Ageing in the critical contact process: a Monte Carlo study

Abstract

The long-time dynamics of the critical contact process which is brought suddenly out of an uncorrelated initial state undergoes ageing in close analogy with quenched magnetic systems. In particular, we show through Monte Carlo simulations in one and two dimensions and through mean-field theory that time-translation invariance is broken and that dynamical scaling holds. We find that the autocorrelation and autoresponse exponents λ Γ and λ R are equal but, in contrast to systems relaxing to equilibrium, the ageing exponents a and b are distinct. A recent proposal to define a non-equilibrium temperature through the short-time limit of the fluctuation-dissipation ratio is therefore not applicable.

Ageing in the contact process: scaling behaviour and universal features

“…In table 1 we compare the values of λ R,C obtained from this scaling relation and the available estimates of δ and z to the results of fitting the asymptotic behavior of the scaling functions f C,R [see equations (1) and (2)] from numerical data [8,9]. The results reported in the last three columns are in quite good agreement both for d = 1 and 2.…”

“…(We comment on the meaning of Ξ in subsection 2.2.) In addition, the numerical results of [8,9] support the applicability of the so-called theory of local scale invariance (LSI) [15,16,17] to the ageing behavior of the CP. This theory tries to use local space-time symmetries to constrain the form of f R (t/s) and has been applied to ageing phenomena in magnetic systems quenched from an initial high-temperature state to and below the critical temperature [16,17] and bosonic reaction-diffusion systems [18].…”

“…In particular it was found that λ C /z = 1.85 (10) and λ R /z = 1.85(10) for d = 1 [8,9], whereas λ C /z = 2.75 (10) and λ R /z = 2.8(3) for d = 2 [9]. These results suggest that λ R = λ C independently of the dimensionality d = 1 and 2.…”

“…The directed percolation (DP) universality class has been recently the subject of renewed interest in the context of ageing phenomena [8,9,10,11], which are characterized by two-time quantities (such as response and correlation functions, see below) which are homogeneous functions of their time arguments and do not display the time-translational invariance which is expected in a stationary state. The two-time quantities of interest are the connected density-density correlation function C x−x ′ (t, s) := n x (t)n x ′ (s) − n x (t) n x ′ (s) and the response function R x−x ′ (t, s) := δ n x (t) /δκ x ′ (s)| κ=0 where κ x ′ (s) is the field conjugate to n x ′ (s) -corresponding to a local spontaneous activation rate of the lattice site x ′ at time s -and .…”

“…In [8,9] the ageing behavior of the critical CP was investigated numerically. A system with a homogenous initial particle density (corresponding to the active phase with Λ ≫ Λ c ) was quenched at t = 0 to the critical point Λ = Λ c and R x=0 (t, s) and C x=0 (t, s) were determined, finding the following scaling behavior (t > s) R x=0 (t, s) = s −1−a f R (t/s), C x=0 (t, s) = s −b f C (t/s),…”