The Markovian Mpemba effect in large many-body systems, and its applications in optimal protocols.*

Finding the optimal heating/cooling protocol is a common task in many applications: from improving the power of a heat engine, through finding the ground state of a complex system, to making fast food . Intuitively, one expects the optimal heating/cooling protocol to include only the extrme temperatures. However, the Mpemba effect suggests that protocols with non-monotonic temperature might be supirior to the naive protocol, and that in some cases optimal protocols might be highly counter-intuitive, e.g. cooling a system can reduce its heating time.
We show that the weak Mpemba effect does not imply that optimal heating/cooling protocols are non-monotonic, but that the strong Mpemba effect does. In addition, we show that in some cases the optimal heating protocol might start with a pre-cooling phase even if there is no Mpemba effect, of any type, in the system.
To demonstrate our results on large many-body system, we developed a projection method, with which one can find optimal protocols or the existing of the strong Mpemba effect through Monte-Carlo simulation, withouth the need to find the slowest component in the Markovina dynamic.