  Micro-mechanical resonator performance is fundamentally limited by the
coupling to a thermal environment. The magnitude of this thermodynamical effect
is typically considered in accordance with a physical temperature, assumed to
be uniform across the resonator's physical span. However, in some
circumstances, e.g. quantum optomechanics or interferometric gravitational wave
detection, the temperature of the resonator may not be uniform, resulting in
the resonator being thermally linked to a spatially varying thermal bath. In
this case, the link of a mode of interest to its thermal environment is less
straightforward to understand. Here, we engineer a distributed bath on a
germane optomechanical platform -- a phononic crystal -- and utilize both
highly localized and extended resonator modes to probe the spatially varying
bath in entirely different bath regimes. As a result, we observe striking
differences in the modes' Brownian motion magnitude. From these measurements we
are able to reconstruct the local temperature map across our resonator and
measure nanoscale effects on thermal conductivity and radiative cooling. Our
work explains some thermal phenomena encountered in optomechanical experiments,
e.g. mode-dependent heating due to light absorption. Moreover, our work
generalizes the typical figure of merit quantifying the coupling of a resonator
mode to its thermal environment from the mechanical dissipation to the overlap
between the local dissipation and the local temperature throughout the
resonator. This added understanding identifies design principles that can be
applied to performance of micro-mechanical resonators.
