I recently reviewed chapter 52. Symmetry in Physical Laws in The Feynman Lectures on Physics, Vol. 1. In section 52-2. Symmetry in space and time, Feynman says, “Next we mention a very interesting symmetry which is obviously false, i.e., reversibility in time. The physical laws apparently cannot be reversible in time, because, as we know, all obvious phenomena are irreversible on a large scale…” [my italics] The passage in italics is the same reasoning I have followed in reference to deterministic ideal gas simulations, as described in the Loschmidt article above. In my 20+ years of studying kinetic theory and statistical mechanics, the unanimous opinion of the many textbooks, papers and articles I have reviewed has been that the laws of motion are reversible, a conclusion usually taken for granted. As far as I have been able to determine, I have been the lone voice in the wilderness proclaiming the irreversibility of the laws of motion. It is gratifying to discover that Feynman came to the same conclusion in 1962, although it is mystifying why this has gone unnoticed for the last 60 years, given his high profile and that of his Lectures on Physics.
He continues with “So far as we can tell, this irreversibility is due to the very large number of particles involved, and if we could see the individual molecules, we would not be able to discern whether the machinery was working forward or backwards.” However, by attributing irreversibility to “the very large number of particles involved”, he begs the question: At what number of particles does a system switch from reversible to irreversible? However, close observation of deterministic ideal gas simulations reveals that second-law behavior holds even for a handful of molecules, in fact, for any number of molecules ≥ 2.
Feynman continues to parallel my line of thought in sections 52-4. Mirror reflections, and 52-5. Polar and axial vectors. He points out the mirror symmetry of motion of the linear momenta in the direction of the three Cartesian coordinates and the angular momentum in two of the directions of spherical coordinates. He calls the third component of spherical motion the polar vector and illustrates mirror symmetry for this vector in Fig. 52-2, showing that a vector pointed in the northeast direction converts to a vector pointed in the northwest direction when rotated 180 degrees around the y axis. However, this is a clear misapplication of mirror symmetry, since the other mirror reversals are rotated around an axis perpendicular to the vector of motion. Therefore, mirror symmetry applied to the polar vector should rotate around an axis perpendicular the direction of the polar vector, resulting in a vector pointed toward the southwest, not northwest.
If the direction of motion is reversed from the positive direction (away from the origin) the direction will be negative (toward the origin) only until the particle reaches the origin, at which time it will become positive again. Since the final direction of motion is the same as the initial direction of motion, the process is irreversible. This is the source of the asymmetry in the laws of motion and the physical basis of the second law. Furthermore, it is apparent that the second law is not fundamental in and of itself, but an epiphenomenon of the more basic law of inertia.
Revised 10/23/22