Ballistic Spreading of Chaos

• Ballistic spreading of chaos is a dynamical regime where local disturbances propagate linearly in time, forming a finite light-cone characterized by a butterfly velocity.
• Analytical methods including Floquet analysis and out-of-time-ordered correlators reveal coexisting ballistic and diffusive channels in both classical and quantum many-body systems.
• This phenomenon underpins insights into operator spreading, superdiffusion, and dynamical phase transitions in models from kicked rotors to nonlinear lattices.

Ballistic spreading of chaos refers to a dynamical regime in which the growth or propagation of a local disturbance—such as a perturbation, information packet, or operator—in a classical or quantum many-body system occurs at a rate linear in time, establishing a sharply defined “light-cone” with finite propagation speed. Unlike normal diffusion, where disturbances spread as the square root of time and the mean-square displacement scales as $t$, ballistic chaos features linear or faster scaling of the front, often characterized by an emergent “butterfly velocity” $v_B$. This phenomenon underpins a broad class of nonequilibrium behaviors in Hamiltonian, quantum, classical, and even automaton systems, manifesting in operator spreading, superdiffusion, and sharp dynamical phase transitions.

1. Ballistic Modes and Accelerator-Mode Fixed Points

In periodically kicked many-body Hamiltonian systems, such as collections of interacting rotors, the phase space admits special families of fixed points—accelerator modes (AM)—where the momentum of each particle grows exactly linearly in the number of kicks. These AM fixed points satisfy

$$p_j(s+1) = p_j(s) + 2\pi w_j, \quad \phi_j(s+1) = \phi_j(s),$$

with integer winding numbers $w_j$. For any finite number of particles $N$ and arbitrarily strong kicking ($K$), AM points can be made linearly stable by proper choice of $w$. This implies that even under strongly chaotic dynamics there exist exact solutions exhibiting ballistic (linear in time) transport in momentum space (Rajak et al., 2020).

The rigorous stability analysis relies on calculation of the Floquet multipliers of the $2N$-dimensional stroboscopic map. It is shown that for both infinite-range and nearest-neighbor interactions, and for any $N$, there exists a window in parameter space where all (or a subset of) directions are linearly stable, establishing long-lived phase-space islands.

2. Isolated Chaotic Zones (ICZ) and Superdiffusion

Surrounding stable or partially stable AM points, simulations reveal “isolated chaotic zones” (ICZ), comprising chaotic trajectories that remain trapped for long times near the AM center. Orbits inside the ICZ display robust ballistic momentum growth,

$$p_j(s) \approx p_j(0) + 2\pi w\,s \implies E_{\text{K}}(s) \sim s^2,$$

where $E_{\text{K}}$ is the mean kinetic energy. When initial ensembles are centered around an AM point, the fraction $f(s)$ that does not escape into the bulk chaotic sea continues to grow ballistically, while the remaining fraction exhibits normal diffusion. Consequently, the ensemble-averaged kinetic energy obeys a superdiffusive law: $$\langle E_{\text{K}}(s)\rangle \sim f(s)\,s^2 + [1 - f(s)]\,s \sim s^\alpha,\quad 1 < \alpha < 2,$$ with $\alpha$ determined by the survival fraction in the ICZ (Rajak et al., 2020). Ballistic spreading thus generically coexists with and dominates over diffusive channels at sufficiently long times.

3. Ballistic Chaos Propagation in Quantum and Classical Models

The ballistic regime is not confined to classical Hamiltonian maps. Quantum systems—including double-kicked rotors, long-range interacting spin chains, and quantum circuits—can also support ballistic spreading of chaos, most rigorously characterized via out-of-time-ordered correlators (OTOCs). In double-kicked quantum rotors, for specific symmetries and quantum resonances, wavepacket variance transitions from $\nu_2(t) \propto t^2$ (ballistic) to $\propto t^3$ (superballistic), with sharply defined crossover times $t_{1,2}$ controlled by detuning and kicking strength (Fang et al., 2016).

In classical integrable and near-integrable systems, numerical diagnostics based on decorrelators or Hamming distances have demonstrated phase transitions from localized (nonspreading) to ballistic chaos phases (Kasim et al., 26 Sep 2025). In particular, deterministic cellular automata such as the momentum-conserving parity check automaton exhibit a well-defined butterfly velocity and sharply demarcated light cone in the chaotic phase, with a continuous approach to localization at critical parameter values.

