Non-equilibrium phase transition and cluster size distribution in aggregation and weighted-fragmentation processes. (English) Zbl 07562485
Summary: Through numerical simulations, we investigate the influence of an index \(\phi\) on the cluster size distribution in aggregation and weighted-fragmentation processes where a decomposed cluster of size \(a\) is selected with a power-law weight \(W(a) \propto a^\phi\). We reveal the existence of a non-equilibrium phase transition from an exponentially decayed phase to a partially condensed phase and calculate the threshold \(\phi_c\) for this phase transition. For \(\phi > \phi_c\), cluster size distribution asymptotically obeys exponential decay, which results in the largest cluster sized on the order of \(\ln N\), where \(N\) is a total cluster number. Exponential decay rates are computed as a function of \(\phi\). It is notable that the curvature of the distribution changes its sign at \(\phi = 1\). For \(\phi < \phi_c\), a partially condensed phase takes place and one big cluster sized on the order of a total monomer number \(M\) emerges and coexists with small clusters.
MSC:
| 82-XX | Statistical mechanics, structure of matter |
References:
| [1] | Schmelzer, J.; Röpke, G.; Mahnke, R., Aggregation Phenomena in Complex Systems (1999), Wiley-VCH: Wiley-VCH Weinheim · Zbl 0989.00003 |
| [2] | Leyvraz, F., Scaling theory and exactly solved models in the kinetics of irreversible aggregation, Phys. Rep., 383, 2-3, 95-212 (2003) |
| [3] | Leyvraz, F., Exact asymptotic solution of an aggregation model with a bell-shaped distribution, Phys. Rev. E, 103, 2 (2021) |
| [4] | Vigil, R. D.; Ziff, R. M.; Lu, B., New universality class for gelation in a system with particle breakup, Phys. Rev. B, 38, 942-945 (1988), URL https://link.aps.org/doi/10.1103/PhysRevB.38.942 |
| [5] | Barrow, J. D., Coagulation with fragmentation, J. Phys. A: Math. Gen., 14, 3, 729-733 (1981), URL https://doi.org/10.1088/0305-4470/14/3/019 · Zbl 0454.70020 |
| [6] | Blatza, P. J.; Tobolsky, A., Note on the kinetics of systems manifesting simultaneous polymerization-depolymerization phenomena, J. Phys. Chem., 49, 2, 77 (1945) |
| [7] | Pöschel, T.; Brilliantov, N.; Frömmel, C., Kinetics of prion growth, Biophys. J., 85, 6, 3460-3474 (2003) |
| [8] | Wattis, J. A.D., An introduction to mathematical models of coagulation-fragmentation processes: A discrete deterministic mean-field approach, Physica D, 222, 1, 1-20 (2006) · Zbl 1113.35145 |
| [9] | Brilliantov, N. V.; Bodrova, A. S.; Krapivsky, P. L., A model of ballistic aggregation and fragmentation, J. Stat. Mech. Theory Exp., 2009, 06, P06011 (2009) |
| [10] | Rotstein, H. G., Cluster-size dynamics: A phenomenological model for the interaction between coagulation and fragmentation processes, J. Chem. Phys., 142, 22, Article 224101 pp. (2015) |
| [11] | Brilliantov, N.; Krapivsky, P.; Bodrova, A.; Spahn, F.; Hayakawa, H.; Stadnichuk, V.; Schmidt, J., Size distribution of particles in Saturn‘s rings from aggregation and fragmentation, Proc. Natl. Acad. Sci. USA, 112, 31, 9536-9541 (2015) |
| [12] | Yamamoto, H.; Ohtsuki, T.; Fujihara, A.; Tanimoto, S.; Yamamoto, K.; Miyazima, S., Asymptotic analysis of the model for distribution of high-tax payers, Japan J. Indust. Appl. Math., 24, 211-218 (2007) · Zbl 1151.91660 |
| [13] | Yamamoto, H.; Ohtsuki, T.; Fujihara, A., Double power-law in aggregation-chipping processes, Phys. Rev. E, 77, Article 061122 pp. (2008) |
| [14] | Yamamoto, H.; Ohtsuki, T., Universal power-law and partial condensation in aggregation-chipping processes, Phys. Rev. E, 81, Article 061116 pp. (2010) |
| [15] | Yamamoto, H.; Sekiyama, M.; Ohtsuki, T., Asymptotic power-law index in aggregation and weighted-chipping processes, J. Phys. Soc. Japan, 84, Article 054003 pp. (2015) |
| [16] | Sekiyama, M.; Ohtsuki, T.; Yamamoto, H., Analytical solution of Smoluchowski equations in aggregation?fragmentation processes, J. Phys. Soc. Japan, 86, Article 104003 pp. (2017) |
| [17] | Pego, R. L.; Velázquez, J. J.L., Temporal oscillations in Becker-Döring equations with atomization, Nonlinearity, 33, 4, 1812-1846 (2020) · Zbl 1444.34065 |
| [18] | Niethammer, B.; Pego, R. L.; Schlichting, A.; Velázquez, J. J.L., Oscillations in a Becker-Döring model with injection and depletion (2021), arXiv:2102.06751 |
| [19] | Brilliantov, N. V.; Otieno, W.; Matveev, S. A.; Smirnov, A. P.; Tyrtyshnikov, E. E.; Krapivsky, P. L., Steady oscillations in aggregation-fragmentation processes, Phys. Rev. E, 98, Article 012109 pp. (2018), URL https://link.aps.org/doi/10.1103/PhysRevE.98.012109 |
| [20] | Budzinskiy, S. S.; Matveev, S. A.; Krapivsky, P. L., Hopf bifurcation in addition-shattering kinetics, Phys. Rev. E, 103, 4 (2021) |
| [21] | Kalinov, A.; Osinsky, A. I.; Matveev, S. A.; Otieno, W.; Brilliantov, N. V., Direct simulation Monte Carlo for new regimes in aggregation-fragmentation kinetics (2021), arXiv:2103.09481 |
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