Memory Stabilizes Spontaneous Particle Aggregation: A Linearized Vlasov--Fokker--Planck Analysis

We perform a linearized stability analysis of the Vlasov--Fokker--Planck equation obtained as the mean-field description of a stochastic spontaneous aggregation model with memory. Memory is represented by a chain of $K$ internal variables. We characterize the spatially homogeneous equilibria and derive a scalar dispersion relation for spatially inhomogeneous perturbations. When all relaxation rates are equal, we show that every Fourier mode that is unstable in the memoryless model possesses a unique critical relaxation parameter: sufficiently long memory stabilizes the mode, whereas it remains unstable for short memory. Although memory may destabilize individual modes associated with negative Fourier coefficients of the sensing kernel, we prove that, for radially symmetric distance-decreasing kernels, it cannot destabilize a homogeneous equilibrium that is stable in the memoryless model. Thus, memory cannot create an overall instability of an otherwise stable homogeneous state, although it may change the set of the unstable modes. We present a numerical example for the normalized top-hat kernel, demonstrating that increasing the effective memory length successively stabilizes the Fourier modes, with higher frequencies being stabilized before lower ones. This is consistent with the coarsening effect observed in recent particle simulations. Finally, under suitable nondegeneracy and regularity assumptions, we use the Crandall--Rabinowitz theorem to obtain branches of spatially inhomogeneous stationary solutions bifurcating from the homogeneous state.