 One can show that, at least in the case of dilute gases, the Boltzmann entropy coincides with the thermodynamic entropy (in the sense that both have the same functional dependence on the basic state variables), and so it is plausible to say that the equilibrium state is the macro-state for which the Boltzmann entropy is maximal (since thermodynamics posits that entropy be maximal for equilibrium states). By assumption the system starts off in a low entropy state, the initial state \(M_p\) (the gas being squeezed into the left half of the box). The problem of explaining the approach to equilibrium then amounts to answering the question: why does a system originally in \(M_p\) eventually move into \(M_{eq}\) and then stay there? (See Figure 6b.)