A representative table illustrates various ballistic chaos systems and their key diagnostics:

Model/Class	Main Diagnostic	Ballistic Chaos Signature
Periodically kicked rotors	Kinetic energy $\langle E_{\text{K}}\rangle$	Superdiffusive $\sim s^\alpha$, $\alpha > 1$
Double-kicked quantum rotor	Momentum variance $\nu_2(t)$	Crossover $\nu_2\propto t^2\to t^3$
Classical and nonreciprocal magnets	Decorrelator $D(x,t)$	Ballistic fronts $x=v_B t$
Quantum long-range chains	OTOC/front propagation	Linear light cone, scaling velocity $v_B$
Classical MCPCA automata (2D)	Hamming distance $H_d(t)$	$H_d\sim t^2$ in ballistic phase

4. Operator Spreading, Butterfly Velocity, and Hydrodynamic Constraints

In quantum systems, ballistic spreading of chaos is rigorously quantified by the growth of the OTOC front, defining the “butterfly velocity” $v_B$. For short-range interactions, front propagation is ballistic ($x=v_B t$); for long-range couplings decaying as $1/r^{1+\alpha}$, the spread transitions from linear to stretched-exponential or instantaneous, depending on $\alpha$ (Zhou et al., 9 May 2025). The noisy Fisher–Kolmogorov–Petrovsky–Piscunov (FKPP) equation provides a unified stochastic-reaction framework connecting microscopic lattice models with nonlocal population dynamics and establishing dualities between discrete and continuum models.

Hydrodynamic theories of chaos spreading (e.g., in spin chains) show that the chaotic front typically outruns (i.e., is much faster than) diffusive spin or energy propagation, with ballistic decorrelator propagation persisting even in the absence of simple conservation laws (Bhatt et al., 15 Dec 2025). Fluctuations of the chaotic front can conform to well-known universality classes, such as the 1D Kardar–Parisi–Zhang (KPZ) scaling for the front in classical Heisenberg spin chains (Das et al., 2017).

5. Ballistic Spreading and Dynamical Phase Transitions

Deterministic cellular automata and strongly disordered nonlinear lattices can exhibit dynamical transitions between localized and ballistic phases of chaotic spreading. In the 2D MCPCA automaton, a second-order phase transition is sharply identified: for sufficient density of “mobile” local charges, the Hamming distance $H_d(t)$ grows ballistically ($\sim t^2$), with a well-defined butterfly velocity $v_B$ characterizing the light cone of influence (Kasim et al., 26 Sep 2025). When frozen regions dominate, ballistic spreading is suppressed and chaos remains localized. The transition is controlled by conserved loop charges, with order-parameter behavior and exponents accurately extracted.

In nonlinear mechanical lattices with multiple degrees of freedom per site, strong nonlinearity and geometric coupling facilitate near-ballistic energy spreading in the presence of persistent, albeit decaying, chaos. Notably, in these systems the finite-time Lyapunov exponent remains positive throughout the ballistic-spreading phase, indicating that the system retains weak chaoticity even as the energy distribution front proceeds nearly at constant velocity (Ngapasare et al., 2022).

6. Impact, Physical Interpretation, and Universality

Ballistic spreading of chaos represents a generic mechanism for the rapid propagation of information, energy, or perturbations in nonequilibrium many-body systems. Its existence implies that localized initial disturbances can engender macroscopic, sharply bounded regions of influence moving with finite velocity (the light-cone structure), even when traditional transport channels are diffusive. The mechanism underlies superdiffusive and anomalous energy transport observed in driven rotors, high-dimensional Floquet systems, classical and quantum spin chains, and automata.

In the context of holographic quantum matter, analysis of shock waves in gravitational duals of strongly coupled field theories yields precise expressions for the butterfly and entanglement velocities. These velocities govern the spread of quantum information and energy, and obey or violate various theoretical bounds depending on symmetries, field content, and renormalization group flow. Notably, flows from UV to IR can result in highly anisotropic ballistic chaos propagation, saturating causal bounds but exceeding simple conformal predictions (Ávila et al., 2018).

The persistence of ballistic chaos in models with nonreciprocal, long-range, or strongly nonlinear interactions, and its apparent insensitivity to the detailed nature of conservation laws or system dimensionality, suggests a universal dynamical phenomenon relevant across diverse platforms. The coexistence of ballistic and diffusive transport, robust up to the thermodynamic limit and strong chaos, points to a fundamental organizing principle of nonequilibrium dynamics in complex many-body systems (Rajak et al., 2020, Ngapasare et al., 2022, Bhatt et al., 15 Dec 2025).