  In nature, objects which are in thermal contact with each other, usually
approach the same temperature, unless a heat source (or sink) cherishes a
persistent flow of heat. Accordingly, in a well-isolated apartment flat, most
items are at a similar temperature. This is a general consequence of
equilibrium thermodynamics, requiring coexisting phases to have identical
temperatures. Opposing this generic situation, here we identify a system
showing different temperatures in coexisting phases, which are separated from
each other by a sharp and persistent temperature gradient. Thermodynamically,
such a "hot" and a "cold" phase are allowed to coexist, as the system we
consider comprises "active" particles which self-propel relative to their
environment and are thus intrinsically out-of-equilibrium. Although these
microparticles are well known to spontaneously phase-separate into a liquid-
and a gas-like state, different kinetic temperatures in coexisting phases occur
if and only if inertia is introduced, which is neglected in standard models
describing active particles. Our results, therefore, exemplify a novel route to
use active particles to create a self-sustained temperature gradient across
coexisting phases, a phenomenon, which is fundamentally beyond equilibrium
physics.
