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Preferential growth: exact solution of the time-dependent distributions
1Department of Theoretical Physics, Institute of Physics, Technical University of Budapest, Budafoki út 8, H-1111 Budapest, Hungary.
Summary
This study models preferential growth, where new particles join clusters based on size probability. We precisely calculated cluster size probabilities and analyzed scaling properties.
Area of Science:
- Physics
- Statistical Mechanics
- Complex Systems
Background:
- Preferential attachment models describe systems where new elements are more likely to connect to larger existing components.
- Understanding cluster dynamics is crucial in various fields, including network science and materials science.
Purpose of the Study:
- To analyze a preferential growth model for particle clusters.
- To derive exact probabilities for cluster sizes over time.
- To investigate the asymptotic and scaling behaviors of cluster size distributions.
Main Methods:
- A probabilistic model of particle addition to clusters was defined.
- Exact calculation of the probability distribution P(i)(k,t) for the size of the ith cluster at time t.
- Asymptotic analysis and scaling property investigation.
Main Results:
- Exact probabilities for cluster sizes were derived.
- Asymptotic behaviors of cluster size distributions were analyzed.
- Scaling properties of the size distribution and mean cluster size were determined.
Conclusions:
- The study provides an exact mathematical framework for preferential cluster growth.
- The findings offer insights into the scaling laws governing such systems.
- Connections between this model and contemporary network models were established.