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1.
Smoluchowski's coagulation equation for irreversible aggregation with constant kernel is considered in its discrete version wherec
t
=c
1
(t) is the concentration ofl-particle clusters at timet. We prove that for initial data satisfyingc
1(0)>0 and the condition 0 c
l
(0) <A (1+)-l
(A >0), the solutions behave asymptotically likec
1
(t)t
–2c(lt–1) ast withlt
–1 kept fixed. The scaling function c() is (1/gr), where
, a conserved quantity, is the initial number of particles per unit volume. An analous result is obtained for the continuous version of Smoluchowski's coagulation equation
wherec(v, t) is the oncentration of clusters of sizev. 相似文献
2.
Dirk Helbing 《The European Physical Journal B - Condensed Matter and Complex Systems》2009,67(3):345-356
A local agglomeration of cooperators can support the survival or spreading of cooperation, even when cooperation is predicted
to die out according to the replicator equation, which is often used in evolutionary game theory to study the spreading and
disappearance of strategies. In this paper, it is shown that success-driven motion can trigger such local agglomeration and
may, therefore, be used to supplement other mechanisms supporting cooperation, like reputation or punishment. Success-driven
motion is formulated here as a function of the game-theoretical payoffs. It can change the outcome and dynamics of spatial
games dramatically, in particular as it causes attractive or repulsive interaction forces. These forces act when the spatial
distributions of strategies are inhomogeneous. However, even when starting with homogeneous initial conditions, small perturbations
can trigger large inhomogeneities by a pattern-formation instability, when certain conditions are fulfilled. Here, these instability
conditions are studied for the prisoner’s dilemma and the snowdrift game. Furthermore, it is demonstrated that asymmetrical
diffusion can drive social, economic, and biological systems into the unstable regime, if these would be stable without diffusion. 相似文献