Relaxation at the Angle of Repose
松弛至休止角

(Received 15 July 1988)
（收到 1988 年 7 月 15 日）

PACS numbers: 46.10.+z, 05.40.+j, 05.70.Jk
PACS 编号：46.10.+z, 05.40.+j, 05.70.Jk

Sandpiles have often been used as a metaphor for other phenomena in physics. As children we have played with sand and it is only natural to believe that we can intuit the processes underlying the shape and dynamics of these structures. Thus de Gennes ${}^1$${}^1$^(1){ }^{1} has compared the motion of vortices in superconductors to avalanches in sandpiles, and Souletie ${}^2$${}^2$^(2){ }^{2} has argued that spin-glasses and sandpiles behave in a similar manner. Dry, noncohesive, granular material like sand can flow, resembling a liquid, but also, like a solid, can sustain under the influence of gravity a finite “angle of repose,” ${\theta }_r$${\theta }_r$theta_(r)\theta_{r}. This is the angle between the horizontal and the free surface of the sandpile after a land slide has restored the pile to a metastable equilibrium slope. As early as 1773 the connection between ${\theta }_r$${\theta }_r$theta_(r)\theta_{r} and the static internal friction of a given material was developed by Coulomb. ${}^3$${}^3$^(3){ }^{3} The analogies mentioned above were made because it is tempting to think of a phase transition occurring at ${\theta }_r$${\theta }_r$theta_(r)\theta_{r} : For slopes such that $\theta <{\theta }_r$$\theta <{\theta }_r$theta < theta_(r)\theta<\theta_{r} no flow of sand can occur and the pile appears to be a solid whereas for $\theta >{\theta }_r$$\theta >{\theta }_r$theta > theta_(r)\theta>\theta_{r} the top layers of sand flow freely downhill. Most recently Bak, Tang, and Wiesenfeld ${}^4$${}^4$^(4){ }^{4} introduced the idea of “self-organized criticality” in terms of a model of how they expected a sandpile to behave. Clearly their ideas may have much wider applicability than to just sandpiles, but it is this analogy which makes their model so intuitively appealing. Their idea rests on the assumption that ${\theta }_r$${\theta }_r$theta_(r)\theta_{r} is a critical angle. If the angle $\theta$$\theta$theta\theta of the free surface is increased continuously (e.g., by adding more material to the top or by tilting the base of the pile), the pile will organize itself such that its average slope will be the angle of repose ${\theta }_r$${\theta }_r$theta_(r)\theta_{r} by unloading excess material through avalanches. Bak, Tang, and Wiesenfeld predict a self-organized state at ${\theta }_r$${\theta }_r$theta_(r)\theta_{r} characterized by long-range spatial and temporal correlations and giving rise to a typical " $1/f$$1/f$1//f1 / f " power spectrum of the fluctuations around the steady-state particle flow. They further suggest an analogy between the nonequilibrium behavior of sandpiles and traditional critical phenomena and find a power-law dependence for the relaxation from a supercritical state $\theta >{\theta }_r$$\theta >{\theta }_r$theta > theta_(r)\theta>\theta_{r} back to the critical state. ${}^5$${}^5$^(5){ }^{5}
沙堆常被用作物理学中其他现象的隐喻。作为孩子，我们玩过沙子，因此自然而然地相信我们可以直观地理解这些结构形状和动力学背后的过程。因此，de Gennes ${}^1$${}^1$^(1){ }^{1} 将超导体中涡流的运动与沙堆中的雪崩相比较，而 Souletie ${}^2$${}^2$^(2){ }^{2} 则认为自旋玻璃和沙堆的行为方式相似。干燥、非粘附性的沙子等颗粒材料可以流动，类似于液体，但像固体一样，在重力的作用下可以维持一个有限的“休止角”， ${\theta }_r$${\theta }_r$theta_(r)\theta_{r} 。这是滑坡将沙堆恢复到亚稳态斜坡后，水平面和沙堆自由表面的夹角。早在 1773 年，Coulomb ${\theta }_r$${\theta }_r$theta_(r)\theta_{r} 就发展了与给定材料的静态内摩擦之间的联系。 ${}^3$${}^3$^(3){ }^{3} 提到的类比是因为人们倾向于认为在 ${\theta }_r$${\theta }_r$theta_(r)\theta_{r} 发生相变：对于斜率 $\theta <{\theta }_r$$\theta <{\theta }_r$theta < theta_(r)\theta<\theta_{r} ，沙子无法流动，沙堆看起来像固体；而对于 $\theta >{\theta }_r$$\theta >{\theta }_r$theta > theta_(r)\theta>\theta_{r} ，沙堆的顶层沙子可以自由地向山下流动。 最近，Bak、Tang 和 Wiesenfeld 在文献[8]中提出了“自组织临界性”的概念，用来说明他们期望沙堆如何表现的一个模型。显然，他们的想法可能比仅仅适用于沙堆有更广泛的应用，但正是这种类比使得他们的模型如此直观地吸引人。他们的想法基于这样一个假设：[9]是一个临界角。如果自由表面的角度[10]连续增加（例如，通过向顶部添加更多材料或倾斜沙堆的底部），沙堆将自行组织，使其平均坡度等于休止角[11]，通过滑坡卸载多余的材料。Bak、Tang 和 Wiesenfeld 预测在[12]处出现一个自组织状态，其特征是长程空间和时间相关性，并产生围绕稳态粒子流的典型“[13]”功率谱。他们进一步提出沙堆的非平衡行为与传统临界现象之间的类比，并发现从超临界状态[14]回到临界状态的弛豫具有幂律依赖性。[15]

