 But Boltzmann knew that simply postulating \(p(M_j) = c\mu(M_j)\) would not solve the problem unless the postulate could be justified in terms of the dynamics of the system. This is where ergodicity enters the scene. As we have seen above, ergodic systems have the property of spending a fraction of time in each part of the phase space that is proportional to its size (with respect to \(\mu)\). As we have also seen, the equilibrium state is the largest macrostate. In fact, the equilibrium state is much larger than the other states. So if we assume that the system is ergodic, then it is in equilibrium most of the time! It is then natural to interpret \(p(M_j)\) as a time average: \(p(M_j)\) is the fraction of time that the system spends in state \(M_j\) over the course of time. We now have the main elements of Boltzmann’s framework in front of us: (a) partition the phase space of the system in macrostates and show that the equilibrium state is by far the largest state; (b) adopt a time average interpretation of probability; and (c) assume that the system in question is ergodic. It then follows that the system is most likely to be found in equilibrium, which justifies (a probabilistic version of) the Second law.