Journal
JOURNAL OF STATISTICAL PHYSICS
Volume 175, Issue 6, Pages 1311-1330Publisher
SPRINGER
DOI: 10.1007/s10955-019-02280-w
Keywords
Fish schools; Bernstein functions; Complete monotonicity; Heavy tails; Convergence to equilibrium
Categories
Funding
- Hausdorff Center for Mathematics
- CRC 1060 on Mathematics of emergent effects, Universitat Bonn
- National Science Foundation [DMS 1514826, 1812573, DMS 1515400, 1812609]
- Simons Foundation [395796]
- Center for Nonlinear Analysis (CNA) under National Science Foundation PIRE Grant
- NSF Research Network Grant [RNMS11-07444]
- Institut de Mathematiques, Universite Paul Sabatier, Toulouse
- Department of Mathematics, Imperial College London under Nelder Fellowship awards
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1812609, 1812573] Funding Source: National Science Foundation
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In a recent study of certain merging-splitting models of animal-group size (Degond et al. in J Nonlinear Sci 27(2):379-424, 2017), it was shown that an initial size distribution with infinite first moment leads to convergence to zero in weak sense, corresponding to unbounded growth of group size. In the present paper we show that for any such initial distribution with a power-law tail, the solution approaches a self-similar spreading form. A one-parameter family of such self-similar solutions exists, with densities that are completely monotone, having power-law behavior in both small and large size regimes, with different exponents.
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