  We consider the Rudvalis card shuffle and some of its variations that were
introduced by Diaconis and Saloff-Coste in \cite{symmetrized}, and we project
them to some stochastic interacting particle system. For the latter, we derive
the hydrodynamic limits and we study the equilibrium fluctuations. Our results
show that, for these shuffles, when we consider an asymmetric variation of the
Rudvalis shuffle, the hydrodynamic limit in Eulerian scale is given in terms of
a transport equation with a constant that depends on the exchange rates of the
system; while for symmetric and weakly asymmetric variations of this shuffle,
in diffusive time scale, the evolution is given by the solution of a martingale
problem.
