Pattern formation with repulsive soft-core interactions: discrete   particle dynamics and Dean-Kawasaki equation
  Brownian particles interacting via repulsive soft-core potentials can spontaneously aggregate, despite repelling each other, and form periodic crystals of particle clusters. We study this phenomenon in low-dimensional situations (one and two dimensions) at two levels of description: performing numerical simulations of the discrete particle dynamics, and by linear and nonlinear analysis of the corresponding Dean-Kawasaki equation for the macroscopic particle density. Restricting to low dimensions and neglecting fluctuation effects we gain analytical insight into the mechanisms of the instability leading to clustering which turn out to be the interplay between diffusion, the intracluster forces and the forces between neighboring clusters. We show that the deterministic part of the Dean-Kawasaki equation provides a good description of the particle dynamics, including width and shape of the clusters, in a wide range of parameters, and analyze with weakly nonlinear techniques the nature of the pattern-forming bifurcation in one and two dimensions. Finally, we briefly discuss the case of attractive forces. 
Jean-Baptiste Delfau, H\'el\`ene Ollivier, Crist\'obal L\'opez, Bernd   Blasius and Emilio Hern\'andez-Garc\'ia
cond-mat.stat-mech nlin.PS