Chaotic behavior in classical Yang-Mills dynamics

Understanding the underlying mechanisms causing rapid thermalization deduced for high-energy heavy ion collisions is still a challenge. To estimate the thermalization time, entropy growth for classical Yang-Mills theories is studied, based on the determination of Lyapunov exponents. Distinct regimes for short, medium and long sampling times are characterized by different properties of their spectrum of Lyapunov exponents. Clarifying the existence of these regimes and their implications for gauge-field dynamics is one of the results of this article. As a phenomenological application we conclude that for pure gauge theories with random initial conditions thermalization occurs within a few $\mathrm{fm}/c$, an estimate which can be reduced by the inclusion of fermions, specific initial conditions, etc.

Figure 1

Time evolution of the distance in SU(2) simulation on $4^3$ lattice. All scales are given in the lattice unit.

Figure 2

The average violation of Gauss’ law $G$ in the SU(2) simulation of Fig. 1. All scales are given in lattice units.

Figure 3

Time evolution in SU(2) simulation on $4^3$, ${10}^3$, and ${20}^3$ lattices with the same energy density.

Figure 4

Distribution of the intermediate Lyapunov exponents. The total number of the eigenvalues is 1152. The right panel is a closeup of the largest 300 eigenvalues.

Figure 5

The spectrum of gauge fields $\tilde{A}(p)$ for different times.

Figure 6

Time evolution in SU(3) simulation on a $4^3$ lattice.

Figure 7

Distribution of the intermediate Lyapunov exponents in SU(3) simulations.

Figure 8

The SU(3) results of the Lyapunov exponents, ${\lambda }_D$, ${\lambda }_{\max }^{\mathrm{LLE}}$, ${\lambda }_{\mathrm{sum}}^{\mathrm{LLE}}$, ${\lambda }_{\max }^{\mathrm{ILE}}$, and ${\lambda }_{\mathrm{sum}}^{\mathrm{ILE}}$. The broken line is ${\epsilon }^{1/4}$.