Dynamical systems and nonlinear transient rheology of the far-from-equilibrium Bjorken flow

Abstract

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.