Lately I found that the entropy and action should be actually the same thing, trust me, I’m final year student your senior…..
Anyhow I will show the proof for the manner of completeness:
- Prove by character .
We know that
Boltzmann Entropy: S = k_B \log{W}
Action: S = \int dt \mathcal{L}
They are both written as S. Think about it mates, while have so much of characters and symbols in Physics, eg. we have the most obvious choices: A for action or E for entropy, why still we choose S to represent both action and entropy huh.
Physicists must not be fool, it must tell something. 
- Prove by Logic
A thermodynamical system is governed by the principle of Maximum entropy.
→ A (trivial) thermodynamical system is consist of particles(trust Boltzmann!).
→Big particle follow Einsteinian dynamics.
→Medium particle follow Newtonian dynamics.
→Small Partical follow {Planck, Bohr, de Brogie, Schrondinger, Heisenberg, Feynman et al.}-ian dynamics
→Each of the governing equation can be derived from least action principle.
Hence, Least action principle = Max Entropy principle,
Thus, Action = - Entropy
- Prove by QED
>[My Prove]
>Q.E.D.
Since the Q.E.D. is written, then it must be correct.
Ok but you may ask, why does it essential, but I will tell you it was so important and so useful.
It comes as follow:
→ Entropy = Information according to E.T Jaynes
→By our result, Action = -Entropy
→ Then Action = - Information.
In words, in order to get action, we need to reduced information! !
Can y’all see how profound this statement is ladies and gentleman? This is at the central of all of our daily life!!
This explain why we always have no action and procrastinating for our project and assignment or even daily manners, we lowkey just have too much information! Hence, our procrastination is not our own fault, it is not we are lack of self-discipline or time management, we just having too much information. It is the fault of this era, blame the TikTok, Douyin, Instagram reels then!
. This is the other topic. But you have good insight mate, yes if you have no action to do something, reduced your information intake, because it reduced the entropy of the system and hence increase the system’s action(but it require energy anyhow)!