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 fluctuationdissipation 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.

利用 X 射线断层扫描获得的粒子轨迹数据,我们确定了循环剪切颗粒系统中的两种有效温度

第一种是通过波动消散定理得到的,该定理与浸入系统中的较轻示踪粒子的扩散和流动性有关。第二个是通过堆积体积涨落定义的 Edward 致密性


Edwards entropy and compactivity in a model of granular matter

Understanding disordered materials far from thermal equilibrium is one of the biggest challenges in condensed matter physics. Many principles of equilibrium statistical mechanics have proven to be valid for characterizing the nonequilibrium behaviors of these systems, with the concept of effective temperature playing an important role, rather than the thermal bath temperature.

Various studies have shown that the effective temperature is useful for understanding the structural relaxation, plasticity, and rheological properties of various nonequilibrium disordered materials.

However, multiple approaches exist to define an effective temperature, and both the physical meaning and the relationship between these temperatures have so far remained elusive.




Granular matter, which is an achetypical disordered system ubiquitous in nature and engineering processes, also needs a statistical mechanics framework that allows the definition of an effective temperature. It is found that under steady external driving, granular systems evolve into stationary packings with a mildly fluctuated packing fraction (or packing volume) irrespective of preparation history, reminiscent of the thermodynamic fluctuation of thermal system at constant temperature.

In the 1990s, Edwards and collaborators introduced a statistical mechanics framework for jammed granular packings based on this observation. According to their approach, volume is postulated to be the quantity equivalent to energy in thermal systems. It is further conjectured that jammed packings of the same volume are equally probable, validated so far only in numerical simulations of frictionless packings.


20 世纪 90 年代,Edwards 及其合作者根据这一观察结果,提出了一种用于堵塞颗粒填料的统计力学框架。根据他们的方法,体积被假定为相当于热系统中能量的量。他们进一步推测,相同体积的受阻填料具有相同的可能性,迄今为止,这一推测仅在无摩擦堆积的数值模拟中得到验证。

Such an approach naturally leads to the definition of entropy as the logarithm of the number of all jammed states in the microcanonical ensemble, and a compactivity $\chi$ that plays the role of the conventional temperature can be defined accordingly. Recent experiments have demonstrated the validity of the Edwards volume ensemble in frictional jammed granular packings.

这种方法自然而然地将熵定义为微观经典集合中所有堵塞态数量的对数,并可相应地定义起常规温度作用的紧凑性 $\chi$。最近的实验证明了 Edward 体积集合在摩擦堵塞颗粒填料中的有效性。

Another approach to define an effective temperature $T_{\text{FD}}$ is based on the fluctuation-dissipation relation (FDR), which has been widely investigated for granular materials in the stationary state. In the FDR, $T_{\text{FD}}$ relates the diffusion and the mobility of a tracer particle over a period of time $t$:

$$ k_{\text{B}}T_{\text{FD}}=\frac{\langle[r(t)-r(0)]^{2}\rangle}{2\langle r(t)-r(0)\rangle/F} $$

where $r$ is the position of the particle and $F$ is a weak external force acting on the particle balanced by the drag force in the stationary directional moving regime.

另一种定义有效温度 $T_{\text{FD}}$ 的方法是基于涨落-耗散关系(FDR),该关系已被广泛用于研究静止状态下的颗粒材料。在 FDR 中,$T_{\text{FD}}$ 与示踪粒子在一段时间内 $t$ 的扩散和流动有关:

$$ k_{\text{B}}T_{\text{FD}}=\frac{\langle[r(t)-r(0)]^{2}\rangle}{2\langle r(t)-r(0)\rangle/F} $$

其中,$r$ 是粒子的位置,$F$ 是作用在粒子上的微弱外力,与静止定向运动状态下的阻力相平衡。

Simulations have shown that in frictionless jammed granular packings $T_{\text{FD}}$ is is equal to $\chi$, so that $\chi$ governs the transport properties of these systems.

