Metastability in the mixing/demixing of two species with reciprocally concentration-dependent diffusivity

Alexander Neiman^1*Xiaochen Dong²Benjamin Lindner^2*

• ¹Ohio University, Athens, Ohio, United States
• ²Bernstein Center for Computational Neuroscience, Humboldt University of Berlin, Berlin, Berlin, Germany

It has been shown before that two species of diffusing particles can separate from each other by the mechanism of reciprocally concentration-dependent diffusivity: the presence of one species amplifies the diffusion coefficient of the respective other one, causing the two densities of particles to separate spontaneously. In a minimal model, this could be observed with a quadratic dependence of the diffusion coefficient on the density of the other species. Here, we consider a more realistic sigmoidal dependence as a logistic function on the other particle's density averaged over a finite sensing radius. The sigmoidal dependence accounts for the saturation effects of the diffusion coefficients, which cannot grow without bounds. We show that sigmoidal (logistic) cross-diffusion leads to a new regime in which a homogeneous disordered (well-mixed) state and a spontaneously separated ordered (demixed) state coexist, forming two long-lived metastable configurations. In systems with a finite number of particles, random fluctuations induce repeated transitions between these two states. By tracking an order parameter that distinguishes mixed from demixed phases, we measure the corresponding mean residence in each state and demonstrate that one lifetime increases and the other decreases as the logistic coupling parameter is varied. The system thus displays typical features of a first-order phase transition, including hysteresis for large particle numbers. In addition, we compute the correlation time of the order parameter and show that it exhibits a pronounced maximum within the bistable parameter range, growing exponentially with the total particle number.