Narayanaswamy's 1971 aging theory and material time

TL;DR

This paper derives Narayanaswamy's aging theory from the nonlinear fluctuation-dissipation theorem, emphasizing material-time translational invariance and proposing tests for this invariance through geometric properties of aging paths.

Contribution

It provides a derivation of Narayanaswamy's aging model based on the Bochkov-Kuzovlev theorem and introduces the unique-triangles property as a test for material-time invariance.

Findings

Only one definition of material time satisfies translational invariance.

The unique-triangles property links aging dynamics to geometric triangle relations.

Proposes computer simulations to test the invariance and geometric properties.

Abstract

The Bochkov-Kuzovlev nonlinear fluctuation-dissipation theorem is used to derive Narayanaswamy's phenomenological theory of physical aging, in which this highly nonlinear phenomenon is described by a linear material-time convolution integral. A characteristic property of the Narayanaswamy aging description is material-time translational invariance, which is here taken as the basic assumption of the derivation. It is shown that only one possible definition of the material time obeys this invariance, namely the square of the distance travelled from a configuration of the system far back in time. The paper concludes with suggestions for computer simulations that test for consequences of material-time translational invariance. One of these is the "unique-triangles property" according to which any three points on the system's path form a triangle such that two side lengths determine the third; this is equivalent to the well-known triangular relation for time-autocorrelation functions of aging spin glasses [L. F. Cugliandolo and J. Kurchan, J. Phys. A: Math. Gen. 27, 5749 (1994)]. The unique-triangles property implies a simple geometric interpretation of out-of-equilibrium time-autocorrelation functions, which extends to aging a previously proposed framework for such functions in equilibrium [J. C. Dyre, cond-mat/9712222].