Simple models for strictly non-ergodic stochastic processes of macroscopic systems

G. George; L. Klochko; A. N. Semenov; J. Baschnagel; J. P. Wittmer

doi:10.1140/epje/s10189-021-00129-3

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2024 Impact factor 2.2

Soft Matter and Biological Physics

Eur. Phys. J. E (2021) 44: 125
https://doi.org/10.1140/epje/s10189-021-00129-3

Regular Article – Flowing Matter

Simple models for strictly non-ergodic stochastic processes of macroscopic systems

G. George, L. Klochko, A. N. Semenov, J. Baschnagel and J. P. Wittmer^e

Institut Charles Sadron, Université de Strasbourg & CNRS, 23 rue du Loess, 67034, Strasbourg Cedex, France

^e joachim.wittmer@ics-cnrs.unistra.fr

Received: 7 July 2021
Accepted: 21 September 2021
Published online: 11 October 2021

Abstract

We investigate simple models for strictly non-ergodic stochastic processes $x_{t}$ $$x_t$$ (t being the discrete time step) focusing on the expectation value v and the standard deviation $δ v$ $\delta v$ of the empirical variance $v [x]$ $v[\mathbf {x}]$ of finite time series $x$ $\mathbf {x}$ . $x_{t}$ $$x_t$$ is averaged over a fluctuating field $σ_{r}$ $\sigma _{\mathbf{r}}$ ( $r$ $\mathbf{r}$ being the microcell position) characterized by a quenched spatially correlated Gaussian field $g_{r}$ $g_{\mathbf{r}}$ . Due to the quenched $g_{r}$ $g_{\mathbf{r}}$ -field $δ v (Δ τ)$ $\delta v(\varDelta \tau )$ becomes a finite constant, $Δ_{ne} > 0$ $\varDelta _{\mathrm {ne}}> 0$ , for large sampling times $Δ τ$ $\varDelta \tau$ . The volume dependence of the non-ergodicity parameter $Δ_{ne}$ $\varDelta _{\mathrm {ne}}$ is investigated for different spatial correlations. Models with marginally long-ranged $g_{r}$ $g_{\mathbf{r}}$ -correlations are successfully mapped on shear stress data from simulated amorphous glasses of polydisperse beads.