Dynamics in a discontinuous field: The smooth Fermi piston

Area of Science:

• Statistical mechanics
• Nonlinear dynamics
• Chaos theory

Background:

• The Fermi piston model explores energy transfer between a particle and a moving boundary.
• Understanding particle dynamics under periodic driving forces is crucial in statistical mechanics.
• Previous studies often focused on continuous driving forces and their impact on particle energy.

Purpose of the Study:

• To analyze the behavior of a Fermi piston with a discontinuous, periodic driving force.
• To investigate the mechanisms of phase space segmentation and energy diffusion.
• To identify conditions under which indefinite energy stochastic driving is not possible.

Main Methods:

• Numerical and analytical techniques were employed to study particle dynamics in one spatial dimension.
• The surface-of-section method was used to analyze the phase space (phi(n), v(n)).
• Investigated the role of discontinuity in the driving force F(t) and its time derivative.

Main Results:

• Indefinite stochastic energy increase is not possible except for specific discontinuous driving force cases.
• A novel mechanism for phase space segmentation, distinct from KAM tori, was identified.
• Resonance between the particle's natural period and the driving force's period is key to phase space segmentation.
• For continuous driving functions, diffusion boundaries arise from parabolic fixed points.

Conclusions:

• Discontinuous driving forces in the Fermi piston model prevent indefinite energy gain.
• Resonance-induced segmentation of phase space offers a new perspective on particle dynamics.
• The findings contribute to understanding energy diffusion and phase space structure in driven systems.