The nonlinear fragmentation equation

We study the kinetics of nonlinear irreversible fragmentation. Here fragmentation is induced by interactions/collisions between pairs of particles, and modelled by general classes of interaction kernels, and for several types of breakage models. We construct initial value and scaling solutions of the fragmentation equations, and apply the ”non-vanishing mass flux ”criterion for the occurrence of shattering transitions. These properties enable us to determine the phase diagram for the occurrence of shattering states and of scaling states in the phase space of model parameters. PACS numbers: 05.20.Dd, 64.60.Ht,61.41.+e, 82.70.-y Submitted to: J. Phys. A: Math. Gen. Fragmentation is a phenomenon of breakup of particles into a range of smaller size particles. It is naturally found in a wide variety of physical systems, ranging from comminution, breakup of grains, bubbles, droplets, polymer degradation, disintegration of atomic nuclei, etc. Fragmentation may occur through external forces, spontaneously, or through interactions/collisions between particles. The subject has been widely studied [1]-[10]. We are mainly interested in collision-induced nonlinear fragmentation as caused by binary interactions. Such systems can be described by the time evolution of c(x, t), which is the number of particles of mass or size x at a given time t, or alternatively by its moments Mn(t) = ∫∞ 0 dxxc(x, t). Quantities with similar properties appear in coagulation processes. In either case the total mass is conserved, M(t) = M1(t) = 1, while the total number of particles, N(t) = M0(t) = ∫ dxc(x, t) is not. In irreversible coagulation, the mean particle mass, s(t) = M/N(t), increases monotonically, and may lead to a finite time singularity at t = tc, the gelation transition, characterized by the appearance of an infinite cluster containing a finite fraction of the total mass, ∆(t) = 1 − M(t) (order parameter), where ∆(t) 6= 0 for t > tc. Alternatively, the gelation transition is characterized by a non-vanishing mass flux ∆̇(t) = −Ṁ(t) from The nonlinear fragmentation equation 2 finite size particles (sol) to the infinite cluster (gel) [11]-[13], i.e. a violation of mass conservation. In irreversible fragmentation the reversed scenario occurs. Here s(t) is monotonically decreasing, while the overall mass is conserved. In these systems a finite time singularity may occur at tc, the shattering transition. It is characterized by a non-vanishing mass flux, ∆̇(t), i.e. the rate at which massive particles are converted into mass-less infinitesimals or fractal dust [1, 2, 3, 7]. If ∆̇(tc) is finite this transition has the character of a continuous phase transition, described by the order parameter ∆(t) = 1−M(t), as in gelation [13]. In case ∆̇(tc) = ∞ all mass instantly ’evaporates’ from the system; the transition is called explosive and also referred to as a first order transition [8]. Smoluchowski’s coagulation fragmentation equation [11] gives the basic mean field description for reversible and irreversible coagulation [2, 3, 13] and fragmentation processes [1]-[10] in terms of the time evolution of c(x, t) in spatially uniform (well stirred) systems. In irreversible fragmentation or coagulation the system is described by a nonlinear coagulation rate, in combination with a spontaneous linear fragmentation rate and/or a collisionor reaction-induced nonlinear fragmentation rate. The system does not reach a steady state, but at asymptotically large times the distribution function c(x, t) approaches under rather general conditions to the standard scaling form, which describes the typical x−dependence around the mean particle size s(t), which is steadily decreasing. The occurrence of shattering has been addressed only partially in the case of collision-induced nonlinear fragmentation. It shows a behavior, qualitatively different from spontaneous (linear) fragmentation. Furthermore, the special cases analyzed so far are not necessarily generic, but appear to be borderline cases. In this letter we study the occurrence of shattering for general classes of fragmentation models within the framework of the nonlinear fragmentation equation and we analyze its peculiarities and point out the parallels with gelation. Collision-induced irreversible fragmentation can be described at the mean field level by the nonlinear fragmentation equation with a collision term I composed of a loss and a gain term[4], ∂c(x, t)/∂t = I(x|c) ≡ −c(x) ∫∞