Non-local representations of the ageing algebra in higher dimensions

TL;DR

This paper generalizes the mathematical representations of the ageing algebra for higher dimensions and arbitrary dynamical exponents, providing tools to analyze complex non-equilibrium systems.

Contribution

It constructs new representations of the ageing algebra for any dimension and dynamical exponent, extending previous results limited to one dimension.

Findings

Derived covariant two-time response functions.

Presented applications to exactly solvable models of phase separation.

Explained the closure mechanism of the Lie algebra with fractional derivatives.

Abstract

The ageing Lie algebra age(d) and especially its local representations for a dynamical exponent z=2 has played an important r\^ole in the description of systems undergoing simple ageing, after a quench from a disordered state to the low-temperature phase. Here, the construction of representations of age(d) for generic values of z is described for any space dimension d>1, generalising upon earlier results for d=1. The mechanism for the closure of the Lie algebra is explained. The Lie algebra generators contain higher-order differential operators or the Riesz fractional derivative. Co-variant two-time response functions are derived. Some simple applications to exactly solvable models of phase separation or interface growth with conserved dynamics are discussed.