The fluctuation-induced, Casimir-like interaction between two parallel rods of length L adsorbed on a fluid membrane is calculated analytically at short separations d L. The rods are modeled as constraints imposed on the membrane curvature along a straight line. This allows to define four types of rods, according to whether the membrane can twist along the rod and/or curve across it. For stiff constraints, all the interaction potentials between the different types of rods are attractive and proportional to L/d. Two of the four types of rods are then equivalent, which yields six universal Casimir amplitudes. Repulsion can occur between different rods for soft constraints. Numerical results obtained for all ranges of d/L show that the attraction potential reaches kBT for d/L 0.2. At separations smaller than dc ≈ L(L/ p) 1/3 , where p is the rod persistence length, two rods with fixed ends will bend toward each other and finally come into contact because of the Casimir interaction.

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 meanfield 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.

We study how universality arises when computing Casimir interactions between arbitrary bodies by discretizing their boundaries into pointlike constraints viewed as pointlike inclusions. Introducing ad hoc cutoff and regularization for the field's correlation function, we find that universality arises when i) the separation δ between the pointlike inclusions is less than the cutoff Λ−1, and ii) the bodies are much larger than the cutoff. A sharp transition from discrete to continuous boundaries occurs at δ = π/Λ in the thermodynamic limit for rods at large separation. We illustrate our findings in two dimensions with rodlike bodies and more complex bodies shaped as moons.