  In this work, we give sharp asymptotic equivalents in the small temperature
regime of the smallest eigenvalues of the generator of some piecewise
deterministic Markov processes (including the ZigZag process and the Bouncy
Particle Sampler process) with refreshment rate $\alpha$ on the one-dimensional
torus T. These asymptotic equivalents are usually called Eyring-Kramers type
formulas in the literature. The case when the refreshment rate $\alpha$
vanishes on T is also considered.
