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arXiv:2504.16857 · Nuclear Physics B

Physical ageing from generalised time-translation-invariance

Malte Henkel

Published May 27, 2025• 8 citations

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 ≤ 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 ≫ 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 ≤ T c . Relevance to experiments is also discussed.

DOI: 10.1016/j.nuclphysb.2025.116968

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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 ≤ 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 ≫ 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 ≤ T c . Relevance to experiments is also discussed.

Systems undergoing phase-ordering kinetics after a quench into the ordered phase with 0 < T < T c from a fully disordered initial state and with a non-conserved order-parameter have the dynamical exponent z = 2 . The long-time behaviour of their single-time and two-time correlators, determined by the noisy initial conditions, is derived from Schrödinger-invariance. It is shown that the generic ageing scaling forms of the single-time and two-time correlators follow from the Schrödinger-covariance of the four-point response functions. The autocorrelation exponent λ is related to the passage exponent ζ p which describes the time-scale for the cross-over into the ageing regime. Both Porod’s law and the bounds d / 2 ≤ λ ≤ d are reproduced in a simple way. The dynamical finite-size scaling in fully finite systems is established. The dynamical scaling of global correlators are found and the low-temperature generalisation λ = d − 2 Θ of the Janssen-Schaub-Schmittmann scaling relation is derived.

For ageing systems, quenched onto a critical temperature

T=T_c

such that the dominant noise comes from the thermal bath, with a non-conserved order-parameter and in addition with dynamical exponent

{z}=2

, the form of the two-time auto-correlator as well as the time-space form of the single-time correlator are derived from Schr\"odinger-invariance, generalised to non-equilibrium ageing. These findings reproduce the exact results in the

1D

Glauber-Ising model at

T=0

and the critical spherical model in

d>2

dimensions.

Abstract For ageing systems, quenched onto a critical temperature T = T c such that the dominant noise comes from the thermal bath, with a non-conserved order-parameter and in addition with dynamical exponent z = 2 , the form of the two-time autocorrelator as well as the time-space form of the single-time correlator are derived from Schrödinger-invariance, generalised to non-equilibrium ageing. These findings reproduce the exact results in the 1 D Glauber–Ising model at T = 0 and the critical spherical model in d > 2 dimensions.

Ageing phenomena and dynamical scaling behaviour have been observed in many out-of-equilibrium systems, but a general framework for the description of such systems is still missing. A first step in this direction is the theory of local scale-invariance (LSI), which attempts to identify generalised forms of spatio-temporal dynamical scaling. For systems with a dynamical exponent z = 2, it has already been known how to treat stochastic partial differential equations and the consequences have been verified in many explicit models. In this thesis a reformulation and extension of LSI for systems with z ? 2 is presented. We infer for the first time generalised Bargmann superselection and discuss extended dynamical symmetries of Langevin equations with z ? 2. We can establish a formalism for the calculation of non-equilibrium correlation -and response functions and the results are confirmed in several new model calculations. Secondly, the ageing behaviour in reaction-diffusion systems is investigated. Although the main features of ageing as seen in magnets are still valid, important differences in exponent relations are found. Explicitly, the contact process is studied through field-theoretical methods and two bosonic models are solved exactly. For the latter, we show how to extend LSI with z = 2 to nonlinear models. Thirdly, the ageing behaviour in semi-infinite magnetic systems close to the surface is considered. The results show that the general scaling picture known from infinite systems remains valid, but some ageing exponents and scaling functions differ from the bulk quantities.

Single-time and two-correlators are computed exactly in the

1D

Glauber-Ising model after a quench to zero temperature and on a periodic chain of finite length

N

, using a simple analytical continuation technique. Besides the general confirmation of finite-size scaling in non-equilibrium dynamics, this allows to test the scaling behaviour of the plateau height

C_{\infty}^{(2)}

to which the two-time auto-correlator converges, when deep into the finite-size regime.

Recent studies on the phenomenology of ageing in certain many-particle systems which are at a critical point of their non-equilibrium steady states are reviewed. Examples include the contact process, the parity-conserving branching-annihilating random walk, two exactly solvable particle reaction models and kinetic growth models. While the generic scaling descriptions known from magnetic systems can be taken over, some of the scaling relations between the ageing exponents are no longer valid. In particular, there is no obvious generalization of the universal limit fluctuation–dissipation ratio. The form of the scaling function of the two-time response function is compared with the prediction of the theory of local scale-invariance.