Central role of entropy in thermodynamics

Providing a simple description of entropy as the foundation of thermodynamics is necessary for researchers in explaining physical phenomena in the universe. Here we dive deep into the meaning of entropy starting from the basics and ending with the Newton's second law of motion which may be rooted in a hidden dimension as the foundation of entropy. Toward this end, after coming up with a general relation for the net entropic energy of a process regardless of its irreversibility, we recover the entropy production relation. Using this relation, we show how the inequality in macroscopic states between work and Helmholtz free energy during the non-equilibrium process converts to equality between reversible microscopic works of microscopic degrees of freedom and Helmholtz free energy change of maximum useful work. In the end, we provide a mathematical solution to this equality that may tell us about the foundation of entropy from underlying degrees of freedom that currently remain unknown to us.