Journal of Statistical Physics

, Volume 174, Issue 2, pp 469–493 | Cite as

Fick and Fokker–Planck Diffusion Law in Inhomogeneous Media

  • D. Andreucci
  • E. N. M. CirilloEmail author
  • M. Colangeli
  • D. Gabrielli


We discuss particle diffusion in a spatially inhomogeneous medium. From the microscopic viewpoint we consider independent particles randomly evolving on a lattice. We show that the reversibility condition has a discrete geometric interpretation in terms of weights associated to un–oriented edges and vertices. We consider the hydrodynamic diffusive scaling that gives, as a macroscopic evolution equation, the Fokker–Planck equation corresponding to the evolution of the probability distribution of a reversible spatially inhomogeneous diffusion process. The geometric macroscopic counterpart of reversibility is encoded into a tensor metrics and a positive function. The Fick’s law with inhomogeneous diffusion matrix is obtained in the case when the spatial inhomogeneity is associated exclusively with the edge weights. We discuss also some related properties of the systems like a non–homogeneous Einstein relation and the possibility of uphill diffusion.


Diffusion Fick’s law Fokker–Planck diffusion law Hydrodynamic limit 

Mathematics Subject Classification

35Q84 82C22 82C31 


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • D. Andreucci
    • 1
  • E. N. M. Cirillo
    • 1
    Email author
  • M. Colangeli
    • 2
  • D. Gabrielli
    • 2
  1. 1.Dipartimento di Scienze di Base e Applicate per l’IngegneriaSapienza Università di RomaRomeItaly
  2. 2.Dipartimento di Ingegneria e Scienze dell’Informazione e MatematicaUniversità degli Studi dell’AquilaL’AquilaItaly

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