The leading Lyapunov exponent in the glasma

TL;DR

This work demonstrates chaotic dynamics in the boost-invariant glasma by introducing infinitesimal perturbations to classical SU(2) gauge fields within the MV framework and tracking their linear evolution to extract a leading Lyapunov exponent. The perturbations exhibit exponential growth in the square root of proper time, $\delta F(\tau) \sim \exp(\lambda\sqrt{g^2\mu\tau})$, with a robust rate $\lambda = 0.39 \pm 0.02$ that is insensitive to the initial momentum structure and to whether electric or magnetic perturbations are used. Across Gaussian, power-law, and shell momentum filters, the exponent remains universal, while amplitudes depend on spectral content, indicating a collective, scale-invariant chaotic dynamics and coupling across polarizations ($E_\eta$, $B_\eta$). Lattice tests show UV and IR independence, confirming that the measured Lyapunov exponent reflects intrinsic glasma dynamics rather than discretization artifacts. The results connect the chaotic growth rate to entropy production and the early thermalization timescale, and point to future extensions to SU(3), full Lyapunov spectra, and links to plasmon/debye-mass scales and Weibel-type instabilities.

Abstract

We show that small perturbations in the boost-invariant color fields of the glasma exhibit an exponential growth with the square root of time. We interpret this growth rate as a Lyapunov exponent, related to entropy production and the thermalization timescale in the earliest stage of heavy-ion collisions. Working in a regime that is linear in this perturbation, we extract the time dependence of this mode as $\sim \exp(0.4\sqrt{g^2μτ})$ for SU($2$), where $g^2μ$ is proportional to the saturation scale and the square-root dependence is caused by the boost-invariant expansion of the system. We show that the growth rate of this mode is, unlike its amplitude, remarkably insensitive to the details of how the perturbations are initialized. In particular, we show that the unstable mode couples to all momentum scales present in the initial perturbation.

Figures (9)

• Figure 1: Rescaled time $g^2\mu\tau$ dependence of the longitudinal electric field perturbation $\langle \mathrm{Tr}(\delta E_\eta)^2\rangle$ with a white noise initial condition as given in Eq. \ref{['Eq:chi_Moments']}, shown both on a linear and logarithmic scale. The solid lines and associated error bands represent the average and statistical errors of the data, with different colors for different values of $\alpha$. The corresponding colored dashed lines are the separate fits to these datasets, and the black dashed line is a fit to the combination of the datasets. The extracted Lyapunov exponent $\lambda$ for each fit is provided in the legend. The fit ranges given by our procedure and the resulting fits are given in Table \ref{['Table: NoFilter']} in Appendix \ref{['Appendix:Fit_Results']}.
• Figure 2: The electric field perturbation $\langle \mathrm{Tr}(\delta E_\eta)^2\rangle$ as a function of $g^2\mu\tau$ with Gaussian filtered noise according to Eq. \ref{['Eq:Gaussian_Filter']}. The evolution is shown for different filter scales $\kappa_\mathrm{g}/(g^2\mu)$ shown in a different color, with the same perturbation amplitude $\alpha = 10^{-4}$. The resulting fit parameters are given in Table \ref{['Table: Gaussian_Filter']} in Appendix \ref{['Appendix:Fit_Results']}.
• Figure 3: The electric field perturbation $\langle \mathrm{Tr}(\delta E_\eta)^2\rangle$ as a function of $g^2\mu\tau$ with power-law filtered noise given in Eq. \ref{['eq:power_filter']}. The evolution is shown for different filter scales $\kappa_\mathrm{p}/(g^2\mu)$ shown in a different color, with the same perturbation amplitude $\alpha = 10^{-4}$. The resulting fit parameters are given in Table \ref{['Table: Powerlaw_Filter']} in Appendix \ref{['Appendix:Fit_Results']}.
• Figure 4: The electric field perturbation $\langle \mathrm{Tr}(\delta E_\eta)^2\rangle$ as a function of time $g^2\mu\tau$ for different momentum shell initializations for $\alpha=10^{-4}$, together with the corresponding fits. The plot on the left shows the full simulation interval, and the one on the right zooms in to the early time behavior.
• Figure 5: Extracted growth rates $\lambda$ using electric field fluctuations for different initial momentum $\kappa_\mathrm{s}/(g^2\mu)$ in the shell filtering from Eq. \ref{['Eq:Shell_Filter']}, where distinct colored marker styles corresponding to different values of $\alpha$, while the black circle marker represent the combined result. The quality of the fit for $\alpha=10^{-4}$ is shown in Fig. \ref{['Fig:shelltimedep']} and is similar for other values of $\alpha$.