Interacting Particle Systems
The (second order) integro-differential equation \[ \begin{array}{rl} \partial_t \rho &= \text{div} \left( D(\rho) \nabla \rho+\sigma(\rho) \nabla V[\rho]\right )\\ V[\rho] & = J * \rho \end{array}\] describes at the macroscopic scale the phase segregation in a gas whose particles are interacting at long ranges, as shown by Giacomin and Lebowitz [1]. This equation not only arises as the limit of the microscopic system but the approximation is good enough to capture both qualitative and quantitative phenomena of the microscopic system [1]. More concretely, the above equation arises as the hydrodynamic limit of an interacting particle system evolving by the so called Kawasaki dynamics. Giacomin and Lebowitz also note that unlike the standard Cahn-Hilliard equation, the above integro-differential equation has been shown rigorously to arise as the macroscopic limit of a microscopic model of interacting particles [1].