However, there have been no experimental studies investigating the relevance between $\chi$ and $T_{\text{FD}}$ for frictional granular systems so far, which is not only critical for the understanding of granular rheology, but also hampers the development of nonequilibrium statistical mechanics in a broader sense.

模拟结果表明,在无摩擦干扰的颗粒填料中,$T_{\text{FD}}$ 等于 $\chi$,因此 $\chi$ 支配着这些系统的传输特性。

然而,迄今为止还没有实验研究调查摩擦粒状系统中 $\chi$ 与 $T_{\text{FD}}$ 之间的相关性,这不仅对理解粒状流变学至关重要,而且阻碍了广义非平衡统计力学的发展。

In this Letter, we use x-ray tomography to determine the compactivity $\chi$ as well as the $T_{\text{eff}}$ of a three-dimensional granular system under quasistatic cyclic shear. In practice, we measure $\chi$ from the microscopic volume fluctuations within the disordered packings.

Using hollow particles (HP) with different lower mass densities as compared to the background particles (BP), we measure $T_{\text{FD}}$ based on Eq. (1) by tracking both the HPs’ directional motion due to buoyancy and diffusive motion. We find that $T_{\text{FD}}$ and $\chi$ agree with each other robustly for all the steady states investigated.

在这封快讯中,我们利用X射线层析成像技术来确定准静态循环剪切作用下三维颗粒系统的致密性 $\chi$ 和 $T_{\text{eff}}$。实际上,我们是通过无序堆积内部的微观体积涨落来测量 $\chi$ 的。

利用与背景粒子(BP)相比质量密度较低的空心粒子(HP),我们根据公式(1),通过跟踪HP因浮力和扩散运动而产生的定向运动来测量 $T_{\text{FD}}$。我们发现,在所有研究的稳定状态下,$T_{\text{FD}}$ 和 $\chi$ 都是一致的。

We 3D print (ProJet MJP $2500$ Plus, $0.032$ mm resolution) the HPs and BPs of the same plastic material (VisiJet M2R-WT, $\rho = 1.12\times 10^{3}$kg/m$^{3}$) and diameter $d = 6$mm.

For both HPs and BPs, we prepare particles with two types of surface properties: a smooth surface and a rough surface realized by uniformly decorating its surface with $150$ hemispheres of radius $0.04d$, which mimics a particle with very large surface friction.

The HPs are lighter than the solid BPs, and their mass difference is $\Delta m = m_{0}-m$, where $m$ and $m_{0} = 0.1263\pm 0.0001$g are the respective masses of HPs and BPs. We prepare $13$ types of HPs for the systems with smooth surfaces with $\Delta m/m_{0}\in [0.017,0.484]$, and five types of HPs for the systems with rough surfaces with $\Delta m/m_{0}\in[0.031,0.500]$.

我们用相同的塑料材料(VisiJet M2R-WT,$\rho = 1.12\times 10^{3}$kg/m$^{3}$)和直径 $d = 6$mm 的 HP 和 BP 进行 3D 打印(ProJet MJP $2500$ Plus,分辨率为 $0.032$mm)。

对于 HP 和 BP,我们制备了具有两种表面特性的粒子:光滑表面和粗糙表面,粗糙表面是通过在粒子表面均匀地装饰半径为 $0.04d$ 的 $150$ 半球来实现的。

HPs 比固体 BPs 轻,它们的质量差为 $\Delta m = m_{0}-m$,其中 $m$ 和 $m_{0} = 0.1263\pm 0.0001$g 分别是 HPs 和 BPs 的质量。我们为表面光滑($\Delta m/m_{0}\in [0.017,0.484]$)的系统准备了 13 种 HP,为表面粗糙($\Delta m/m_{0}\in [0.031,0.500]$)的系统准备了 5 种 HP。

Figure 1(a) is the schematic of our experiment setup. The shear cell has a cuboid shape with a size of $120$ mm $(x) \times 120$ mm $(y)\times 140$ mm $(z)$, where the bottom and side walls are rendered rough by gluing a layer of hemisphere particles at random positions to prevent crystallization.

