  The thermal Hall conductance is a universal and topological property which
characterizes the fractional quantum Hall (FQH) state. The quantized value of
the thermal Hall conductance has only recently been measured experimentally in
integer quantum Hall (IQH) and FQH regimes, however, the existing setup is not
able to detect if the thermal current is counter-propagating or co-propagating
with the charge current. Furthermore, although there is experimental evidence
for heat transfer between the edge modes and the bulk, the current theories do
not take this dissipation effect in consideration. In this work we construct
phenomenological rate equations for the heat currents which include
equilibration processes between the edge modes and energy dissipation to an
external thermal bath. Solving these equations in the limit where temperature
bias is small, we compute the temperature profiles of the edge modes in a FQH
state, from which we infer the two terminal thermal conductance of the state as
a function of the coupling to the external bath. We show that the two terminal
thermal conductance depends on the coupling strength, and can be non-universal
when this coupling is very strong. Furthermore, we propose an experimental
setup to examine this theory, which may also allow the determination of the
sign of the thermal Hall conductance.
