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Full-Text Articles in Physical Sciences and Mathematics
Kinetic Theory Of Random Graphs: From Paths To Cycles, E. Ben-Naim, P.L. Krapivsky
Kinetic Theory Of Random Graphs: From Paths To Cycles, E. Ben-Naim, P.L. Krapivsky
Eli Ben-Naim
Structural properties of evolving random graphs are investigated. Treating linking as a dynamic aggregation process, rate equations for the distribution of node to node distances (paths) and of cycles are formulated and solved analytically. At the gelation point, the typical length of paths and cycles, l, scales with the component size k as l ~ k^{1/2}. Dynamic and finite-size scaling laws for the behavior at and near the gelation point are obtained. Finite-size scaling laws are verified using numerical simulations.
The Inelastic Maxwell Model, E. Ben-Naim, P.L. Krapivsky
The Inelastic Maxwell Model, E. Ben-Naim, P.L. Krapivsky
Eli Ben-Naim
Dynamics of inelastic gases are studied within the framework of random collision processes. The corresponding Boltzmann equation with uniform collision rates is solved analytically for gases, impurities, and mixtures. Generally, the energy dissipation leads to a significant departure from the elastic case. Specifically, the velocity distributions have overpopulated high energy tails and different velocity components are correlated. In the freely cooling case, the velocity distribution develops an algebraic high-energy tail, with an exponent that depends sensitively on the dimension and the degree of dissipation. Moments of the velocity distribution exhibit multiscaling asymptotic behavior, and the autocorrelation function decays algebraically with …
Kinetics Of Clustering In Traffic Flows, E. Ben-Naim, P. L. Krapivsky, S. Redner
Kinetics Of Clustering In Traffic Flows, E. Ben-Naim, P. L. Krapivsky, S. Redner
Eli Ben-Naim
We study a simple aggregation model that mimics the clustering of traffic on a one-lane roadway. In this model, each ``car'' moves ballistically at its initial velocity until it overtakes the preceding car or cluster. After this encounter, the incident car assumes the velocity of the cluster which it has just joined. The properties of the initial distribution of velocities in the small velocity limit control the long-time properties of the aggregation process. For an initial velocity distribution with a power-law tail at small velocities, $\pvim$ as $v \to 0$, a simple scaling argument shows that the average cluster size …