Emergence of fluctuation relations in UNO

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>W</a:mi></a:math> needed for one player's deck to change from <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>x</b:mi></b:math> to <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>y</c:mi></c:math> 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 <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>W</d:mi></d:math> 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>x</e:mi><e:mo>→</e:mo><e:mi>y</e:mi></e:mrow></e:math>. Our paper contributes to extending the scope of fluctuation theorems beyond their usual thermodynamic setting.