Impact of memory on clustering in spontaneous particle aggregation

The effect of short-term and long-term memory on spontaneous aggregation of organisms is investigated using a stochastic agent-based model. Each individual modulates the amplitude of its random motion according to the perceived local density of neighbors. Memory is introduced via a chain of $K$~internal variables that allow agents to retain information about previously encountered densities. The parameter $K$ controls the effective length of memory. A formal mean-field limit yields a macroscopic Fokker--Planck equation, which provides a continuum description of the system in the large-population limit. Steady states of this equation are characterized to interpret the emergence and morphology of clusters. Systematic stochastic simulations in one- and two-dimensional spatial domains reveal that short- or moderate-term memory promotes coarsening, resulting in a smaller number of larger clusters, whereas long-term memory inhibits aggregation and increases the proportion of isolated individuals. Statistical analysis demonstrates that extended memory reduces the agents' responsiveness to environmental stimuli, explaining the transition from aggregation to dispersion as $K$ increases. These findings identify memory as a key factor controlling the collective organization of self-driven agents and provide a bridge between individual-level dynamics and emergent spatial patterns.