The cyclic shear is generated by a step motor attached to the bottom plate of the shear cell. See Ref. [28] for further details of the setup. For each measurement, a packing in the shear cell contains ∼$12 000$ particles of a height ∼$22d$.

For better statistics, we manually immerse 100 HPs with a specific $\Delta m$ uniformly in the packing within a height interval of $8–12d$ initially. The HPs are spaced sufficiently apart from each other to avoid any collective effect.

Furthermore, to investigate the influence of pressure $p$ on the dynamics of the system, we perform experiments on the packings either with a free upper surface or covered by a lid of mass $M$ of $1.95$ or $3.85$ kg.

Correspondingly, the average imposed pressure at the HPs’ height is directly measured to be $p = 0.35\pm 0.07, 1.67\pm 0.10, and 2.86\pm 0.17\times 10^{3}$ Pa(see Supplemental Material [29] for more details).

图 1(a) 是我们的实验装置示意图。剪切池呈立方体,尺寸为 $120$ mm $(x) \times 120$ mm $(y)\times 140$ mm $(z)$,其底部和侧壁通过在随机位置粘贴一层半球形颗粒而变得粗糙,以防止结晶

如何理解颗粒物质在循环剪切过程中的"结晶化"? 是指的完全进入堵塞态吗?

循环剪切由连接在剪切池底板上的步进电机产生。参见参考文献 [28]。有关设置的更多详情,请参阅文献[28]。每次测量时,剪切池中的填料包含高度为 ∼$22d$ 的 ∼$12 000$ 颗粒。

为了获得更好的统计结果,我们手动将 $100$ 个具有特定 $\Delta m$ 的 HPs 均匀地浸入填料中,初始高度间隔为 $8-12d$。这些 HP 之间有足够的间距,以避免任何集体效应


此外,为了研究压强 $p$ 对系统动力学的影响,我们对上表面自由的填料或由质量 $M$ 为 $1.95$ 或 $3.85$ 千克的盖子覆盖的填料进行了实验。

相应地,直接测得的 HP 高度处的平均外加压力分别为:$p = 0.35\pm 0.07, 1.67\pm 0.10$ 和 $2.86\pm 0.17×10^{3}$ Pa(详见补充材料[29])。

The cyclic shear is applied with a strain rate of $\dot{\gamma} = 0.33s^{-1}$ and different strain amplitudes $\gamma = 0.03, 0.05, 0.08, 0.12, $ and $0.20$.

The inertial number is on the order of $10^{-4}$ for all cases investigated, so that the shear is quasistatic. For each value of $\gamma$, we shear the initial system for hundreds of cycles until a steady state is reached. A large $\gamma$ can lead to a lower steady state packing fraction.

After the system reaches a steady state, we obtain its packing structure via a medical CT scanner (UEG Medical Group Ltd., $0.2$ mm spatial resolution) after every ten shear cycles. A total of $200$ CT scans for the systems with smooth surfaces and $150$ CT scans for the systems with rough surfaces are taken for each $\gamma$ and $\Delta m$.

循环剪切的应变速率为 $\dot{\gamma} = 0.33s^{-1}$,应变振幅为 $\gamma = 0.03、0.05、0.08、0.12 和 0.20$。

在所有研究案例中,惯性数都在 $10^{-4}$ 的数量级上,因此剪切是准静态的。对于每个 $\gamma$ 值,我们对初始系统进行数百次剪切,直到达到稳定状态。较大的 $\gamma$ 会导致较低的稳态堆积分数。

在系统达到稳定状态后,我们通过医用 CT 扫描仪(UEG 医疗集团有限公司,空间分辨率为 $0.2$ 美元毫米)在每十次剪切循环后获取其堆积结构。对于表面光滑的系统,每组 $\gamma$ $\Delta m$ 总共需要 $200$ 次 CT 扫描;对于表面粗糙的系统,每组 $\gamma$ 和 $\Delta m$ 总共需要 $150$ 次 CT 扫描。

To further improve the statistics, we repeat the measurements for each type of HPs $3$ or $4$ times. Following the image processing procedures and tracking algorithms of our previous study [28], one can obtain the centroid coordinates with an error less than $3\times 10^{-3}d$ of each particle. In the following, only particles that are at least $3d$ away from the boundary of the shear cell are analyzed.

