Experimental observation of violent relaxation

The observable universe's structures, like galaxies and globular clusters, appear static but aren't in thermodynamic equilibrium; their stellar velocity distributions aren't Maxwellian. Chandrasekhar (1941) showed these objects take far longer to reach thermal equilibrium than their age, confirmed by observations showing them far from equilibrium. Lynden-Bell's 1967 proposal of "violent relaxation" explains the formation of these out-of-equilibrium, quasi-stationary states, evolving much faster than full thermodynamic equilibrium. This mechanism is common in Hamiltonian systems with long-range interaction potentials (non-integrable due to large-scale extension). It's similar to Landau damping in plasmas (energy exchange between electromagnetic waves and plasma particles), observed in experiments and space plasmas. Violent relaxation, however, remains elusive, lacking repeatable or controllable experimental observation or in-situ detection. Experimental observation is hindered by: (1) systems where it's potentially present but destroyed by noise (e.g., cold atoms in optical traps); and (2) systems where it exists but timescales are too large to observe (e.g., galaxies, regardless of classical or quantum dark matter composition, where timescales are millions of years). Numerical simulations of classical N-body systems with nonlocal interactions (e.g., gravitational), governed by Vlasov-Poisson equations, confirm Lynden-Bell's process but offer no experimental guidance. This paper reports the experimental observation of violent relaxation in an optical setting, observing an optical beam's evolution under a self-generated long-range interaction, leading to phase-mixing and relaxation to a quasi-stationary state. This optical dynamics is analogous to (dark matter) galaxy formation via violent relaxation, linked by the underlying Vlasov-Poisson equation—the semi-classical limit of the quantum description of dark matter evolution based on the Newton-Schrödinger equation (NSE), describing both classical dark matter and the optical experiments. NSE has been experimentally realized in nonlinear optical experiments probing gravitational lensing, tidal forces, and analogous quantum processes like Boson star evolution.

The literature extensively discusses violent relaxation theoretically and through simulations, but experimental verification has been lacking. Early work by Chandrasekhar established the timescale problem for reaching equilibrium in stellar systems. Lynden-Bell's seminal paper introduced the concept of violent relaxation as a mechanism for faster equilibration. Subsequent theoretical work explored its generality in Hamiltonian systems with long-range interactions. In contrast to the abundant literature on Landau damping, which has seen experimental verification, experimental observations of violent relaxation have remained elusive due to technical challenges involving noise and timescales. Numerical simulations, based on the Vlasov-Poisson and Newton-Schrödinger equations, provided valuable insights into the process, but these did not lead to a practical experimental realization. This paper bridges this gap by demonstrating violent relaxation experimentally using an optical analogue system.

The experiment uses a slab of thermo-optically nonlinear glass or crystal. An intense laser beam propagating through the crystal induces a nonlocal interaction (heating), creating a heat profile that acts on the beam, emulating a self-attractive gravitational force. In the paraxial approximation, a monochromatic laser beam's propagation (amplitude E(r,z)) in a thermally focusing nonlinear medium is described by: i∂zE + ∇²E + k0βΔnE + iαE = 0, ∇²Δn = |E(r,z)|² where r⊥ = (x,y) is the transverse position, z is the propagation direction, ∇² is the 2D Laplacian, k0 is the wave number, β is the thermo-optic coefficient, κ is thermal conductivity, and α is the absorption coefficient. The last term (absorption) has minimal effect and is neglected. Equation (2) is analogous to the Newton-Schrödinger equation (NSE) describing self-gravitating dark matter: ih∂tψ + ħ²/2m ∇²ψ + mφψ = 0 ∇²φ = −4πG|ψ|² The similarity between equations (1) and (2) allows for observation of 2D violent relaxation. The main difference is the potential shape (logarithmic in 2D optical, compared to 3D gravitational). The underlying physics—mode or particle mixing in an evolving potential—remains the same. The optical equivalent of the semi-classical regime (ħ/m ≤ 1) is achieved when χ = ξ/s = 1 (ξ is soliton size, s is system size). The experiment uses a Gaussian laser beam (λ = 532 nm) propagating through three aligned lead-doped glass slabs (total length L = 30 cm). The beam width (s = 350 µm) ensures the semi-classical regime (χ ≈ 10²). Off-axis digital holography measures the output field's intensity and phase. Varying input power (0.2 W to 5.5 W) effectively simulates propagation along z. Violent relaxation is characterized by the chemical potential density U(r,z) = |∇E(r,z)|²/2k0 − k0Δn(r,z) and the Wigner transform F(r₁, k₁, z), showing phase-space mixing. The experiment's setup consists of a continuous-wave (CW) laser (λ = 532 nm) split into reference and target beams. The target beam, shaped with a waist of 350 µm, is directed through three aligned lead-doped glass slabs. The output beam is imaged using off-axis digital holography to reconstruct its amplitude and phase. Power is varied from 0.2 W to 5.5 W in steps of 0.25 W.

The experiment successfully observed violent relaxation in an optical setting. The initial beam collapse is followed by the formation of a central high-intensity peak (analogous to a solitonic core in gravitational systems) surrounded by a broader, lower-amplitude field (analogous to a galaxy). This stabilization, despite oscillations, signifies a quasi-stationary state. The chemical potential density distribution shows a significant variation in the region where the potential is most dynamic (0-2 W), indicating violent relaxation. After the collapse (P ≈ 3 W), the distribution stabilizes, representing the quasi-stationary state. Analysis of the Wigner distribution reveals phase-space mixing, a hallmark of violent relaxation. The initial Gaussian distribution twists and forms filaments as power increases. Comparison with a system exhibiting only mixing (Snyder-Mitchell model) shows distinct differences in evolution, confirming the presence of violent relaxation. The experimental results closely match numerical simulations based on the optical system's governing equations. Furthermore, a direct comparison with a numerical N-body simulation of a self-gravitating dark matter system shows striking similarity in the resulting density profiles, strengthening the analogy between the optical experiment and the astrophysical process.

The experimental observation of violent relaxation confirms its existence beyond theoretical predictions and numerical simulations, validating Lynden-Bell's original hypothesis and highlighting its prevalence in diverse long-range interaction systems. The optical analogue provides a controllable and repeatable platform to study violent relaxation, overcoming the limitations of astrophysical observations. The striking resemblance between the experimental results and N-body simulations underscores the power of using optical systems to model complex astrophysical phenomena. The findings suggest that violent relaxation might be a more common phenomenon than previously anticipated, potentially playing a significant role in various systems beyond the astrophysical context.

This work presents the first experimental evidence of violent relaxation, using a table-top optical system that directly simulates the process. The results validate theoretical models and demonstrate the potential for using optical analogues to study complex phenomena. Future research could explore different parameter regimes, investigate the role of noise and nonlocality in detail, and extend the analogy to other types of long-range interactions.

The experiment is a 2D analogue of a 3D system, and some aspects of the 3D behavior might not be fully captured. The optical system’s nonlocal interaction might not perfectly mimic the gravitational interaction in all aspects. The choice of parameters ensures operation within the semi-classical regime, maximizing the generation of the background field representing the galaxy, but this specific regime might limit the generality of some findings.