  By Liouville's Theorem, if all the microstates (up to a set of measure zero) in an initial macrostate evolve into the same final macrostate, then the Boltzmann entropy of the final macrostate cannot be less than the Boltzmann entropy of the initial macrostate. This is a macroscopically deterministic process. A macroscopically indeterministic process is one in which microstates starting in the same initial macrostate end up in different final macrostates. Penrose (1970, Chapters V, VI) analysed the problem of the change in Boltzmann entropies for these kinds of processes in detail. After attempting a justification of probabilities being proportional to phase space volume, he argued that even the average of the Boltzmann entropy can decrease for macroscopically indeterministic processes. For a system initially in a macrostate with Boltzmann entropy SB, and evolving, with probability pi, into a macrostate i with Boltzmann entropy SBi, it is possible that ∑i piSBi &lt; SB (it is even possible that ∀i SBi &lt; SB). However, Penrose also showed that this decrease is bounded: ∑i pi (SBi  − k ln pi) ≥ SB. He suggested that, when there is a probability distribution over distinct macrostates, a modified, statistical entropy, SP  =  ∑i pi (SBi  − k ln pi), should be used. This statistical entropy is non-decreasing even for macroscopically indeterministic processes.