为了进一步改进统计结果,我们对每种类型的 HP 重复测量 $3$ 或 $4$ 次。按照我们之前的研究[28]中的图像处理程序和跟踪算法,我们可以得到每个粒子的中心点坐标,误差小于 $3\times 10^{-3}d$。下面只分析距离剪切单元边界至少 $3d$ 的粒子。

For the system with a free upper surface and of particles with smooth surfaces, the trajectory of a HP ($\Delta m/m_{0} = 0.376$) under cyclic shear ($\gamma$) is shown in the inset of Fig. 1(b).

It is clear that a HP displays both diffusive motion and vertical directional motion under cyclic shear, analogous to the dynamics of a Brownian particle subject to a directional force in a thermal liquid.

The one-dimensional diffusive motion of a thermal Brownian particle can be characterized by the mean squared displacement (MSD), $\langle [z(t)-z(0)]^{2}\rangle = 2Dt$, where $D$ is the self-diffusion coefficient.

对于具有自由上表面和光滑表面的粒子系统,图 1(b) 的插图显示了 HP($\Delta m/m_{0} = 0.376$)在循环剪切力($\gamma$)作用下的运动轨迹。

很明显,在循环剪切力的作用下,HP 同时表现出扩散运动和垂直方向运动,类似于热液体中受方向力作用的布朗粒子的动力学。

热布朗粒子的一维扩散运动可以用平均位移平方(MSD)来表征,即 $\langle [z(t)-z(0)]^{2}\rangle = 2Dt$,其中 $D$ 是自扩散系数。

If an external force $F$ is applied to the particle, the particle will experience a viscous drag force from the liquid and the balance of the two forces will lead to a long-term average directional motion with constant velocity $\langle z(t) - z(0)\rangle /t = BF$, where $B$ is the mobility.

According to the Einstein relation, a temperature independent of $F$ and the Brownian particle can be obtained by $k_{\text{B}}T_{\text{FD}} = D/B$. The diffusive and vertical directional motion of HPs in our sheared granular system can be connected by a $T_{\text{FD}}$ basically the same way. In the following, we denote $t$ the shear cycle number and set $k_{B} = 1$ for brevity.

如果对粒子施加外力 $F$,粒子将受到来自液体的粘性阻力,两种力的平衡将导致粒子以恒定的速度做长期平均定向运动 $\langle z(t) - z(0)\rangle /t = BF$,其中 $B$ 是流动性。

根据爱因斯坦关系,可以通过 $k_{\text{B}}T_{\text{FD}} = D/B$ 得到温度(与 $F$ 和布朗粒子无关)。在我们的剪切颗粒系统中,HP 的扩散运动和垂直方向运动基本上可以通过 $T_{\text{FD}}$ 以相同的方式联系起来。在下文中,我们用 $t$ 表示剪切周期数,为简洁起见,设 $k_{B} = 1$。

We note that the vertical displacements of HPs include both the buoyant motion with respect to the BPs and a convective flow. This is manifested by the significantly enhanced vertical displacements of the BPs compared with those along the other two directions.

To obtain the mean relative displacement $\Delta z_{\text{rela}}$ of the HPs, we subtract this convective motion of the BPs, that is, their average vertical displacement. The resulting $\Delta z_{\text{rela}}$ depends on the time interval $\Delta t$ for all HPs linearly, from which a mobility $B$ can be properly defined:

$$ \Delta z_{\text{rela}}(\Delta t) = \langle z_{\text{rela}}(t + \Delta t) - z_{\text{rela}}(t_{0})\rangle = BF\Delta t, $$

where $\langle \dots\rangle$ denotes the averages over all HPs and different starting shear cycle number $t_{0}$, $F = \Delta mg$ is the effective buoyancy force acting on the HPs.

