Ageing in one-dimensional coagulation–diffusion processes and the Fredrickson–Andersen model

Abstract

We analyse the ageing dynamics of the one-dimensional Fredrickson–Andersen (FA) model in the nonequilibrium regime following a low temperature quench. Relaxation then effectively proceeds via diffusion-limited pair coagulation (DLPC) of mobility excitations. By employing a familiar stochastic similarity transformation, we map exact results from the free fermion case of diffusion-limited pair annihilation to DLPC. Crucially, we are able to adapt the mapping technique to averages involving multiple time quantities. This relies on knowledge of the explicit form of the evolution operators involved. Exact results are obtained for two-time correlation and response functions in the free fermion DLPC process. The corresponding long-time scaling forms apply to a wider class of DLPC processes, including the FA model. We are thus able to exactly characterize the violations of the fluctuation–dissipation theorem (FDT) in the ageing regime of the FA model. We find nontrivial scaling forms for the fluctuation–dissipation ratio (FDR) X = X(tw/t), but with a negative asymptotic value X∞ = −3π/(6π − 16) ≈ −3.307. While this prevents a thermodynamic interpretation in terms of an effective temperature, it is a direct consequence of probing FDT with observables that couple to activated dynamics. The existence of negative FDRs should therefore be a widespread feature in non-mean-field systems.