Exact relations between elastic and electrical response of d-dimensional percolating networks with angle-bending forces

Abstract

Arguments are presented to demonstrate that exact equality relations exist between the critical exponents which characterize the macroscopic conductivity σ e and the macroscopic elastic stiffness moduli C e of percolating systems of any dimensionality. Using the notation σ e ∝Δp t , C e ∝Δp T for the critical behavior of a randomly diluted system slightly above the percolation threshold p c , (Δp≡p−p c >0) and σ e ∝|Δp|−s , C e ∝|Δp|−S for the critical behavior of a random mixture of normal and perfectly conducting or normal and perfectly rigid constituents slightly below that threshold, (Δp≡p−p c <0) we show that T=t+2ν and S=s, where ν is the percolation correlation length critical exponent ξ∝|Δp|−ν (Δp≷0).