 So far we have only dealt with equilibrium, and things get worse once we turn to non-equilibrium. The main problem is that it is a consequence of the formalism that the Gibbs entropy is a constant! This precludes a characterisation of the approach to equilibrium in terms of increasing Gibbs entropy, which is what one would expect if we were to treat the Gibbs entropy as the SM counterpart of the thermodynamic entropy. The standard way around this problem is to coarse-grain the phase space, and then define the so-called coarse grained Gibbs entropy. Put simply, course-graining the phase space amounts to putting a grid on the phase space and declare that all points within one cell of the grid are indistinguishable. This procedure turns a continuous phase space into a discrete collection of cells, and the state of the system is then specified by saying in which cell the system’s state is. If we define the Gibbs entropy on this grid, it turns out (for purely mathematical reasons) that the entropy is no longer a constant and can actually increase or decrease. If one then assumes that the system is mixing, it follows from the so-called convergence theorem of ergodic theory that the coarse-grained Gibbs entropy approaches a maximum. However, this solution is fraught with controversy, the two main bones of contention being the justification of coarse-graining and the assumption that the system is mixing.