Thermalization of rate-independent processes by entropic regularization
  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. 
T. J. Sullivan, M. Koslowski, F. Theil, M. Ortiz
math.PR math-ph math.DS math.MP