Models.

Dynamical urn models, such as the Ehrenfest model, have played an important role in the early days of statistical mechanics. Dynamical many-urn models generalize the former models in two respects: the number of urns is macroscopic, and thermal effects are included. These many-urn models are exactly solvable in the mean-field geometry. They allow analytical investigations of the characteristic features of nonequilibrium dynamics referred to as aging, including the scaling of correlation and response functions in the two-time plane and the violation of the fluctuation-dissipation theorem. This review paper contains a general presentation of these models, as well as a more detailed description of two dynamical urn models, the backgammon model and the zeta urn model. Prologue Urns containing balls, just as dice or playing cards, are ubiquitous in writings on probability theory, reminding us that this branch of mathematics owes its early developments to practical questions arising in playing games. Dynamical urn models, such as the Ehrenfest model, have played an important role in the elucidation of conceptual problems in statistical mechanics. More recently, dynamical many-urn models have been investigated in the mean-field geometry. These exactly solvable models exhibit characteristic features of nonequilibrium dynamics referred to as aging, such as the scaling of the correlation and response functions in the two-time plane and the violation of the fluctuation-dissipation theorem. This paper contains a didactic introduction to urn models (Section 1), a presentation of static and dynamical properties of many-urn models (Section 2), a reminder of the main characteristic features of aging (Section 3), and an overview of recent results on two dynamical urn models, namely the backgammon model (Section 4) and the zeta urn model (Section 5). 1. Urn models 1.1. The Ehrenfest urn model This model was devised by P and T Ehrenfest in 1907 [1] in their attempt to critically review Boltzmann's H−theorem. It is defined as follows: N balls are distributed in