  The Boltzmann entropy, SB, is not  guaranteed to be non-decreasing. While decreases of  SB are known to be possible through the reversibility and recurrence objections to Boltzmann's  H-theorem, such decreases would be seen as surprising if they occurred in practice. While an individual microstate can evolve from a high volume macrostate to a low volume macrostate, Liouville's Theorem guarantees that only a fraction of the microstates from the larger macrostate can end up in the smaller macrostate under a Hamiltonian evolution. From the logarithmic form of Boltzmann entropy the proportion, by volume, is p ≤ eΔSB/k (with ΔSB the reduction in the  Boltzmann entropy between the two macrostates). If it can be assumed that the probability of the microstate being in a given subregion of the macrostate is proportional to the phase space volume of the subregion, this gives Einstein's fluctuation formula. Although widespread use is made of this assumption, its justification is one of the more significant challenges for the foundations of statistical mechanics.