Memory as activity: pattern formation in a conserved scalar field

We explore the concept of memory in scalar active matter, focusing on the collective dynamics of particles whose interactions depend on their evolutionary history rather than solely on their current configuration. We introduce the idea of an active particle whose velocity includes an active contribution that depends on its past trajectory suitably weighted by a memory kernel. The memory kernel is unrelated to the thermal noise acting on the particle, meaning that the dynamics breaks detailed balance at the microscopic level. We show that the number density of these active particles is described by a Cahn-Hilliard equation, which typically describes passive phase separation, suitably modified to account for this particular non-equilibrium effect. We establish the novel emergent features of the model and use the example of time delayed interactions to highlight its novel pattern-forming abilities through theory and simulations.