At the connection of Entropy and H-Theorem some basic assumption are not correct:
– First of all: the positions, x, and the velocities, v, are not intrinsic properties of particles.
– x and v are never known exactly: definite initial conditions, definite assumptions about x(t), v(t) at some time exact t is impossible.
– The infinity (infinite large and infinite small) distances belong not to physical descriptions. The particles cannot be too close to each other.
– Intrinsic properties of particles are two kinds of conserved charges qi and gi,; they are elementary electric and elementary gravitational charges.
– qi and gi generate two fundamental interactions; the interactions are non-conservative in presence of charges. The interactions propagate with c. The interactions between particles can be attractive or repulsive.
– Never exist particles without charges; never exist particles without interactions.
– In small distances the electromagnetic interaction dominates. In large distance the gravity dominates, however, electromagnetism is also always present.
– Elastic scattering of particles can never occur. The particles are always correlated.
– Particles and fields appear always together.
– What is the equilibrium of matter composed of particles and fields? Is the equilibrium the state of neutrinos or the neutron stars? Or somewhat other?
Almost each basic assumption at the foundation of entropy is incorrect.