# Single-Parameter Aging in the Weakly Nonlinear Limit

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The TN Formalism and Single-Parameter Aging

## 3. Calculation of a Generalized Fragility

## 4. Solving the Jump Differential Equation to First Order in the Temperature Change $\mathsf{\Delta}\mathit{T}$

## 5. Numerical Results for a Binary Lennard–Jones Model

#### 5.1. The Relevant Fluctuation–Dissipation Theorem

#### 5.2. Simulation Results

## 6. Summary and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Results from computer simulations of the Kob–Andersen binary Lennard–Jones model. The figures show the normalized relaxation function $R\left(t\right)$ (Equations (25) and (29)) defined from the potential energy U after a temperature jump at $t=0$ starting from a state of thermal equilibrium (blue filled circles): (

**a**,

**b**) show results for magnitude 0.10 temperature up and down jumps to the reference temperature ${T}_{0}=0.60$; (

**c**,

**d**) show results for magnitude 0.05 temperature up and down jumps to ${T}_{0}=0.60$; (

**e**,

**f**) show results for magnitude 0.03 temperature up and down jumps to ${T}_{0}=0.60$. The orange filled circles are the first-order predictions of the jump differential equation Equation (9) (given in Equation (25) in which ${R}_{0}\left(t\right)$ is the normalized equilibrium potential-energy time-autocorrelation function at ${T}_{0}=0.60$, ${R}_{1}\left(t\right)$ is given by Equation (29), and $\mathsf{\Lambda}$ is given by Equation (39)). For reference, in all figures ${R}_{0}\left(t\right)$ is plotted as small black filled circles.

**Figure 2.**Test of time–temperature superposition for the normalized potential-energy time-autocorrelation function ${R}_{0}\left(t\right)$ at the temperatures indicated in the legends: (

**a**) shows the simulation data and (

**b**) shows the same data empirically scaled on the time axis. We conclude that TTS applies except at the shortest times.

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**MDPI and ACS Style**

Mehri, S.; Costigliola, L.; Dyre, J.C.
Single-Parameter Aging in the Weakly Nonlinear Limit. *Thermo* **2022**, *2*, 160-170.
https://doi.org/10.3390/thermo2030013

**AMA Style**

Mehri S, Costigliola L, Dyre JC.
Single-Parameter Aging in the Weakly Nonlinear Limit. *Thermo*. 2022; 2(3):160-170.
https://doi.org/10.3390/thermo2030013

**Chicago/Turabian Style**

Mehri, Saeed, Lorenzo Costigliola, and Jeppe C. Dyre.
2022. "Single-Parameter Aging in the Weakly Nonlinear Limit" *Thermo* 2, no. 3: 160-170.
https://doi.org/10.3390/thermo2030013