Exploring Alternative Definitions and Applications of Entropy

[fluidistic]

After having completed 2 years of a Bachelor's degree, the only definition of entropy I know is $$\Delta S = \int_1^2 \frac{dQ}{T}$$. I realize it's the change of entropy rather than the entropy of a system.

My question is "Is this the only definition of entropy"? I've seen in wikipedia and Fundamentals of Physics (Resnick-Halliday) the definition $$S=k \ln \Omega$$ but I never learned it nor do I understand what it means.

In the case of 2 isolated iron spheres in the Universe that are separated by a distance d. When they get closer and closer to each other, does the entropy of the system (the 2 spheres) increases? With the definition I have from entropy, heat is not involved so the formula is not useful.
Because I realize that the 2 spheres will only get closer and closer and if I could record a film of the motion, I'd realize instantly if the film passes reversely or not.
Assuming that yes, entropy increases in the example... I have another question:

Do you buy that it's IMPOSSIBLE for a sphere not to get closer to the other?

It's different than in the case of having a gas confined into a 1 m^3 cube we want to know whether the film passes reversely or not. Because there is a small probability that say 10^24 particles are confined into 10 cm^3 rather than in 1m^3 which is the volume of the container. Hence looking at the film, although I could almost always be right in telling the direction of the film, I could still be wrong (ok, I realize this won't happen within a very large time, even much greater than the current age of the Universe but I consider this as a possibility-improbability and not as an impossibility).
I wrote the last paragraph to show a distinction between what I consider impossible and what I consider improbable but possible. I would like to know if it is possible but improbable that the 2 spheres reduce their acceleration even for a very short time when they're getting closer to each other. I believe it's impossible and would violate Newton's laws (I'm pretty sure that it would also violate Relativity ones).

Thanks for all.

 

[sweet springs]

Hi.
The definitions of entropy in thermodynamics and of statistical mechanics are different but they give identical value to a system.
Both thermodynamics and statistical mechanics are for systems of great many particles. Both are not applicable in your case of only two bodies involved.　Gravity between the bodies could make them closer.
Regards.

 

[Gerenuk]

My question is "Is this the only definition of entropy"? I've seen in wikipedia and Fundamentals of Physics (Resnick-Halliday) the definition $$S=k \ln \Omega$$ but I never learned it nor do I understand what it means.

The second can be used to derive the first, but no vice versa (to my knowledge).
I posted an outline of the proof in
https://www.physicsforums.com/showthread.php?t=353528&page=2

You can also see
https://www.physicsforums.com/showthread.php?t=365823
first, to understand what the number of possible realizations $\Omega$ means.

You are right, that the first thermodynamics definition is useful for common thermodynamics only.

In the case of 2 isolated iron spheres in the Universe that are separated by a distance d. When they get closer and closer to each other, does the entropy of the system (the 2 spheres) increases? With the definition I have from entropy, heat is not involved so the formula is not useful.

In this example neither the entropy increases, nor is there an arrow of time. The iron sphere might well be together but have an outwards momentum, which would reverse the whole movie.

Entropy arguments only work for macroscopic systems and even then fail to a small extend. See the references given in
https://www.physicsforums.com/showthread.php?t=364225&page=2

And moreover a bunch of particles released in the corner of and empty room will return there after a very very long time. Movies may well be reversed, but it just happens that almost all of the possible movies increase entropy.

It's different than in the case of having a gas confined into a 1 m^3 cube we want to know whether the film passes reversely or not. Because there is a small probability that say 10^24 particles are confined into 10 cm^3 rather than in 1m^3 which is the volume of the container. Hence looking at the film, although I could almost always be right in telling the direction of the film, I could still be wrong (ok, I realize this won't happen within a very large time, even much greater than the current age of the Universe but I consider this as a possibility-improbability and not as an impossibility).

This is correct. For any other process it is even worse in the sense that the state-return-time is much shorter.

 

[fluidistic]

Ok thank you both. It's much clearer in my mind now.