entropy of final equilibrium state is not lower than entropy of initial equilibrium state.

>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.