Comminution is the process of grainsize reduction due to grain crushing, which is common in both natural and industrial problems that involve brittle granular media. Grain crushing is a stochastic process, where the strength of individual grains is determined both by their own size and mineralogy, and the local arrangement of their neighbouring particles, here termed grainsize fabric. The relationship between these effects was described previously using a stochastic lattice model of comminution, whose most general form further includes segregation and mixing. While the stochastic segregation and mixing dynamics have simple analogous differential equations for describing the equivalent continuum behaviour, until now the stochastic comminution dynamics has lacked a smooth representation. Here we resolve this gap by developing a homogeneous continuum model that is based on the same stochastic physics. We show that the new model yields the same results as the stochastic lattice model in the limit of indefinitely large lattices. Given a time varying stress state, the model describes the time-evolution of the grainsize distribution by representing the state of the system in terms of a two-dimensional joint distribution over grainsize and local-average grainsize. The model has a scalable numerical implementation with a simple deterministic rule for time-evolution.

粉碎是由于晶粒破碎而导致的晶粒尺寸减小，这在涉及脆性颗粒介质的自然和工业问题中都很常见。颗粒破碎是一个随机过程，其中单个颗粒的强度取决于其自身的大小和矿物学以及相邻颗粒的局部排列（在此称为颗粒织物）。这些效果之间的关系已使用随机粉碎晶格模型进行了描述，其最一般的形式还包括分离和混合。尽管随机的分离和混合动力学具有简单的类似微分方程式来描述等效的连续性行为，但到现在为止，随机的粉碎动力学还缺乏平滑的表示形式。在这里，我们通过开发基于相同随机物理学的齐次连续谱模型来解决这一差距。我们表明，在无限大晶格的限制下，新模型产生的结果与随机晶格模型相同。给定随时间变化的压力状态，该模型通过在晶粒尺寸和局部平均晶粒尺寸上的二维联合分布来表示系统状态，从而描述了晶粒尺寸分布的时间演变。该模型具有可扩展的数值实现，并带有用于时间演化的简单确定性规则。