Chaos in violent relaxation dynamics

Disentangling micro- and macro-chaos in numerical experiments of dissipationless collapse

¹ Consorzio RFX (CNR, ENEA, INFN, Università di Padova, Acciaierie Venete SpA), Corso Stati Uniti 4, 35127 Padova, Italy
² Centro Ricerche Fusione, Università di Padova, Padova, Italy
³ Consiglio nazionale delle Ricerche, Istituto dei Sistemi Complessi Via Madonna del piano 10, 50019 Sesto Fiorentino, Italy
⁴ Istituto Nazionale di Fisica Nucleare – Sezione di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Italy
⁵ Istituto Nazionale di Astrofisica – Osservatorio Astrofisico di Arcetri, Piazzale E. Fermi 5, 50125 Firenze, Italy
⁶ Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
⁷ Istituto Nazionale di Fisica Nucleare – Sezione di Trieste, 34127 Trieste, Italy
⁸ Istituto Nazionale di Astrofisica – IASF Via Alfonso Corti 12, 20133 Milano, Italy
⁹ Istituto Nazionale di Astrofisica – Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy

^★ Corresponding authors: simone.sartorello@phd.unipd.it; pierfrancesco.dicintio@cnr.it; aatrani@gmail.com

Received: 28 March 2025
Accepted: 11 April 2025

Abstract

Aims. Violent relaxation is often regarded as the mechanism that leads stellar systems to collisionless meta equilibrium via rapid changes in the collective potential.

Methods. We investigate the role of chaotic instabilities on single particle orbits in leading to nearly invariant phase-space distributions, aiming at disentangling their role from that of the chaos induced by collective oscillations in the self-consistent potential.

Results. We explore, as a function of the system’s size (i.e. number of particles N), the chaoticity in terms of the largest Lyapunov exponent of test trajectories in a simplified model of gravitational cold collapse, mimicking an N-body calculation via a time-dependent smooth potential and a noise-friction process accounting for the discreteness effects. A new numerical method to evaluate effective Lyapunov exponents for stochastic models is presented and tested.

Conclusions. We find that the evolution of the phase-space of independent trajectories reproduces rather well what is observed in self-consistent N-body simulations of dissipationless collapses. The chaoticity of test orbits rapidly decreases with N for particles that remain weakly bounded in the model potential, while it decreases with different power laws for more bound orbits, consistently with what was observed in previous self-consistent N-body simulations. The largest Lyapunov exponents of ensembles of orbits starting from initial conditions uniformly sampling the accessible phase-space are somewhat constant for N ≲ 10⁹, while decreases towards the continuum limit with a power-law trend. Moreover, our numerical results appear to confirm the trend of a specific formulation of dynamical entropy and its relation with Lyapunov timescales.