Glass transition in systems without static correlations: a microscopictheory

Abstract

We present a first step toward a microscopic theory for the glass transition insystems with trivial static correlations. As an example we have chosenN infinitely thin hardrods with length L,fixed with their centres on a periodic lattice with lattice constanta. Startingfrom the N-rodSmoluchowski equation we derive a coupled set of equations for fluctuations of reducedk-roddensities. We approximate the influence of the surrounding rods on the dynamics of a pairof rods by introduction of an effective rotational diffusion tensor D(Ω1, Ω2)and in this way we obtain a self-consistent equation for D.This equation exhibits a feedback mechanism leading to a slowing down ofthe relaxation. It involves as an input the Laplace transform υ0 (l/r)at z = 0,l = L/a,of a torque–torque correlator of an isolated pair of rods with distanceR = ar. Ourequation predicts the existence of a continuous ergodicity-breaking transition at a criticallength lc = Lc /a.To estimate the critical length we perform an approximate analytical calculation of υ0 (l/r)based on a variational approach and obtain lcvar ∼ = 5.68,4.84 and 3.96 for an sc, bcc and fcc lattice. We also evaluate υ0 (l/r) numericallyexactly from a two-rod simulation. The latter calculation leads to lcnum ∼ = 3.45,2.78 and 2.20 for the corresponding lattices. Close tolcthe rotational diffusion constant decreases as D(l) ∼ (lc − l)γ withγ = 1and a diverging timescale tϵ ∼ |lc − l|−δ, δ = 2, appears. Onthis timescale the t-and l-dependenceof the one-rod density is determined by a master function depending only on t/tϵ. Incontrast to present microscopic theories our approach predicts a glass transitiondespite the absence of any static correlations.