  A similar argument was developed from first principles by Penrose (1970, Chapters V and VI, with particular reference to VI.3), who had deduced that the Boltzmann entropy could go down on average, but only during macroscopically indeterministic processes. This reduction was bounded, such that, if a macroscopically indeterministic process started in a determinate macrostate of Boltzmann entropy S0, and produced distinct macrostates of Boltzmann entropy Si, with probability pi,  then S0 − ⟨ Si ⟩ ≤ −k ⟨ ln pi ⟩. He further argued this meant that a process which started with distinct macrostates of Boltzmann entropy Si, each occurring with probability pi, could not end up in a macrostate with Boltzmann entropy Sf with certainty, unless Sf − ⟨ Si ⟩ ≥ −k ⟨ ln pi ⟩  as, otherwise, the two processes in succession could lead to a macroscopically deterministic process with decreasing Boltzmann entropy (Sf &lt;S0).