285 research outputs found
Genuine Non-Self-Averaging and Ultra-Slow Convergence in Gelation
In irreversible aggregation processes droplets or polymers of microscopic
size successively coalesce until a large cluster of macroscopic scale forms.
This gelation transition is widely believed to be self-averaging, meaning that
the order parameter (the relative size of the largest connected cluster)
attains well-defined values upon ensemble averaging with no sample-to-sample
fluctuations in the thermodynamic limit. Here, we report on anomalous gelation
transition types. Depending on the growth rate of the largest clusters, the
gelation transition can show very diverse patterns as a function of the control
parameter, which includes multiple stochastic discontinuous transitions,
genuine non-self-averaging and ultra-slow convergence of the transition point.
Our framework may be helpful in understanding and controlling gelation.Comment: 8 pages, 10 figure
Discontinuous percolation transitions in real physical systems
We study discontinuous percolation transitions (PT) in the diffusion-limited
cluster aggregation model of the sol-gel transition as an example of real
physical systems, in which the number of aggregation events is regarded as the
number of bonds occupied in the system. When particles are Brownian, in which
cluster velocity depends on cluster size as with
, a larger cluster has less probability to collide with other
clusters because of its smaller mobility. Thus, the cluster is effectively more
suppressed in growth of its size. Then the giant cluster size increases
drastically by merging those suppressed clusters near the percolation
threshold, exhibiting a discontinuous PT. We also study the tricritical
behavior by controlling the parameter , and the tricritical point is
determined by introducing an asymmetric Smoluchowski equation.Comment: 5 pages, 5 figure
Percolation Transitions in Scale-Free Networks under Achlioptas Process
It has been recently shown that the percolation transition is discontinuous
in Erd\H{o}s-R\'enyi networks and square lattices in two dimensions under the
Achlioptas Process (AP). Here, we show that when the structure is highly
heterogeneous as in scale-free networks, a discontinuous transition does not
always occur: a continuous transition is also possible depending on the degree
distribution of the scale-free network. This originates from the competition
between the AP that discourages the formation of a giant component and the
existence of hubs that encourages it. We also estimate the value of the
characteristic degree exponent that separates the two transition types.Comment: 4 pages, 6 figure
Finite-size scaling theory for explosive percolation transitions
The finite-size scaling (FSS) theory for continuous phase transitions has
been useful in determining the critical behavior from the size dependent
behaviors of thermodynamic quantities. When the phase transition is
discontinuous, however, FSS approach has not been well established yet. Here,
we develop a FSS theory for the explosive percolation transition arising in the
Erd\H{o}s and R\'enyi model under the Achlioptas process. A scaling function is
derived based on the observed fact that the derivative of the curve of the
order parameter at the critical point diverges with system size in a
power-law manner, which is different from the conventional one based on the
divergence of the correlation length at . We show that the susceptibility
is also described in the same scaling form. Numerical simulation data for
different system sizes are well collapsed on the respective scaling functions.Comment: 5 pages, 5 figure
- …