我们注意到,HP 的垂直位移包括相对于 BP 的浮力运动和对流。这表现在于, 与沿其他两个方向的垂直位移相比,BP 的垂直位移明显增大。

为了得到 HPs 的平均相对位移 $\Delta z_{\text{rela}}$,我们要减去 BP 的对流运动,即它们的平均垂直位移。由此得到的 $\Delta z_{\text{rela}}$ 与所有 HP 的时间间隔 $\Delta t$ 成线性关系,由此可以正确定义移动量 $B$:

$$ \Delta z_{\text{rela}}(\Delta t) = \langle z_{\text{rela}}(t + \Delta t) - z_{\text{rela}}(t_{0})\rangle = BF\Delta t, $$

其中,$\langle \dots\rangle$ 表示所有 HP 和不同起始剪切循环数 $t_{0}$ 的平均值, $F = \Delta mg$ 是作用在 HP 上的有效浮力。

Figure 2(a) demonstrates that HPs with larger $\Delta m$ have a steeper increasing of $\Delta z_{\text{rela}}$ versus $\Delta t$ and the resulting velocity is linear in $\Delta m$ if $\Delta m/m_{0} > 0.079$, thus allowing to obtain the mobility $B$.

Similarly, the diffusion constant $D$ of HPs can be extracted from their MSD curves:

$$ z_{\text{rela}}(\Delta t)^{2} = \langle[z_{\text{rela}}(t_{0} + \Delta t)-z_{\text{rela}}(t_{0})]^{2}\rangle = 2D\Delta t. $$

图 2(a) 表明,具有较大 $\Delta m$ 的 HPs 在 $\Delta t$ 上具有更陡的 $\Delta z_{\text{rela}}$ 增加率,则如果 $\Delta m/m_{0} > 0.079$,得到的速度与 $\Delta m$ 成线性关系,从而可以得到流动性 $B$。

类似地,可以从 HP 的 MSD 曲线中提取扩散常数 $D$:

$$ z_{\text{rela}}(\Delta t)^{2} = \langle[z_{\text{rela}}(t_{0} + \Delta t)-z_{\text{rela}}(t_{0})]^{2}\rangle = 2D\Delta t. $$

In Fig. 2(b), the MSD curves show a crossover from the initial subdiffusion to the long-term normal diffusion and hence we only include data in the diffusive regime to calculate $B$ and $D$ [insets of Figs. 2(a) and 2(b)].

Note that $F$ is estimated to be only about 1% of the average contact force between particles, and therefore the buoyancyinduced directional motions of the HPs are too small to modify their MSD behaviors in this experiment. As a result, a temperature $T_{\text{FD}}$ based on the Einstein relation can be obtained for $\Delta m/m_{0} \geq 0.079$:

$$ T_{\text{FD}} = \frac{D}{B} = \frac{z_{\text{rela}}(\Delta t)^{2}}{2\Delta z_{\text{rela}}(\Delta t)/F} $$

在图 2(b)中,MSD 曲线显示了从初始亚扩散到长期正常扩散的交叉,因此我们只包含扩散体系中的数据来计算 $B$ 和 $D$[图 2(a) 和 2(b)的插入部分]。

请注意,据估计 $F$ 仅占粒子间平均接触力的 1% 左右,因此在本实验中,由浮力引起的 HP 定向运动太小,不足以改变它们的 MSD 行为。因此,在 $\Delta m/m_{0}\geq 0.079$ 时,可以得到基于爱因斯坦关系的温度 $T_{\text{FD}}$:

$$ T_{\text{FD}} = \frac{D}{B} = \frac{z_{\text{rela}}(\Delta t)^{2}}{2\Delta z_{\text{rela}}(\Delta t)/F} $$

FD = Fluctuation-Dissipation, 即涨落耗散定理定义的温度.

In Fig. 3(a), we clearly observe that for all $\Delta m$ the fluctuations and responses collapse on the same linear relationship, the slope of which is just $T_{\text{FD}}$.

This indicates that the effective temperature obtained by the fluctuationdissipation theorem is a well-defined temperature-like quantity, which reflects the states and characteristics of the driven granular system, irrespective of the various mass difference of the HPs.

