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

• The dynamics of a 1-D dissipative impact system is studied.
• Statistical properties for the average velocity, root mean square velocity and its deviation are characterized.
• A thermodynamics formalism is developed as function of the statistical variable.
• The formalism shows itself to be robust, and we can predict numerical values without doing numerical simulations.

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.