  We consider fixed points of steady solutions and flow directions using the
boson Boltzmann equation that is a one-dimensionally reduced kinetic equation
after the angular integration. With an elastic collision integral of the
two-to-two scattering process, in the dense (dilute) regime where the
distribution function is large (small), the boson Boltzmann equation has
approximate fixed points with a power-law spectrum in addition to the thermal
distribution function. We argue that the power-law fixed point can be exact in
special cases. We elaborate a graphical presentation to display evolving flow
directions similarly to the renormalization group flow, which explicitly
exhibits how fixed points are connected and parameter space is separated by
critical lines. We discuss that such a flow diagram contains useful information
on thermalization processes out of equilibrium.
