Chaos in Classical D0-Brane Mechanics

TL;DR

This work analyzes chaos in the classical limit of the D0-brane matrix quantum mechanics by computing Lyapunov exponents and the full spectrum in the large-$N$ limit. The authors introduce a gauge-invariant, symplectically reduced framework to extract physical exponents and demonstrate that the leading exponent converges to $\lambda_L \approx 0.2925\,\lambda_{ m eff}^{1/4} T$, with a finite-$N$ correction of order $N^{-2}$, while the spectrum forms a smooth density at large $N$. They show a classical analogue of fast scrambling with scrambling time $t_* \sim \frac{1}{4\lambda_L}\log N^2$, and confirm this via both trajectory growth and Poisson-bracket correlators. The results support the $k$-locality of matrix interactions and raise questions about the quantum interpretation of the Lyapunov spectrum, the Kolmogorov-Sinai entropy at large $N$, and potential holographic connections to bulk chaos and entanglement dynamics.

Abstract

We study chaos in the classical limit of the matrix quantum mechanical system describing D0-brane dynamics. We determine a precise value of the largest Lyapunov exponent, and, with less precision, calculate the entire spectrum of Lyapunov exponents. We verify that these approach a smooth limit as $N \rightarrow \infty$. We show that a classical analog of scrambling occurs with fast scrambling scaling, $t_* \sim \log S$. These results confirm the k-locality property of matrix mechanics discussed by Sekino and Susskind.

Figures (11)

• Figure 1: Lyapunov exponent as a function of thermalization time $t_0$.
• Figure 2: Time evolution of $|\delta X(t)|$ for $N=16$. Here $t=0$ is the time of the initial perturbation.
• Figure 3: The exponent estimated using the Sprott algorithm as a function of time, for $N=20$ and at $\lambda_{\rm eff}^{1/4} \, T = 1$. The band represents the statistical fluctuations of different samples.
• Figure 4: Leading Lyapunov exponent for $N=6,8,12,16,20,24$ at $\lambda_{\rm eff}^{1/4} \, T = 1$. The error bars are statistical.
• Figure 5: Last phase of the evolution of $|\delta X(t)|/\sqrt{N}$ for $N=6$ (black) through $N=16$ (blue). At long times, $|\delta X(t)|$ converges to a value that scales as $\sqrt{N}$.