# Broken versus Non-Broken Time Reversal Symmetry: Irreversibility and Response
## 2 Standard Picture for Irreversible Relaxation to Equilibrium  
We begin illustrating [PERSON]'s approach to irreversible relaxation to equilibrium that has been considered for a gas that expands from an initially compressed state until it fills its container. The main physical ingredients of this approach are: (a) the enormous number of particles that constitute the gas; (b) the fact that these particles interact, so that their velocities are randomized; and (c) the interaction energy is negligible compared to the kinetic energy of the system. The mathematical theory assumes, cf. Reference [3; 4], that the microscopic constituents of the gas are described by Hamiltonian dynamics in a phase space \(\mathcal{M}\) and that the measurement of an observable \(O\) selects a certain subvolume of \(\mathcal{M}\), namely the set of phases that are compatible with the experimentally obtained numerical values. It turns out that volumes corresponding to the equilibrium values of the observables of interest are overwhelmingly larger than volumes corresponding to non-equilibrium values. Therefore, a system evolving spontaneously and without constraints in \(\mathcal{M}\) finds it practically impossible to avoid the equilibrium region or to exit from it. Because the dynamics continue forever, the phase space trajectory may fluctuate outside the equilibrium region, but these are going to be exceedingly rare and quickly vanishing events. Equilibrium microstates are thus called _typical_, in the sense that they constitute almost entirely the phase space, i.e., with the exception of phases of vanishing volume. Because the Liouville measure on \(\mathcal{M}\), \(\mu\) say, is preserved by these conservative dynamics, typicality also means that the exceptions to the most common behaviour constitute a set of zero \(\mu\)-measure [The roles of phase space volumes and of the Liouville measure, however, ought not to be confused [5]. In addition, observe that this notion of typicality is mathematically convenient, but it does not need to be exactly verified by systems of physical interest; in particular, [PERSON]'s ergodic theory implies that a proper description of a macroscopic object may include a small but positive volume of trajectories with different behaviours [6]]. To explain how this can be the case, combinatorial calculus over finite state spaces is often invoked. For instance, consider a system whose states are represented by strings of \(N\) symbols, \(X^{(N)}=(s_{1},s_{2},...,s_{N})\), each of which belongs to a finite \"alphabet\", \(\mathcal{A}=\{\alpha_{1},\alpha_{2},...,\alpha_{L}\}\). One may think of \(N\) as the number of particles, \(s_{i}\) as the state of particle \(i\), e.g., its position and velocity, and \(X^{(N)}\) as one microscopic phase of the system. The empirical distribution of the symbols in a given string may then be viewed as one macrostate:  
\[\
u\left(X^{(N)}\right)=\left\{\
u_{1},\
u_{2},...,\
u_{L}\right\},\]  
where \(\
u_{j}\) is the number of occurences of symbol \(\alpha_{j}\) in \(X^{(N)}\) divided by \(N\). Different microstates may nevertheless yield the same empirical distribution, hence they may belong to the same macrostate \(\
u\). This represents the fact that, depending on how many particles are in certain positions with given velocities, the mass density, the temperature, etc., take certain values rather than other values. However, the same mass density, temperature, etc. belong to different microscopic arrangements. One may then count how many such arrangements have the same \(\
u\). Denote by  
\[M(\
u)=\{X_{1}^{(N),\
u},X_{2}^{(N),\
u},...,X_{n}^{(N),\
u}\}\]  
the set of strings of \(N\) symbols with the same empirical distribution \(\
u\).