Physical ageing from generalised time-translation-invariance

TL;DR

The paper proposes a unified framework for classical ageing by introducing a generalised time-translation-invariance via a representation shift $X_n \mapsto X_n = e^{\xi \ln t} X_n^{\mathrm{equi}} e^{-\xi \ln t}$. This leads to explicit scaling forms for two-time observables and a consistent set of exponents, including the equality $\lambda_C=\lambda_R$, and extends the Janssen-Schaub-Schmittmann relation to all $T\le T_c$. It also derives finite-size scaling laws for both local and global observables, and provides a criterion for when nonlinear terms become irrelevant in the long-time dynamics. The framework is illustrated through phase-ordering kinetics and non-equilibrium critical dynamics, with results matching known exact solutions and numerical data, and it emphasizes potential experimental relevance through measurable plateaux and finite-size effects. Overall, the work offers a coherent, symmetry-based route to derive and connect the hallmark features of ageing across temperatures below criticality and various geometries.

Abstract

A generalised form of time-translation-invariance permits to re-derive the known generic phenomenology of ageing, which arises in classical many-body systems after a quench from an initially disordered system to a temperature $T\leq T_c$, at or below the critical temperature $T_c$. Generalised time-translation-invariance is obtained, out of equilibrium, from a change of representation of the Lie algebra generators of the dynamical symmetries of scale-invariance and time-translation-invariance. Observable consequences include the algebraic form of the scaling functions for large arguments of the two-time auto-correlators and auto-responses, the equality of the auto-correlation and the auto-response exponents $λ_C=λ_R$, the cross-over scaling form for an initially magnetised critical system and the explanation of a novel finite-size scaling if the auto-correlator or auto-response converge for large arguments $y=t/s\gg 1$ to a plateau. For global two-time correlators, the time-dependence involving the initial critical slip exponent $Θ$ is confirmed and is generalised to all temperatures below criticality and to the global two-time response function, and their finite-size scaling is derived as well. This also includes the time-dependence of the squared global order-parameter. The celebrate Janssen-Schaub-Schmittmann scaling relation with the auto-correlation exponent is thereby extended to all temperatures below the critical temperature. A simple criterion on the relevance of non-linear terms in the stochastic equation of motion is derived, taking the dimensionality of couplings into account. Its applicability in a wide class of models is confirmed, for temperatures $T\leq T_c$. Relevance to experiments is also discussed.

Figures (6)

• Figure 1: Schematic free energy of a simple ferromagnet (a) before a temperature quench and (b) after such a quench, either onto $T=T_c$ or else into $T<T_c$. The circle represents the system's disordered initial state.
• Figure 2: Physical ageing illustrated through the characteristic data collapses. Panel (a) shows a typical behaviour of a single-time correlator $C(s;r)$ for different times $s_1<s_2<s_3$, which collapse in (b) onto a single curve when distances $r=|\boldsymbol{r}|$ are measured in units of the dynamical length scale $\ell(s)$. Similarly, panel (c) illustrates the two-time auto-correlator $C(s+\tau,s)$ in dependence of $\tau=t-s$, for different waiting times $s_1<s_2<s_3$ which collapse in (d) when replotted as a function of $y=t/s$. The inset shows the asymptotic power-law form $f_C(y)\sim y^{-\lambda/z}$.
• Figure 3: Finite-size scaling of the two-time auto-correlator $C(ys,s;N^{-1})$ in a fully finite volume (for phase-ordering with $b=0$). (a) Change of the plateau height $C_{\infty}^{(2)}$ with varying $N$ for fixed waiting time $s$. (b) Change of the plateau height $C_{\infty}^{(2)}$ with varying $s$ for fixed size $N$. The dashed line gives the infinite-system auto-correlator with its asymptotic power-law $C(ys,s)\sim y^{-\lambda/\mathpzc{z}}$.
• Figure 4: Time-dependence of the non-equilibrium magnetisation $m(t)$.
• Figure D1: Auto-response function of the $3D$ spherical model, for (a) the fully finite case $d^*=3$ and (b) the case of a plaque with $d^*=1$. The parameters $N=[8,16,32]$ from top to bottom and $s=10$ are used. The dashed line indicates the infinite-system auto-response, with the asymptotics $R(ys,s)\sim y^{-\lambda/\mathpzc{z}}$. See eq. (\ref{['D7']}) for the decay of $R$ when $d^*<d$.