# Classical analog to the Airy wave packet

## TL;DR

This paper constructs a classical analog to the quantum accelerating Airy wave packet by solving the Liouville equation for free particles, demonstrating complex behaviors like acceleration in classical ensembles.

## Contribution

It introduces a classical solution to the Liouville equation that mimics quantum Airy wave packet dynamics, including acceleration, and compares it with the quantum case using the Wigner representation.

## Key findings

- Classical ensembles can exhibit acceleration similar to quantum Airy packets.
- The classical solution explains complex wave packet behaviors through velocity distributions.
- Comparison with quantum solutions highlights the analogy between classical and quantum dynamics.

## Abstract

The solution of the Liouville equation for the ensemble of free particles is presented and the classical analog to the quantum accelerating Airy wave packet is constructed and discussed. Considering the motion of various classical packets -- with an infinite and restricted distribution of velocities of particles -- and also the motion of their fronts, we demonstrate in the simplest and most definite way why the packet can display a more sophisticated behavior (even acceleration) as compared with a free individual particle that moves at a fixed velocity. A comparison of this classical solution with the quantum one in the Wigner representation of quantum mechanics which provides the closest analogy is also presented.