Heat pump without particle transport or external work on the medium achieved by differential thermostatting of the phase space

We propose a new mechanism that enables heat flow from a colder region to a hotter region without necessitating either particle transport or external work on the conductor, thereby bypassing the compressor part of a classical heat pump cycle. Our mechanism relies on thermostatting the kinetic and configurational temperatures of the same particle differently. We keep the two ends of a conductor, which in the present study is a single dimensional $\phi^4$ chain, at the same kinetic temperature $T_0$, but at different configurational temperatures - one end hotter and the other end colder than $T_0$. While external energy is needed within the thermostatted regions to achieve this differential thermostatting, no external work is performed on the system itself. We show that the mechanism satisfies the statistical form of the second law of thermodynamics (the fluctuation theorem). The proposed mechanism reveals two interesting findings - (i) contrary to traditional thermodynamics where only the kinetic temperature is thought to govern heat conduction, configurational temperature can also play an important role, and (ii) the relative temperature difference between the kinetic and configurational variables governs the direction of heat flow. The challenge, however, is in developing experimental techniques to thermostat the kinetic and configurational variables of the same particle at different values.