Equivalence of Fluctuation-Dissipation and Edwards' Temperature in Cyclically Sheared Granular Systems

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Equivalence of Fluctuation-Dissipation and Edwards’ Temperature in Cyclically Sheared Granular Systems

Zhikun Zeng, Shuyang Zhang, Xu Zheng, Chengjie Xia, Walter Kob, Ye Yuan, and Yujie Wang

Phys. Rev. Lett. 129, 228004 – Published 22 November 2022

Abstract

Using particle trajectory data obtained from x-ray tomography, we determine two kinds of effective temperatures in a cyclically sheared granular system. The first one is obtained from the fluctuation-dissipation theorem which relates the diffusion and mobility of lighter tracer particles immersed in the system. The second is the Edwards compactivity defined via the packing volume fluctuations. We find robust agreement between these two temperatures, independent of the type of the tracers, cyclic shear amplitudes, and particle surface roughness, giving therefore the first experimental evidence that the concept of effective temperature is valid in driven frictional granular systems.

Received 17 September 2022
Accepted 26 October 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.228004

Physics Subject Headings (PhySH)

Granular packing Nonequilibrium statistical mechanics

Granular materials

X-ray techniques

Polymers & Soft MatterStatistical Physics & Thermodynamics

Authors & Affiliations

Zhikun Zeng ¹, Shuyang Zhang ¹, Xu Zheng ³, Chengjie Xia ⁴, Walter Kob ⁵, Ye Yuan ^1,*, and Yujie Wang ^1,2,3,†

¹School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
²State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu 610059, China
³Department of Physics, College of Mathematics and Physics, Chengdu University of Technology, Chengdu 610059, China
⁴School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China
⁵Laboratoire Charles Coulomb, University of Montpellier and CNRS, 34095 Montpellier, France

^*Corresponding author. yuanyeoct20@sjtu.edu.cn
^†Corresponding author. yujiewang@sjtu.edu.cn

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Issue

Vol. 129, Iss. 22 — 23 November 2022

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Images

Figure 1
(a) Schematic of the experimental setup. (b) Average displacements $⟨ r ⟩$ in three directions for hollow particles (HPs) and background particles (BPs) as a function of $t$ . Inset: trajectory (length $t = 1000$ ) of a HP under cyclic shear. Results in (b) are for the systems with smooth surfaces, free top surface, $γ = 0.05$ , and $Δ m / m_{0} = 0.376$ .
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Figure 2
(a) Relative vertical displacements $Δ z_{rela}$ as a function of $Δ t$ with $Δ m / m_{0} = 0. 484$ , 0.376, 0.312, 0.243, 0.200, 0.130, 0.079, 0.047, 0.022, and 0 (from top to bottom). The solid lines denote a linear fit to the data. Inset: relative vertical speeds $Δ z_{rela} / Δ t$ as a function of $Δ m / m_{0}$ . The solid lines are the linear fits to $v_{rela}$ for $Δ m / m_{0} \geq 0.079$ . (b) MSDs for different $Δ m / m_{0}$ as a function of $Δ t$ , where subdiffusion and normal diffusion regions are marked by the two solid lines. Inset: self-diffusion coefficient $D$ as a function of $Δ m / m_{0}$ . Results in (a) and (b) are for the HPs in systems with smooth surfaces, free top surface, and $γ = 0.05$ .
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Figure 3
(a) Parametric plot of diffusion versus mobility for the systems with smooth surfaces and different $Δ m / m_{0}$ (legend). Inset shows the same parametric plot for the systems with rough surfaces. (b) $T_{FD}$ for both systems as a function of $Δ m / m_{0}$ . Inset: $T_{FD}$ as a function of pressure $p$ for the SS system. Results in (a) and (b) are for systems with $γ = 0.05$ .
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Figure 4
(a) PDFs of local volumes $v$ for systems with smooth (blue triangles) and rough (red circles) surfaces under cyclic shear ( $γ = 0.05$ ) and their respective reference RLP states. Their global packing fractions are included in the legend. Inset: the ratio of PDFs of $v$ between sheared packings and their respective RLP states. (b) Effective temperature calculated via the FDR versus Edwards’ compactivity for both systems. Solid line is the linear fit: $p χ / T_{FDT} = 1.1 \pm 0.15$ . Inset: $T_{FDT}$ and $p χ$ at different $ϕ$ , tuned by $γ = 0.03$ , 0.05, 0.08, 0.12, 0.20 for systems with smooth surfaces and $γ = 0.03$ , 0.012 for systems with rough surfaces.
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Figure 5
The average contact number in the upper and lower hemisphere, $Z_{up}$ and $Z_{low}$ , as a function of $Δ m / m_{0}$ for both HPs and BPs in systems with smooth surfaces and free top surface. Solid line is the linear fit to $Z_{up}$ of HPs for $Δ m / m_{0} \geq 0.079$ . The inset is a schematic diagram describing how the viscouslike drag force on the HPs (red) is generated by the asymmetric distribution of the BPs (gray) in direct contact.
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