团聚-破碎过程中的振荡
Oscillations in aggregation-shattering processes
摘要：
我们观察到，在经历聚合和碰撞控制粉碎的系统中，存在着永无止境的振荡。具体地说，我们研究了具有聚集核K_{i,j} =（i/j）^a+（j/i）^a和破碎核F_{i,j} =λ K_{i,j}的聚集-破碎过程，其中i和j是簇的大小，参数λ量化了破碎的强度。当0<a<1/2时，对于所有的λ值，没有振荡，系统单调地接近稳态;在这个区域，我们得到了稳态团簇尺寸分布的解析解。速率方程的数值解表明，振荡出现在1/2<a<1的范围内。当破碎率足够大时，振荡衰减并最终消失，而当破碎率λ<λ c（a）时，振荡似乎永远存在。因此，在没有粒子汇和粒子源的封闭的聚集-破碎过程中，可以产生无休止的振荡。
英文摘要：
We observe never-ending oscillations in systems undergoing aggregation and collision-controlled shattering. Specifically, we investigate aggregation-shattering processes with aggregation kernels K_{i,j} = (i/j)^a+(j/i)^a and shattering kernels F_{i,j}=λK_{i,j}, where i and j are cluster sizes and parameter λquantifies the strength of shattering. When 0<a<1/2, there are no oscillations and the system monotonically approaches to a steady state for all values of λ; in this region we obtain an analytical solution for the stationary cluster size distribution. Numerical solutions of the rate equations show that oscillations emerge in the 1/2<a<1 range. When the shattering rate is sufficiently large oscillations decay and eventually disappear, while for λ<λ_c(a) oscillations apparently persist forever. Thus never-ending oscillations can arise in closed aggregation-shattering processes without sinks and sources of particles.
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