Moreover, $T_{\text{FD}}$ is clearly different for systems with distinct surface properties for the same $\gamma = 0.05$, see Fig. 3(b). This implies that the packing structures and the underlying mechanism of exploring the configurational space of the systems with different surface roughness are rather different for a same $\gamma$.

在图 3(a)中,我们可以清楚地观察到,对于所有 $\Delta m$,涨落和响应都坍缩在相同的线性关系上,其斜率恰为 $T_{\text{FD}}$。

这表明,由涨落耗散定理得到的有效温度是一个定义明确的类温量,它反映了被驱动粒状系统的状态和特性,而与 HP 的不同质量差无关。

此外,在相同的 $\gamma = 0.05$ 条件下,具有不同表面性质的系统的 $T_{\text{FD}}$ 明显不同,见图3(b)。这意味着在相同的 $\gamma$ 条件下,具有不同表面粗糙度的系统的堆积结构和探索构型空间的基本机制是相当不同的。

We also check the dependence of $T_{\text{FD}}$ on pressure at $\gamma = 0.05$ by varying the weight of the top lid. We find that, although the resulting disordered packings remain similar, $T_{\text{FD}}$ shows a clear linear dependence on $p$, see Fig. 3(b).

Note that the ratio $T_{\text{FD}}/p$ has the dimension of volume(in unit of $d^{3}$), which implies that granular packings under cyclic shear relax through free volume, reminiscent of compactivity in the Edwards ensemble.

我们还通过改变顶盖的重量来检验 $T_{text{FD}}$ 与 $\gamma = 0.05$ 时压强的关系。我们发现,尽管产生的无序堆积仍然相似,但 $T_{\text{FD}}$ 与 $p$ 呈明显的线性关系,见图 3(b)。

请注意,比值 $T_{\text{FD}}/p$ 具有体积维度(单位为 $d^{3}$),这意味着循环剪切作用下的粒状堆积会通过自由体积松弛,这让人联想到爱德华兹簇中的致密性。

In the Edwards ensemble, the volume fluctuation of a steady-state granular packing can be characterized by a Boltzmann distribution which defines a temperaturelike quantity, the compactivity $\chi$.

In practice, $\chi$ can be obtained by a histogram overlapping method, i.e., by calculating the ratio between probability distribution functions (PDFs) of the local volume $v$ in a packing and its reference random loose packing (RLP).

As shown in Fig. 4(a), the RLP states of the systems with distinct surface properties are different due to their different friction. The logarithm of the ratio depends linearly on $v$, the slope of which is the compactivity $\chi$ in unit of the particle volume [inset of Fig. 4(a)], and its slope depends on the type of particle.

在爱德华兹集合中,稳态颗粒填料的体积涨落可以用波尔兹曼分布来表征,波尔兹曼分布定义了一个类似温度的量–密实度 $\chi$.


实际上,$\chi$ 可以通过直方图重叠法获得,即通过计算堆积中局部体积 $v$ 的概率分布函数(PDF)与其参考随机松散堆积(RLP)之间的比率。

RLP(Random Loose Packing)随机松散堆积的数值计算方法.

如图 4(a)所示,由于摩擦力不同,表面性质不同的系统的 RLP 状态也不同。该比值的对数与 $v$ 成线性关系,其斜率是以颗粒体积为单位的密实度 $\chi$[图 4(a) 插图],其斜率取决于颗粒的类型。

The logarithm of the ratio depends linearly on $v$, the slope of which is the compactivity $\chi$ in unit of the particle volume [inset of Fig. 4(a)], and its slope depends on the type of particle.

To further examine the quantitative relationship between $T_{\text{FD}}$ and $\chi$, we perform additional experiments with different cyclic shear amplitude $\gamma$ under the same free surface condition.

For all systems investigated, we find $T_{\text{FD}}$ varies linearly with $\chi$, with a proportionality factor $T_{\text{FD}}/p\chi = 0.91\pm 0.12$, see Fig. 4(b). We thus draw the conclusion that $T_{\text{FD}}/p$ equals the compactivity $\chi$ within the experimental uncertainty.

