Phase Transition and Diffusion in an Adiabatic Bouncer Model

Abstract: This study examines anomalous diffusion and dynamical phase transitions in a nonlinear bouncer model with short-range interactions leading to velocity-dependent (adiabatic) collisions. By varying a control parameter, transitions between superdiffusion, normal diffusion, subdiffusion, and complete dynamical freezing are observed. The phase behavior is characterized via scaling laws and critical exponents, providing a robust framework for understanding the underlying dynamics.

• The paper demonstrates that modulating the adiabatic parameter transitions diffusion regimes from superdiffusion to subdiffusion and eventually to a dynamically frozen state.
• The model employs modified discrete mapping equations to capture finite-time, velocity-dependent collisions that reflect realistic particle-surface interactions.
• Key results include scaling laws for mean square velocity growth that align with theoretical predictions across chaotic and localized dynamical behaviors.

Analysis of Phase Transition and Diffusion in an Adiabatic Bouncer Model

This paper explores the phenomena of anomalous diffusion and dynamical phase transitions within a nonlinear bouncer model characterized by short-range interactions leading to velocity-dependent, or adiabatic, collisions. It systematically analyzes the changes in diffusion behaviors through variation of a control parameter, offering insight into the transitions between superdiffusion, normal diffusion, subdiffusion, and complete dynamic freezing.

Model and Methodology

The study uses a modified bouncer model, which traditionally involves a particle colliding elastically with an oscillating surface under the influence of gravity. Here, the model is augmented with a velocity-dependent restitution characteristic that simulates molecular-scale interactions, providing a more physically relevant framework for investigating energy transport phenomena. The model's innovation lies in accounting for finite-time interactions influencing collision outcomes, as opposed to idealized, instantaneous interactions. Such effects are quantified by incorporating an adiabatic parameter that modulates the particle-surface interaction.

The mathematical foundation of this model is laid through discrete mapping equations derived from the classical bouncer framework, extended here to consider the influence of the interaction duration and effect on velocity updates. The paper employs rigorous analytical techniques to derive maps governing the transition between chaotic and localized dynamics, characterized by the interplay between continuous control parameters.

Key Findings

The main result is that the nature of diffusion can be tuned by modulating the adiabatic parameter. Scaling laws describe the system's behavior in terms of the mean square velocity (MSV) growth rate: normal diffusion ($\mu = 1$) aligns with linear growth, while superdiffusion ($\mu > 1$) and subdiffusion ($\mu < 1$) denote deviations above or below this baseline. Numerically, the paper verifies these results against theoretical predictions, capturing subdiffusive regimes, normal diffusive behaviors, and superdiffusion, eventually culminating in a frozen state when parameters reach a critical threshold.

Of particular note is the introduction of a dynamical trapping criterion at low velocities, providing a novel insight into how the system transitions into a dynamically frozen phase. This frozen state is attributed to insufficient perturbation under critical parameter conditions, preventing any substantial change in a particle's trajectory.

Implications and Future Directions

This research illustrates the intricate relationship between control parameter variations and resulting diffusion behaviors, with the potential to generalize these findings across other nonlinear dynamical systems exhibiting adiabatic interactions. The theoretical underpinning of the phase transition also suggests possible applications in fields exploring thermodynamic parallels, such as heat transfer and gas dynamics.

Looking forward, this work paves the way for further exploration in time-dependent systems, nonlinear maps, and their diffusive characteristics. It suggests the investigation of multidimensional extensions or the inclusion of stochastic elements and external forces to examine the robustness of diffusion dynamics and phase transitions in more complex, realistic systems.

In sum, the paper makes a substantial contribution to the understanding of phase transitions and diffusion in nonlinear dynamical systems and opens avenues for developing adiabatically controlled transport processes, potentially impacting physics and material science studies into diffusion phenomena at various scales.