Diffusion on hierarchical systems of weakly-coupled networks
Grzegorz Siudem , Janusz Hołyst
AbstractWe analyzed diffusion dynamics on weakly-coupled networks (interconnected networks)by means of separation of time scales. Using an adiabatic approximation we reduced thesystem dynamics to a Markov chain with aggregated variables and derived a transportequation that is analogous to Fick’s First Law and includes a driving force. Entropy produc-tion is a sum of microscopic entropy transport, which results from the particle’s migrationbetween networks of different topologies and macroscopic entropy production of theMarkov chain. Equilibrium particles partition between different sub-networks dependsonly on internal sub-network parameters. By changing structure of networks one can notonly modify diffusion constants but can also induce or reverse the direction of the particles’flow between different networks. Our framework, confirmed by numerical simulations,is also useful for considering diffusion in nested systems corresponding to hierarchicalnetworks with several different time scales thus it can serve to uncover hidden hierarchylevels from observations of diffusion processes.
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