该比率的对数与 $v$ 成线性关系,其斜率就是以颗粒体积为单位的密实度 $\chi$[图 4(a) 插图],其斜率取决于颗粒的类型。

为了进一步研究 $T_{\text{FD}}$ 和 $\chi$ 之间的定量关系,我们在相同的自由表面条件下,用不同的循环剪切振幅 $\gamma$ 进行了额外的实验。

对于所研究的所有系统,我们发现 $T_{\text{FD}}$ 与 $\chi$ 呈线性变化,比例因子为 $T_{\text{FD}}/p\chi = 0.91\pm 0.12$,见图 4(b)。因此,我们得出结论:在实验不确定性范围内,$T_{\text{FD}}/p$ 等于致密度 $\chi$。

$10$% 的实验误差, 这是可以接受的吗?

This equivalence between $T_{\text{FD}}$ and the effective temperature defined via the Edwards ensemble has been identified in theoretical calculations of an aging glass with low heat bath temperature.

For these systems, the effective temperatures are closely related to the configurational entropy (or complexity) defined via the local minima (or inherent structures) of the energy landscape.

The analogy of these systems can be justified by the fact that a jammed frictionless granular packing can be mapped onto a local minimum of the free-energy landscape of hardsphere glasses.

在对热浴温度较低的老化玻璃进行理论计算时,发现了 $T_{\text{FD}}$ 与通过爱德华兹集合定义的有效温度之间的这种等价关系。



The equivalence of two effective temperatures therefore signals that a new type of ergodicity exists when the system enters the glassy landscape regime. This equivalence persists in our frictional granular systems, suggesting that the same ergodicity is preserved despite the complex frictional forces and dissipative interactions between granular particles.

From this perspective, many thermodynamic concepts defined on glass systems are still applicable for dense granular materials.



Despite equivalence between $T_{\text{FD}}$ and $\chi$ identified above in our system, we point out that there exists a qualitative difference between the microscopic mechanisms of the drag force for the Brownian motion in a granular packing and an ordinary liquid.

In the inset of Fig. 2(a), the steadystate relative speed $\Delta z_{\text{rela}}/\Delta t$ of the HPs for $\Delta m/m_{0}\geq 0.079$ shows a linear relationship with the applied buoyancy force, which is analogous to the viscous behavior of a thermal liquid.

However, we also note a distinct behavior of a simple liquid as $v_{\text{rela}}$ increases more quickly if $\Delta m/m_{0} < 0.079$.

在图 2(a)的插图中,在 $\Delta m/m_{0}\geq 0.079$ 条件下,HP 的稳态相对速度 $\Delta z_{\text{rela}}/\Delta t$ 与所施加的浮力呈线性关系,这类似于热液体的粘性行为。

尽管在我们的系统中,$T_{\text{FD}}$ 和 $\chi$ 是等价的,但我们提醒: 颗粒堆积中布朗运动的阻力微观机制, 与普通液体中的阻力微观机制存在质的区别。

We speculate this to be a result of the different microscopic origin of the drag force in a granular “fluid”. Unlike particles in thermal fluids in which the viscous drag force is generated by the variation of velocity distribution of the liquid surrounding the HPs, in a granular packing, the particles interact through static contact forces.

Simply varying the relative contact sliding speeds would leave the frictional force unmodified and therefore it cannot be responsible for the speed-dependent drag force. Instead, we conjecture that the viscous force originates from the asymmetric distribution of contacts on the HP surface.

We characterize this asymmetric distribution by the average contact number in the upper and lower hemisphere, $Z_{\text{up}}$ and $Z_{\text{low}}$, respectively, for both HPs and BPs, see Fig. 5. The results show that for $\Delta m/m_{0} < 0.079$, the contact distributions of HPs and BPs are the same.

