Let us consider a closed system whose degrees of freedom can be divided into two distinguishable sets – call them S and B, e.g., electrons and phonons – and we are interested in the dynamics of only one of the two, say, the electrons. Call the set of degrees of freedom S the “system”. These two sets of degrees of freedom are mutually interacting, but do not exchange particles, namely the number of particles of S is fixed.

Let us also suppose that the other set of degrees of freedom B is so large that we are not interested in its microscopic dynamics, or it is simply impossible to calculate. For both mathematical and physical reasons, by “large” I mean infinitely large. This set of degrees of freedom then acts as an environment for the system S (Sec. 1.2).

Given an initial condition, the dynamics of S + B is reversible, so that it is generated by a group of unitary operators U (t) (Eq. 1.18) on the Hilbert space of S + B, that depends on one parameter: the time t. On the other hand, if we follow only the degrees of freedom of S, by considering the environment B infinite (hence with an infinite Poincaré recurrence time – Sec. 1.2.1), we impose a preferential direction of time because, due to the interaction of S with B, during time evolution some correlations in the system S are “lost” into the degrees of freedom of B without the possibility to recover them (see Sec. 2.8).