  We consider the thermal relaxation of a particle in a piecewise-constant
potential landscape subject to thermal fluctuations in the overdamped limit. We
study the connection between the occurrence of the Mpemba effect, the presence
of metastable states, and phase transitions as a function of the potential. We
find that the Mpemba effect exists even in cases without metastable states. In
the considered physical system, the borders of the areas where the effect
happens correspond to either eigenvector changes of direction or to phase
transitions. Finally, we discuss the topological aspects of the strong Mpemba
effect and propose using topology to search for the Mpemba effect in a physical
system.
