On the dynamics of the interaction between triggered active inclusions

In a one-dimensional elastic medium with finite correlation length and purely relaxational dynamics, we calculate the time dependence of the elastic force F(t) exchanged between two active inclusions that trigger an elastic deformation at t=0. We consider (i) linear inclusions coupling to the field with a finite force, and (ii) non-linear inclusions imposing a finite deformation. In the non-linear case, the force exhibits a transient maximum much larger than the equilibrium force, diverging as ~L^(-2) at separations L shorter than the field's correlation length. Both the mean-field and the Casimir component of the interaction are calculated. We also discuss the typical appearance time and equilibration time of the force, comparing the linear and the non-linear cases. The existence of a high transient force in the non-linear case should be a generic feature of elastically-mediated interactions.