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

In relativistic kinetic theory, the one-particle distribution function is approximated by an asymptotic perturbative power series in the 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 multiparameter transseries which makes the constitutive relations inherit the same structure. A new nonperturbative dynamical renormalization scheme is born out of this formalism that goes beyond the linear response theory. We show that there is a Lyapunov function, also known as dynamical potential, which is, in general, a function of the moments and time satisfying Lyapunov stability conditions along renormalization group 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 nonperturbative 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 nonhydrodynamic 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 nonhydrodynamic transport theory.

Figure 1

Five lowest nonhydrodynamic modes computed by transseries, leading bare asymptotic expansion and exact numerics. The blue shaded area depicts the variation in the transseries solution due to the standard deviation of ${\sigma }_i$. For each moment, the transseries is truncated and renormalized by transasymptotic matching including all the transmonomials up to 15th order. For example $c_{01}^{\text{Renorm}}=C_{01,1}/w$, where $C_{01,1}$ is constructed by including up to 15th-order transmonomials.

Figure 2

Eigenvalues of the Jacobian matrix around asymptotic UV fixed points in the truncated system. Red and blue points denote eigenvalues of the maximally oblate point and the spiral source, respectively. In (a), the maximally oblate point behaves like a spiral sink because $\mathrm{Re}(b_i)$ are all negative. For the maximally prolate point, however, $\mathrm{Re}(b_i)$ are all positive, a property that classifies it as a spiral source. In (b), both of these points are shown to be able to have positive and negative $\mathrm{Re}(b_i)$ at the same time; hence, they are in general spiral saddle points. From the perspective of transseries, upon choosing those integration constants ${\sigma }_i=0$ associated with the monomials with $b_i$ satisfying $\mathrm{Re}(b_i)>0$, we can make the stability of the UV fixed points change to a source in the general case $N>0$.

Figure 3

The phase portraits for the system truncated at $N=0$, $L=3$. The initial value of the flow is taken to be $c_{nl}(w_0)=c_{nl}(w_0;{\xi }_0)$. In the UV (early time), the flows are coming out of a spiral source as seen on the left where anisotropy parameters are close to $-1$ that corresponds to $c_{01}\sim 6.6$ [41]. In the range closer to the maximally oblate point, that is $10<{\xi }_0<1000$, the flow lines do not converge since they are initiated outside of the basin of attraction (of the invariant manifold). The maximally oblate point is exactly located on the boundary of the basin. Note that the flow lines are shown to be spiraling in the UV as $w\to 0$, a symptom of being in the vicinity of a spiral saddle. We can see the forward attractor in the IR for the flows initiated in the basin of attraction.

Figure 4

The phase space portraits for the system truncated at $N=1$, $L=3$. In the UV (early time), the flows are coming out of a spiral source as seen on the left where anisotropy parameters are close to $-1$. In the range closer to the maximally oblate point, that is $10<{\xi }_0<1000$, the flow lines are shown to be spiraling without any convergence happening in the UV as $w\to 0$, a symptom of being in the vicinity of a spiral saddle. We can see the forward attractor in the IR for the flows initiated in either range of anisotropy parameters.

Figure 5

Evolution of several normalized moments ${\overline{M}}^{nm}$ as a function of $w$. We compare the exact numerical expressions for the normalized moments obtained by solving RTA BE (e8) (black lines) alongside the soft limit (117) (green dash-dotted lines) and the $\mathrm{soft}+\mathrm{semihard}$ limit (118) (blue dashed lines). For the initial conditions of the exact RTA BE solutions, ${\tau }_0=0.25\mathrm{fm}/\mathrm{c}$ is chosen, and the initial temperature is set to $T_0=600\mathrm{MeV}$, with the initial anisotropy parameter being ${\xi }_0=10$.

Figure 6

Saturation time $w_c$ versus $m$ for different values of $n=\{0,1,2,3\}$ (top right, top left, bottom right and bottom left panels, respectively). The black dots are computed using the exact RTA BE solver; the blue squares and red triangles represent the values obtained in the soft and $\mathrm{soft}+\mathrm{semihard}$ regimes, respectively. We also see that the normalized moments $M_{\text{exact}}^{nm}$ equilibrate later as both $n$ and $m$ are increased.

