Local thermal current from cold to hot in a classical harmonic system with multi-path geometry

The second law of thermodynamics forbids an overall flow of heat from a cold object to a hot one. However, it does not rule out a local heat flow from cold to hot in a multi-path geometry.
We show an atypical, local steady-state thermal current from cold to hot emerge in a classical harmonic system of Hookean springs and masses driven by two Langevin Reservoirs at different temperatures. The simulations were performed by the standard molecular dynamics, and physical quantities such as the thermal current were averaged over random realizations in the steady-state regime. Our results show that the atypical thermal current depends explicitly on the system-reservoir coupling, in addition to the spring constants and masses. Including non-linear on-site potentials which models the system-substrate coupling further tunes the regime where the atypical local thermal current can survive. Our theoretical framework is universal and applies to classical systems following Newtonian dynamics. If we consider possible realizations using nano-mechanical systems, the device with a tunable local thermal current may function as a thermal switch or memory element.