Fermi acceleration and adiabatic invariants for non-autonomous billiards

Area of Science:

• Mathematical Physics
• Dynamical Systems
• Statistical Mechanics

Background:

• Particle energy growth in dynamic containers is a key problem in physics.
• Understanding particle behavior in non-autonomous systems is crucial for various applications.

Purpose of the Study:

• To summarize and discuss recent findings on particle energy growth in containers with moving walls.
• To investigate the conditions leading to exponential versus quadratic energy growth.
• To present a unified view of particle acceleration in diverse billiard systems.

Main Methods:

• Review of existing theoretical results, including Anosov-Kasuga averaging theory.
• Numerical simulations to corroborate theoretical predictions.
• Development of a stochastic description for particle dynamics.
• Analysis of non-integrable and non-ergodic billiard models.

Main Results:

• For smooth-boundary breathing domains, particle acceleration is at most exponential.
• Averaged particle energy in periodically perturbed ergodic and mixing billiards grows quadratically.
• Exponential acceleration is proven to occur in non-integrable breathing billiards.
• New non-ergodic billiards robustly admit exponentially accelerating particles.

Conclusions:

• Particle acceleration dynamics depend significantly on the system's integrability and ergodicity.
• Stochastic descriptions provide insights into averaged particle energy growth.
• The existence of exponentially accelerating particles is robust in specific non-ergodic billiard systems.