Fixed points and flow analysis on off-equilibrium dynamics in the boson Boltzmann equation

Understand

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.