Dynamical thermalization in time-dependent billiards

Area of Science:

• Statistical mechanics
• Nonlinear dynamics
• Computational physics

Background:

• Understanding particle behavior in complex systems is crucial.
• Billiard models with inelastic collisions are used to study statistical properties.

Purpose of the Study:

• To investigate the statistical evolution of particle speeds in a time-dependent billiard with inelastic collisions.
• To identify and characterize different statistical regimes of speed evolution.
• To analytically derive these regimes and compare them with numerical experiments.

Main Methods:

• Numerical experiments simulating particle ensembles in a time-dependent billiard.
• Analysis of speed distributions (Gaussian-like to Boltzmann-like).
• Analytical derivation using velocity-space diffusion analysis.

Main Results:

• Identified three statistical regimes: diffusion plateau, normal growth/exponential decay, and stagnation.
• Linked these regimes to transitions in speed distribution shapes.
• Derived analytical expressions for key parameters like root mean square speed and growth/decay rates.
• Achieved agreement between analytical calculations and numerical experiments.

Conclusions:

• The study demonstrates a dynamical thermalization mechanism in this system.
• Inelastic collisions and high-dimensional phase space drive bounded diffusion in velocity space.
• A stationary distribution function is reached, with 'reservoir temperature' dependent on boundary oscillations and restitution coefficient.