Emergence of Fluctuation Relations in UNO

TL;DR

This paper demonstrates that fluctuation relations, typically found in thermodynamics, also appear in the card game UNO, revealing non-Markovian effects and extending the scope of fluctuation theorems beyond physics.

Contribution

It reports the first observation of fluctuation relations in a non-thermodynamic, non-physical process, specifically in the game of UNO, highlighting non-Markovian dynamics.

Findings

W obeys a fluctuation relation similar to Crooks' theorem

Deviations indicate non-Markovianity and finite bath effects

Temperature parameter varies with state transition

Abstract

In the last two decades, fluctuation theorems have been proved formally and demonstrated experimentally for several variables (such as entropy production, work, or flux) and different noises causing the fluctuations (of either thermal or other origin; Markovian or non-Markovian). Here we report the observation of a detailed fluctuation relation in a statistical process outside thermodynamics and physics: the card game UNO. As the fluctuating variable, we consider the number of steps $W$ needed for one player's deck to change from $x$ to $y$ number of cards. The other players and the remaining cards play the role of a finite non-Markovian bath. Numerical simulations of runs of the game show that $W$ obeys a fluctuation relation analogous to Crooks' theorem. While the observed behavior shares some common features with infinite random walks, it also exhibits deviations that are clear signatures of non-Markovianity and the finiteness of the bath: Notably, the parameter corresponding to temperature depends strongly on the transition $x\rightarrow y$. Our paper contributes to extending the scope of fluctuation theorems beyond their usual thermodynamic setting.