  We consider the effective behaviour of a rate-independent process when it is
placed in contact with a heat bath. The method used to "thermalize" the process
is an interior-point entropic regularization of the Moreau--Yosida incremental
formulation of the unperturbed process. It is shown that the heat bath destroys
the rate independence in a controlled and deterministic way, and that the
effective dynamics are those of a non-linear gradient descent in the original
energetic potential with respect to a different and non-trivial effective
dissipation potential.
