When chaos meets thermodynamics: from Fermi acceleration to thermalisation

A particle colliding with a moving wall provides one of the simplest mechanical settings in which a small system can exchange energy with its surroundings. In a chaotic time-dependent billiard, repeated elastic impacts may produce Fermi acceleration, with the characteristic kinetic energy growing without bound. This result poses an instructive thermodynamic puzzle: a physical wall maintained at finite temperature should act as a reservoir and drive a dilute gas toward a stationary temperature rather than heat it indefinitely. We use this puzzle to connect nonlinear dynamics with elementary thermodynamics. Under suitable decorrelation assumptions, collision sequences generate effective diffusion in velocity space and the square-root growth associated with normal diffusion. We then introduce two ingredients absent from the conservative accelerator: phase randomisation, which simplifies the statistics of energy exchange, and dissipation, which transfers energy back to unresolved degrees of freedom of the boundary. Their competition replaces unbounded acceleration by a stationary kinetic-energy scale. A recurrence for the second velocity moment yields a stationary kinetic-temperature scale $T_\infty=(mA^2/8k_B)(1+γ)/(1-γ)$, where $A$ measures the boundary forcing and $γ$ is a restitution coefficient. We compare relaxation in collision number with a simple Fourier-inspired heat-transfer model in physical time, emphasising that the two clocks are not interchangeable during transients. The resulting picture shows how fluctuations, dissipation, coarse graining and relaxation can produce an effective thermal state from chaotic dynamics while also clarifying the limits of the thermodynamic analogy.