  We analyze the effect of temperature on the yielding transition of amorphous
solids using different coarse-grained model approaches. On one hand we use an
elasto-plastic model, with temperature introduced in the form of an Arrhenius
activation law over energy barriers. On the other hand, we implement a
Hamiltonian model with a relaxational dynamics, where temperature is introduced
in the form of a Langevin stochastic force. In both cases, temperature
transforms the sharp transition of the athermal case in a smooth crossover. We
show that this thermally smoothed transition follows a simple scaling form that
can be fully explained using a one-particle system driven in a potential under
the combined action of a mechanical and a thermal noise, the
stochastically-driven Prandtl-Tomlinson model. Our work harmonizes the results
of simple models for amorphous solids with the phenomenological $\sim T^{2/3}$
law proposed by Johnson and Samwer [Phys. Rev. Lett. 95, 195501 (2005)] in the
framework of experimental metallic glasses yield observations, and extend it to
a generic case. Finally, our results strengthen the interpretation of the
yielding transition as an effective mean-field phenomenon.