Figure 7

Evolution of the normalized moments ${\overline{M}}^{n0}$ as a function of $w$. In this figure, the exact numerical expressions for the normalized moments obtained by solving RTA BE (e8) (black lines) are compared against Eqs. (119a) (green dot-dashed lines) and hard mode limit (119b) (blue dashed lines). For the initial conditions of the exact RTA BE solutions, ${\tau }_0=0.25\mathrm{fm}/\mathrm{c}$ is chosen, and the initial temperature is set to $T_0=600\mathrm{MeV}$, with the anisotropy parameter being ${\xi }_0=10$.

Figure 8

Saturation time $w_c$ versus $n$ for the moments ${\overline{M}}^{n0}$. The black circles are computed using the exact RTA BE solutions; the blue squares and red triangles represent the asymptotic hard limits (119a) and (119b), respectively. The deviation between the exact and $c_{10}+c_{20}$ results is obviously due to truncation effects.

Figure 9

Time evolution of the deviation from the equilibrium distribution function (120) for the exact RTA (black lines), the $s$ (green dot-dashed lines) and the $s+sh$ (blue dashed lines) regimes. For $\delta f_{exact}$, the initial ratio of $p_0^{\tau }/T_0$ is set to be $\{0.5,2.5,4,6\}$, and the initial temperature for the exact RTA BE solver is set at $T_0=0.6\mathrm{GeV}$, ${\tau }_0=0.25\mathrm{fm}/\mathrm{c}$. Finally, we initialize the solver with ${\xi }_0=0$.

Figure 10

Five lowest nonhydrodynamic moments for the $\mathrm{\Delta }=1$ system are computed using transseries, leading asymptotics and the exact solution. The blue shaded area depicts the probable variation in the transseries due to the standard deviation of integration constants ${\sigma }_i$. Each moment is calculated up to its leading asymptotic order and renormalized by transasymptotic matching including all the transmonomials up to 15th order.

Figure 11

Time evolution of several normalized moments ${\overline{M}}^{nm}$ ($n>0,l\ge 1$) in the case where the power law dependence of the relaxation time is a constant, i.e., ${\tau }_r={\theta }_0$. In this plot we present the numerical results of the exact RTA solution (e8) (black line), soft (117) (green dot-dashed line) and $\mathrm{soft}+\mathrm{semihard}$ (118) (blue dashed line) limits. The initial conditions are the same as in Fig. 5.

Figure 12

Time evolution of different normalized moments ${\overline{M}}^{n0}$ in the case where the power law dependence of the relaxation time is a constant, i.e., ${\tau }_r={\theta }_0$. In this plot we present the numerical results of the exact RTA solution (e8) (black line)and the hard limits given by Eqs. (119a) (green dot-dashed line) and (119b) (blue dashed line). The saturation bounds are set to at the same level as before. The initial conditions are the same as in Fig. 5.

Figure 13

Saturation value of $w_c$ versus $m$ for different values of $n=\{0,1,2,3\}$ (top right, top left, bottom right and bottom left panels, respectively) for the exact RTA (black circle), soft (blue square) and $\mathrm{soft}+\mathrm{semihard}$ (red triangle) regimes.

Figure 14

Saturation value of $w_c$ versus $n$ of the moments ${\overline{M}}^{n0}$ for the exact RTA (black circle) and hard limits, Eqs. (119a) (blue square) and (119b) (red triangle).

Figure 15

Time evolution of the deviation from the distribution function (e1) for the exact RTA BE solution when ${\tau }_r={\theta }_0$. The black lines represent the exact result (e1) while the green dot-dashed and blue dashed lines correspond to the soft limit (121a) and $\mathrm{soft}+\mathrm{semi}\mathrm{hard}\lim \mathrm{it}$ (121b) limits. The initial conditions are the same as in (121b).

Figure 16

Impact of truncation on $c_{01},\dots ,c_{06}$. The exact solution to the RTA BE is shown with a black line. Dashed lines correspond to $c_{0l}$ in a truncated system of $L$ moments.

Figure 17

Exact solutions for $c_{nl}$ of both the RTA BE and the truncated system. Dashed lines represent $c_{nl}$ in a truncated system composed of $N=5,L$ moments. The initial conditions for the exact $c_{nl}$ (e6) are ${\tau }_0=0.25\mathrm{fm}/\mathrm{c}$, $T_0=0.6\mathrm{GeV}$ and ${\xi }_0=1000$.