我们推测这是由于颗粒 “流体” 中阻力的微观来源不同造成的。与热流体中的颗粒不同,在热流体中,粘滞阻力是由 HP 周围液体的速度分布变化产生的,而在颗粒填料中,颗粒是通过静态接触力相互作用的。

简单地改变相对接触滑动速度将使摩擦力保持不变,因此它不可能是随速度变化的阻力的原因。相反,我们推测粘性力来源于 HP 表面上接触点的不对称分布。

我们通过 HP 和 BP 上下半球的平均接触数(分别为 $Z_{\text{up}}$ 和 $Z_{\text{low}}$)来描述这种不对称分布,见图 5。结果表明,当 $\Delta m/m_{0}< 0.079$ 时,HP 和 BP 的接触分布相同。

我们能够将 接触数 替换理解为 配位数 吗?

However, for $\Delta m/m_{0}\geq 0.079$, $Z_{\text{up}}$ of the HPs starts to increase linearly with the mass difference $\Delta m$ while $Z_{\text{low}}$ of of the HPs remains at the same value as that of the BPs.

We can therefore speculate that an external “effective drag force” is produced as follows: owing to its buoyant tendency, HPs tend to pile up the BPs on their top, hence accumulating a denser “cap” on the upper hemisphere, as shown in the inset of Fig. 5.

然而,当 $\Delta m/m_{0}\geq 0.079$ 时,HPs的 $Z_{text{up}}$ 开始随质量差 $\Delta m$ 线性增加,而HPs的 $Z_{text{low}}$ 与 BPs 保持相同。

因此,我们可以推测外部的 “有效阻力 “是这样产生的:由于 HP 具有浮力趋势,它往往会将 BP 堆积在其顶部,从而在上半球积累了一个更密集的 “盖”,如图 5 的插图所示。

Assuming that each contact possesses similar contact and frictional force, this will generate an average “effective drag force” proportional to the asymmetry of contact distribution. This can also naturally explain why there exists a qualitative difference of the viscous behavior when $\Delta m/m_{0}$ is below or above $0.079$.

When $\Delta m/m_{0}$ is small, there exists no significant asymmetry in the contact structure and the viscous drag force is mainly generated by mobilizing the frictional contacts; when $\Delta m/m_{0}$ is sufficiently large, all the frictional contacts around the HP are mobilized, and the viscous force is then generated by the asymmetric distribution of frictional contacts on the particle surfaces.

假设每个接触点都具有相似的接触力和摩擦力,这将产生与接触点分布不对称成比例的平均 “有效阻力”。这也自然解释了为什么当 $\Delta m/m_{0}$ 低于或高于 $0.079$ 时,粘性行为存在质的差异。

当 $\Delta m/m_{0}$ 较小时,接触结构不存在明显的不对称性,粘滞阻力主要是通过移动摩擦接触产生的;当 $\Delta m/m_{0}$ 足够大时,HP 周围的所有摩擦接触都被调动起来,此时粘滞力是通过颗粒表面摩擦接触的不对称分布产生的。

In summary, using x-ray tomography, we have presented the first experimental evidence that the two types of effective temperatures derived from the FDR and Edwards volume ensemble, coincide with each other in a driven frictional granular system. This finding is robust with respect to different types of hollow particles, cyclic shear amplitudes, and even particle surface roughness.

Since granular materials belong to the wide class of disordered materials, the presented validation of the concept of effective temperature consolidates the very foundation of related rheological theories, like shear transformation zone or soft glass rheology, in which the effective temperature plays a crucial role in connecting the microscopic or mesoscopic information with the macroscopic plasticity or complex flowing.

In a broader sense, our Letter enlightens the relationship between disordered materials and their transport properties using a statistical mechanics framework for general nonequilibrium systems.

总之,我们利用 X 射线层析成像技术首次提出了实验证据,证明在驱动摩擦颗粒系统中,从 FDR 和 Edwards 体积集合中得出的两类有效温度相互吻合。这一发现对于不同类型的空心颗粒、循环剪切振幅甚至颗粒表面粗糙度都是可靠的。

不同的质量和粗糙度所造成的摩擦-阻力类型完全不同, 两个温度定义却出现高度吻合, 这也是值得大胆类比的原因.