  Information storage is a key element of autonomous, out-of-equilibrium
dynamics, especially for biological and synthetic active matter. In synthetic
active matter however, the implementation of internal memory in self-propelled
systems is often absent, limiting our understanding of memory-driven dynamics.
Recently, a system comprised of a droplet generating its guiding wavefield
appeared as a prime candidate for such investigations. Indeed, the wavefield,
propelling the droplet, encodes information about the droplet trajectory and
the amount of information can be controlled by a single scalar experimental
parameter. In this work, we show numerically and experimentally that the
accumulation of information in the wavefield induces the loss of time
correlations, where the dynamics can then be described by a memory-less
process. We rationalize the resulting statistical behaviour by defining an
effective temperature for the particle dynamics where the wavefield acts as a
thermostat of large dimensions, and by evidencing a minimization principle of
the generated wavefield.
