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. 1 A slightly more general class of models for the relaxation time approximation have been studied in Ref. [60]. 2 Blaizot and Li studied the time evolution of similar moments for a constant relaxation time [63] and a more general nonlinear collisional kernel in the small angle approximation [64]. In their case, the authors were interested in the details of the longitudinal momentum anisotropy which in our notation this corresponds to those moments with n = 0. Up to some normalization factor, the main difference between the moments L l (see Eq. (2.8) in Ref. [63]) and our moments c 0l is that the former are dimensionful. 3 It is customary to use the following tensor decomposition of the energy-momentum tensorwhere P 0 is the equilibrium pressure and π µν is the shear viscous tensor. Nonetheless, for highly anisotropic systems, Mólnar et. al [55,65] showed that using T µν (5) is convenient. It should be noted that both formulations are equivalent [55] and explicit examples for the Bjorken [65] and Gubser flows [42,49] have already been discussed in the literature.

Global flow structure and exact formal transseries of the Gubser flow in kinetic theory

“…A definite answer to this difficult problem can at least be given for certain kinetic models of expanding plasmas undergoing Bjorken flow. The longitudinal boost-invariant systems are often described in terms of the longitudinal proper time τ = (x 0 ) 2 − (x 3 ) 2 and they present an unresolvable singularity in the Boltzmann equation at τ = 0 [40]. It has been known for a while that certain interesting features of the hydrodynamic expansion of the Bjorken flow are better understood if we use the variable w = τ T (T being the temperature) [12].…”

“…3 It is possible for this continuation to not generate a unique critical line in case both the UV fixed points are "saddle", a fact that does not obviously hold in the configuration space of the Bjorken flow. Note that a critical line is really a complete flow line that basically lies on the boundary of basin of attraction [40]. 4 The variable w does not capture any UV information from the original system and rather serves as a toy model for the Bjorken flow in τ .…”

Exact solutions and attractors of higher-order viscous fluid dynamics for Bjorken flow

“…In this article, we employ the theory of Lyapunov exponents to study the attractors for MIS theory and two other, improved versions of causal relativistic dissipative hydrodynamics (see Refs. [32,47] for earlier related work).…”

Early- and Late-Time Behavior of Attractors in Heavy-Ion Collisions

“…Recently, such attractors have received attention in the context of ultrarelativistic heavy-ion collisions. Their form is of interest for understanding the onset of fluid-dynamic behavior [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and the origin of the nonthermal fixed-point behavior in far-from-equilibrium dynamics [1,19,[22][23][24][25]. For the phenomenology of heavy-ion collisions, these studies are needed to clarify to what extent different observables inform us either about the details of the initial conditions or about the material properties of the system.…”