Perturbative Expansions and Critical Phenomena in Random Structured Media

  • Simon GluzmanEmail author
  • Dmitry A. Karpeyev
Conference paper
Part of the Trends in Mathematics book series (TM)


We present constructive solutions for the effective properties for three problems in the field of random structured media. They are all based on truncated series and on a constructive investigation of their behavior near divergence points where the physical percolation or phase transitions occur. (1) Effective conductivity of 2D conductors with arbitrary contrast parameters is reconstructed from the expansion at small concentrations and of the critical behavior at high concentrations. (2) Effective shear modulus of perfectly rigid spherical inclusions randomly embedded into an incompressible matrix is reconstructed given its expansion at small concentrations and critical behavior. In addition, the critical index S of super-elasticity is estimated. (3) We also employ a truncated Fourier expansion to study spontaneous directional ordering in models of planar fully-connected suspensions of active polar particles. The main result is the discovery of a discontinuous, abrupt transition from an ordered to a disordered state. It is a macroscopic effect caused by a mesoscopic self-quenching noise. The relaxation time remains finite at the critical point, therefore the effect of self-quenching is to strongly suppress the critical slowing down and improve the reaction time to external stimuli.


Effective conductivity Effective shear modulus Active suspensions Discontinuous transition 

Mathematics Subject Classification (2010)

Primary 40C 41-06; Secondary 74B 74Q 92B 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsPennsylvania State UniversityState CollegeUSA
  2. 2.TorontoCanada
  3. 3.Computation InstituteUniversity of ChicagoChicagoUSA
  4. 4.Mathematics and Computer Science DivisionArgonne National LaboratoryArgonneUSA
  5. 5.EvanstonUSA

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