Statistical investigation and thermal properties for a 1-D impact system with dissipation

TL;DR

This paper investigates the statistical and thermal properties of a 1-D dissipative impact system using numerical, analytical, and probabilistic methods, demonstrating a robust formalism for estimating statistical variables.

Contribution

It introduces a comprehensive approach combining three techniques to analyze a 1-D dissipative impact system, enabling estimation of statistical variables without extensive simulations.

Findings

Analytical and numerical results are consistent, validating the formalism.

The approach allows estimation of temperature and velocity statistics efficiently.

Extension to other dynamical systems, including time-dependent billiards, is feasible.

Abstract

The behavior of the average velocity, its deviation and average squared velocity are characterized using three techniques for a 1-D dissipative impact system. The system -- a particle, or an ensemble of non interacting particles, moving in a constant gravitation field and colliding with a varying platform -- is described by a nonlinear mapping. The average squared velocity allows to describe the temperature for an ensemble of particles as a function of the parameters using: (i) straightforward numerical simulations; (ii) analytically from the dynamical equations; (iii) using the probability distribution function. Comparing analytical and numerical results for the three techniques, one can check the robustness of the developed formalism, where we are able to estimate numerical values for the statistical variables, without doing extensive numerical simulations. Also, extension to other dynamical systems is immediate, including time dependent billiards.