Irreversible dynamics of the phase boundary in U(Ru$_{1-x}$Rh$_x$)$_2$Si$_2$

Specific heat and magneto-caloric effect measuments on the systems: U(Ru$_{1-x}$Rh$_x$)$_2$Si$_2$ and Yb$_{1-x}$Y$_x$In Cu$_4$ were carried out at high magnetic fields. Thermodynamic similarities are found at the phase boundary into the magnetic field-induced phase (phase II) of the former and at the valence transition of the latter. We show that the irreversible properties of the transition into the phase II of U(Ru$_{1-x} $Rh$_x$)$_2$Si$_2$ are very similar to that associated with the valence transition of Yb$_{1-x}$Y$_x$In Cu$_4$. To explain these results, we propose a bootstrap mechanism by which lattice degrees of freedom within phase II become coupled to the 5f-electron hybridization, giving raise to a valence change at the transition.