  In relativistic kinetic theory, the one-particle distribution function is
approximated by an asymptotic perturbative power series in Knudsen number which
is divergent. For the Bjorken flow, we expand the distribution function in
terms of its moments and study their nonlinear evolution equations. The
resulting coupled dynamical system can be solved for each moment consistently
using a multi-parameter transseries which makes the constitutive relations
inherit the same structure. A new non-perturbative dynamical renormalization
scheme is born out of this formalism that goes beyond the linear response
theory. We show that there is a Lyapunov function, aka dynamical potential,
which is, in general, a function of the moments and time satisfying Lyapunov
stability conditions along RG flows connected to the asymptotic hydrodynamic
fixed point. As a result, the transport coefficients get dynamically
renormalized at every order in the time-dependent perturbative expansion by
receiving non-perturbative corrections present in the transseries. The
connection between the integration constants and the UV data is discussed using
the language of dynamical systems. Furthermore, we show that the first
dissipative correction in the Knudsen number to the distribution function is
not only determined by the known effective shear viscous term but also a new
high energy non-hydrodynamic mode. It is demonstrated that the survival of this
new mode is intrinsically related to the nonlinear mode-to-mode coupling with
the shear viscous term. Finally, we comment on some possible phenomenological
applications of the proposed non-hydrodynamic transport theory.