Motivated by the above conjectures we have investigated the nature of particle flow along the free surface in
受上述猜想启发，我们研究了粒子沿自由表面的流动性质
a model system for granular materials: “sandpiles” consisting of spherical glass beads or rough aluminum-oxide particles. We report here on the power spectrum for fluctuations as well as on relaxation properties.
一个颗粒材料模型系统：“沙堆”，由球形玻璃珠或粗糙的氧化铝颗粒组成。在此，我们报告了波动功率谱以及弛豫特性。

We have performed the experiment in two basic configurations [see Fig. 1(a)]. The average slope of the free surface was varied either by turning a semicircular drum (i.e., tilting the free surface of the pile) partially filled with granular material, or by randomly adding particles to the top surface of a pile contained in a box with one open side. The semicircular drum of width 8 cm and radius 5 cm had an angular velocity $\mathrm{\Omega }={1.3}^{\circ }/\min$$\mathrm{\Omega }={1.3}^{\circ }/\min$Omega=1.3^(@)//min\Omega=1.3^{\circ} / \mathrm{min}. Several box geometries were used having a width of ap-
我们进行了两种基本配置的实验[见图 1(a)]。自由表面的平均斜率通过旋转一个直径 8 厘米、半径 5 厘米的半圆形鼓（即倾斜部分填充颗粒材料的堆体自由表面）或随机向一个开口一侧的箱子中堆体的顶部添加颗粒来改变。该半圆形鼓的角速度为 $\mathrm{\Omega }={1.3}^{\circ }/\min$$\mathrm{\Omega }={1.3}^{\circ }/\min$Omega=1.3^(@)//min\Omega=1.3^{\circ} / \mathrm{min} 。使用了具有不同箱形几何形状的几个，宽度为 ap-

