The transition from a microscopic model for the movement of many particles to a macroscopic continuum model for a density flow is studied. The microscopic model for the free flow is completely deterministic, described by an interaction potential that leads to a coherent motion where all particles move in the same direction with the same speed known as a flock. Interaction of the flock with boundaries, obstacles and other flocks leads to a temporary destruction of the coherent motion that macroscopically can be modeled through density dependent diffusion. The resulting macroscopic model is an advection-diffusion equation for the particle density whose diffusion coefficient is density dependent. Examples describing ⅰ) the interaction of material flow on a conveyor belt with an obstacle that redirects or restricts the material flow and ⅱ) the interaction of flocks (of fish or birds) with boundaries and ⅲ) the scattering of two flocks as they bounce off each other are discussed. In each case, the advection-diffusion equation is strictly hyperbolic before and after the interaction while the interaction phase is described by a parabolic equation. A numerical algorithm to solve the advection-diffusion equation through the transition is presented.

中文翻译：

---

粒子系综与边界相互作用的密度相关扩散模型

研究了从许多粒子运动的微观模型到密度流的宏观连续模型的转变。自由流动的微观模型是完全确定的，由相互作用势描述，导致相干运动，其中所有粒子以相同的速度沿相同方向移动，称为群。羊群与边界、障碍物和其他羊群的相互作用导致连贯运动的暂时破坏，宏观上可以通过密度相关扩散建模。由此产生的宏观模型是粒子密度的平流扩散方程，其扩散系数与密度有关。描述 ⅰ) 传送带上的物料流与改变方向或限制物料流的障碍物的相互作用和 ⅱ) 鸡群（鱼或鸟）与边界的相互作用和 ⅲ) 两个鸡群反弹时的散射互相讨论。在每种情况下，对流扩散方程在相互作用之前和之后都是严格双曲线的，而相互作用阶段由抛物线方程描述。提出了一种通过过渡求解对流扩散方程的数值算法。