# Mathematics Calendar

Wednesday, November 21, 2018 - 10:45

Weierstraß-Institut

Hausvogteiplatz 11A, 10117 Berlin, Konferenzraum 3.13

Seminar Interacting Random Systems

Experiments have shown that there is a sharp phase transition in polymerization: it takes long time to have small amount of stable polymers and once some amount of stable polymers appear, very quickly all particles are polymerized. Moreover, the lag time usually have a very high variance. We propose a growth-fragmentation model with a critical mass to explain these phenomenon. Particles having a mass less than this critical mass are unstable -- they are fragmented much more quicker than the larger particles. A scaling approach is used, by taking the initial total mass N as a scaling parameter and assuming that the ratio of the unstable fragmentation rates to stable fragmentation rates are of order Phi(N), which is a non-decreasing function of N. We study the time evolution of this infinite dimension process under a certain class of fragmentation distributions. We show that 1) with a proper scaling parameter, the time (T) spent for the stable polymers that generating from small particles is asymptotically exponential distributed; 2) the time for the growth of stable particles has a much smaller order than T. The exponential distribution explains the high variance and the different time scales explain the sharp phase transition. These results are proved via stochastic calculus, estimations for occupation measures on different time scales, some coupling techniques and branching processes. It is a joint work with Philippe Robert.

submitted by vdsand (vandesand@wias-berlin.de, 030 20372559)