Closed Form Single Particle Gas Explorer

Live Script exploring closed-form expressions for a 1D single particle gas compressed by a plunger.

An ideal gas compressed in a cylinder behaves like a spring: collisions of molecules with the walls produce a pressure that rises as the volume falls. The pressure-volume relation encodes the number of molecular degrees of freedom through the adiabatic index. Remarkably, simulation shows that a single particle in these circumstances exhibits the same effective adiabatic index as an ideal gas governed by thermodynamics even though the motion is not ergodic.[1] One way to understand this apparent coincidence is through the notion of an adiabatic invariant in classical mechanics.[2]

This Live Script shows that the motion during adiabatic compression can be evaluated in closed form from which proxies for the pressure and temperature can be calculated confirming simulation results for adiabatic compression. It also examines single particle "isothermal" compression and processes which are neither adiabatic nor isothermal by introducing a coefficient of restitution into the collision kinematics. Adiabatic processes correspond to elastic collisions with unit coefficient of restitution. Isothermal transformations correspond to collisions which leave the particle speed unchanged and that condition is shown to require a speed dependent coefficient of restitution with a value that can exceed one, corresponding to heat transfer from the plunger to the particle. Vanishing restitution coefficient corresponds to a degenerate case where the moving wall simply absorbs any initial kinetic energy immediately.

This script will interest students and instructors of physics and related fields. "Try this" suggestions, coding "Challenges", hyperlinks, and references are supplied for further exploration. Additional educational Live Scripts by the author are here.

The included files SingleParticleGasRestitutionMath.tex and its compiled SingleParticleGasRestitutionMath.pdf step through the analytic results used here.