  We quantitatively analyze a basic question: what is the stationary solution
of the background plasma temperature profile around a black hole (BH)? One may
naively expect that the temperature profile continuously decreases from the
Hawking temperature at the surface of the BH towards an outer region. We show
analytically and numerically that this is not the case because local thermal
equilibrium cannot be maintained near the surface of the BH and also because
the high-energy particles emitted from Hawking radiation cannot be
instantaneously thermalized into the background plasma. The temperature profile
has a plateau within a finite distance from the BH, and even the overall
amplitude of background temperature at a distance far away from the BH is
significantly suppressed compared with the naive expectation. The main reason
for these counterintuitive results comes from the fact that the size of the BH
is too small that particles of Hawking radiation goes far away within the
typical time scale of interactions.
