Boundary Thermalization in Superdiffusive Energy Transport

We study energy transport in a finite one-dimensional unpinned harmonic chain with stochastic nearest-neighbor momentum exchanges and Langevin heat baths at its endpoints. Such systems are known to exhibit superdiffusive transport driven by long-wavelength acoustic modes, leading to fractional macroscopic behavior. While fractional heat equations have been rigorously derived for infinite chains, the corresponding boundary conditions for finite systems in contact with heat baths remain unclear due to the nonlocality of the fractional Laplacian. Under the superdiffusive time scaling $t\sim n^{3/2}$, where $n$ is the system size, we prove that the averaged microscopic energy profile converges, as $n\to+\infty$, to a temperature field solving a fractional heat equation on $[0,1]$,
with the generator given by a Neumann fractional Laplacian and
additional nonlocal boundary terms induced by the heat baths. Our results provide a
rigorous derivation of macroscopic boundary conditions for
superdiffusive heat transport in open chains and introduce new
boundary conditions for fractional Laplacians, that are motivated by
a physical model.