# Broken versus Non-Broken Time Reversal Symmetry: Irreversibility and Response
## 1 Introduction  
Temporal irreversibility is one of the classic questions of physics, particularly because it deals with the most common experience of our daily life, but it looks at odds with our understanding of the properties of the microscopic constituents of matter. Since men first observed movement and mutation in nature, they learned that things generally happen in a preferential direction, like they are carried on by some _active principles_ that follow a rigorous arrow of time. A stone rolling down the hill would never be seen at the top again (unless an external agent takes it back), just like a hot body in contact with a colder one will cool down by transferring heat to the other and not vice versa. Possibly more evident is the psychological arrow of time: we can remember the past but not the future, and physicists of course make no exception to that. Nevertheless, the standard picture of the microscopic dynamics, concerning atoms and molecules of which matter is made, is time reversal invariant. Given a container filled up with a gas of \(N\) interacting molecules, each characterized by position \(q\) and momentum \(p\), this picture assumes that one can invert (at least in principle) all the momenta of these particles keeping their absolute values fixed. The ensuing evolution would then trace back the forward trajectory with reversed momenta. This means that we can consider as equally possible two opposite evolutions, merely resulting from different initial conditions. Therefore, there seems to be no hint of irreversibility at this level of description of the physical world. Around 1865, [PERSON] arguably made the first attempt to characterize irreversibility as a physical phenomenon, introducing the concept of entropy [1], a totally new thermodynamic variable providing a constraint for spontaneous macroscopic processes. Entropy never decreases in an isolated system--it either increases or remains the same--and its growth then constitutes a measure of irreversibility at the macroscopic level. This idea proved to be physically extremely accurate and practically veryuseful, hence [PERSON] tried to frame it within the atomistic description of matter, which he strongly supported. His famous \(H\)-theorem was meant to prove that the second law of thermodynamics is consistent and can actually be derived from the microscopic point of view. In our discussion of time irreversibility, two things seem particularly important in [PERSON]'s argument: first of all, the totally new statistical flavour of the molecular chaos hypothesis used by [PERSON] in deriving his renowned kinetic equation. [PERSON] argued that, for a rarefied gas of particles, the joint probability distribution of positions and momenta can be factorized in the product of single-particle distributions, which means that correlations between interacting particles must decay after some short time due to collisions. This merely statistical idea of loss of memory, being alien to classical mechanics, was hardly acceptable for most of the scientists of that period. Indeed, [PERSON]'s probabilistic approach apparently abandoned the rigorous laws of mechanics, believed to explain all natural phenomena, in order to accept the apparently more vague laws of probability, to account for the fact that we are just limited observers in the world. In addition, the existence of different scales of observation implied by this approach could have (erroneuosly) raised the doubt that a degree of subjectivity was introduced in physics. This was particularly hard to accept at a time in which the very existence of atoms was still debated. The second important aspect of [PERSON]'s reasoning is outlined by [PERSON] [2]: the validity of the kinetic equation is based on the requirement of smoothness of the distribution function at the initial time, and on the assumption that this smoothness persists in time.