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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.
Stationary States And Energy Cascades In Inelastic Gases, E. Ben-Naim, J. Machta
Stationary States And Energy Cascades In Inelastic Gases, E. Ben-Naim, J. Machta
Eli Ben-Naim
We find a general class of nontrivial stationary states in inelastic gases where, due to dissipation, energy is transfered from large velocity scales to small velocity scales. These steady-states exist for arbitrary collision rules and arbitrary dimension. Their signature is a stationary velocity distribution f(v) with an algebraic high-energy tail, f(v) ~ v^{-sigma}. The exponent sigma is obtained analytically and it varies continuously with the spatial dimension, the homogeneity index characterizing the collision rate, and the restitution coefficient. We observe these stationary states in numerical simulations in which energy is injected into the system by infrequently boosting particles to high …