  Non-radiating protons in the radio lobes have an essential role to form the
jet morphology which is shown by recent radio and X-ray observations. However,
since protons and electrons are not always in energy equilibrium due to weak
Coulomb coupling, it is difficult to estimate the energy contribution of
protons for inflation of radio lobes. The main focus of this study is to
examine the effect of the variable model for electron heating by turbulence and
shock waves on the thermal energy distribution of electron and proton. We
performed two-temperature three-dimensional magnetohydrodynamic simulations of
sub-relativistic jets in the galaxy cluster while varying jet magnetization
parameters. Because the energy partition rate between electrons and protons in
shock and turbulence is determined by plasma kinetic scale physics, our global
simulations include electron instantaneous heating sub-grid models for shock
waves and turbulence. We find that most of the bulk kinetic energy of the jet
is converted into thermal energy of protons through both shocks and turbulence.
Thus, protons are energetically dominant. Meanwhile, thermal electrons stored
in the lobe evolve toward energy equipartition with magnetic energy through
turbulent dissipation. We further estimated the radio power and the mechanical
jet power of radio lobes following the same method as used for radio and X-ray
observations, and compared these powers with that of the observed radio jets.
The two-temperature model quantitatively explains the radiatively inefficient
radio cavities, but cannot reproduce the radiatively efficient cavity, even for
strong magnetized jets. This implies that a significant population of
pair-plasma is needed in the radiatively efficient radio cavities.
