File:PhysRevX.9.021060 fig10.png

English: Lower bound for the probability of the strong Mpemba effect—the probability of the parity being positive for the direct Mpemba index [see Eq. (15)] for the case of REM with random barriers. The number of energy levels is L=10 and the bath temperatures are kBTb=0.1 (left) and kBTb=1.0 (right). The energies are IIDs from Eq. (59) with variance σ2Elog2L, and the barriers are IIDs from Eq. (60) with variance σ2Blog2L. Each point on the density plot is averaged over 105 realizations. The condensation phase transition is at kBTcritical=σE/√2ln2≈0.84σE. We notice that the probability of having a direct strong Mpemba effect is finite and even high for certain regions of the σEσB− plane for Tb<Tcritical (left panel).