Fragmentation with a Steady Source

Fragmentation underlies numerous natural phenomena [1–5]. The quantity being “split” can be the mass, momentum, or the area, and typically, fragments continue splitting independently of each other. Examples include polymer degradation [6], breakup of liquid droplets [7] and atomic nuclei [8], martensitic transformations [9,10], shattering of solid objects [11,12], and meteor impacts. Fragmentation also arises in several topics of computer science [13–15]. The simplest fragmentation models assume that the rate by which fragments are produced is a function of their size only [16–20]. In this study, we focus on the classic “random scission model” where the cutting is uniform and hence a fragment is cut with a rate proportional to its size. In particular, we are interested in situations where the system is subject to a steady input of fragments. Such “open” systems occur in industrial applications where coarse material is continuously fed into a grinding apparatus to produce a fine powder [21–23]. Fragmentation in open systems has received less attention than fragmentation in closed systems [24,25]. We will show, however, that fragmentation with input is actually simpler than the classical counterpart as the system reaches a stationary state which is remarkably robust. Specifically, fragmentation with a steady source is characterized by an algebraic divergence of the size distribution in the small size limit, and this behavior is independent of the particular form of the input. Additionally, the time dependent behavior, obtained analytically for arbitrary inputs, follows a scaling behavior. These two features are shown to be closely related. We start with a one-dimensional fragmentation process subject to constant input of segments. Here “onedimensional” means that the fragments are characterized by a single variable which we shall call “length” (the fragments can be viewed as segments). Let the system be initially empty and intervals whose length is within the range (x, x+ dx) are added with rate f(x) dx. Additionally, intervals are cut with a constant spatially homogeneous rate; we set this rate equal to unity without loss of generality. Fragmentation with input has a natural geometric interpretation. Consider the segments as part of an infinite line. The fragmentation process is equivalent to deposition of point “cracks” on the line. The line is initially “immune” to fragmentation, but then segments of length x become “susceptible” to fragmentation with rate f(x). Hence, fragmentation with input is equivalent to inhomogeneous fragmentation of a growing line. The density P (x, t) of intervals of length x at time t evolves according to the following rate equation