FIG. 1. (a) Schematic diagram of the three experimental configurations (left to right): A rotating semicircular drum, an open box with sand added from the top, and a closed cylindrical drum. In the first two the sand falls through a pair of capacitor plates. (b) The time trace of a typical sequence of avalanches in a rotating drum $(\mathrm{\Omega }={1.3}^{\circ }/\min )$$\left(\mathrm{\Omega }={1.3}^{\circ }/\min \right)$(Omega=1.3^(@)//min)\left(\Omega=1.3^{\circ} / \mathrm{min}\right) with spherical glass beads. $\mathrm{\Delta }C$$\mathrm{\Delta }C$Delta C\Delta C is the change in capacitance due to the flow of the beads through the capacitor. © The power spectrum, $S(f)$$S(f)$S(f)S(f), for the above time trace. Dashed line shows a $1/f$$1/f$1//f1 / f power spectrum for comparison.
图 1. (a) 三种实验配置的示意图（从左到右）：一个旋转的半圆形鼓、一个从顶部添加沙子的开放式箱子和一个封闭的圆柱形鼓。在前两种配置中，沙子通过一对电容器板落下。(b) 旋转鼓中典型雪崩序列的时间轨迹 $(\mathrm{\Omega }={1.3}^{\circ }/\min )$$\left(\mathrm{\Omega }={1.3}^{\circ }/\min \right)$(Omega=1.3^(@)//min)\left(\Omega=1.3^{\circ} / \mathrm{min}\right) ，其中 $\mathrm{\Delta }C$$\mathrm{\Delta }C$Delta C\Delta C 是球状玻璃珠通过电容器流动引起的电容变化。©上述时间轨迹的功率谱 $S(f)$$S(f)$S(f)S(f) 。虚线显示用于比较的 $1/f$$1/f$1//f1 / f 功率谱。
proximately 5 cm and aspect ratios (width/length) ranging from $\frac{1}{3}$$\frac{1}{3}$(1)/(3)\frac{1}{3} to 3 . In the drum we used spherical glass beads of average diameter $0.54\pm 0.08\mathrm{mm}$$0.54\pm 0.08\mathrm{mm}$0.54+-0.08mm0.54 \pm 0.08 \mathrm{~mm}. For the open box we used smaller beads, $0.07\pm 0.01-\mathrm{mm}$$0.07\pm 0.01-\mathrm{mm}$0.07+-0.01-mm0.07 \pm 0.01-\mathrm{mm} average diameter, to keep the momentum transfer as low as possible when the particles hit the surface of the pile. The results in both configurations were insensitive to the particle size although for the very small particles the experiments were performed in a drybox since special care had to be taken to prevent cohesion due to moisture. We also performed the experiment with rough aluminumoxide particles of diameter $0.5\pm 0.1\mathrm{mm}$$0.5\pm 0.1\mathrm{mm}$0.5+-0.1mm0.5 \pm 0.1 \mathrm{~mm}.
大约 5 厘米，宽高比（宽度/长度）从 $\frac{1}{3}$$\frac{1}{3}$(1)/(3)\frac{1}{3} 到 3。在鼓中，我们使用了平均直径为 $0.54\pm 0.08\mathrm{mm}$$0.54\pm 0.08\mathrm{mm}$0.54+-0.08mm0.54 \pm 0.08 \mathrm{~mm} 的球形玻璃珠。对于开口盒子，我们使用了直径更小的珠子，平均直径为 $0.07\pm 0.01-\mathrm{mm}$$0.07\pm 0.01-\mathrm{mm}$0.07+-0.01-mm0.07 \pm 0.01-\mathrm{mm} ，以尽可能降低粒子撞击堆表面时的动量传递。在这两种配置下，结果对粒子大小不敏感，尽管对于非常小的粒子，实验是在干燥箱中进行的，因为需要特别注意防止由于水分引起的粘附。我们还使用直径为 $0.5\pm 0.1\mathrm{mm}$$0.5\pm 0.1\mathrm{mm}$0.5+-0.1mm0.5 \pm 0.1 \mathrm{~mm} 的粗糙氧化铝粒子进行了实验。
The flow of particles over the open edge of the drum or box was detected by a parallel-plate capacitor through which the particles flowed. The corresponding time sequence of capacitance changes, $\mathrm{\Delta }C$$\mathrm{\Delta }C$Delta C\Delta C, was detected by a bridge circuit and fed into a spectrum analyzer. Capacitance changes directly correspond to changes in the flow rate of particles. The resolution was sufficient to detect even the passage of only a few particles at a time.
粒子在鼓或箱开口处的流动通过粒子流过的平行板电容器被检测到。相应的电容变化时间序列， $\mathrm{\Delta }C$$\mathrm{\Delta }C$Delta C\Delta C ，通过电桥电路检测并输入到频谱分析仪。电容变化直接对应于粒子流速的变化。分辨率足以检测每次只有几个粒子的通过。

We show, in Fig. 1(b), a typical time trace and, in Fig. 1 ©, the corresponding power spectrum obtained with the rotating drum using the spherical glass beads. Similar results were obtained with the open box and for the case of rough aluminum-oxide particles. Each inverted spike in Fig. 1(b) corresponds to a major avalanche, in almost all cases spanning the whole free surface of the system. The avalanches start at an angle ${\theta }_m\equiv {\theta }_r+\delta$${\theta }_m\equiv {\theta }_r+\delta$theta_(m)-=theta_(r)+delta\theta_{m} \equiv \theta_{r}+\delta (where $\delta$$\delta$delta\delta is typically a few degrees) and return the pile to its metastable state at ${\theta }_r$${\theta }_r$theta_(r)\theta_{r}. The distributions for the $\mathrm{\Delta }t$$\mathrm{\Delta }t$Delta t\Delta t interval between avalanches and their width $\tau$$\tau$tau\tau are sharply peaked near their average values as shown in Fig. 2. The nearly uniform spacing of avalanches, $⟨\mathrm{\Delta }t⟩$$⟨\mathrm{\Delta }t⟩$(:Delta t:)\langle\Delta t\rangle $=⟨\mathrm{\Delta }\theta ⟩/\mathrm{\Omega }$$=⟨\mathrm{\Delta }\theta ⟩/\mathrm{\Omega }$=(:Delta theta:)//Omega=\langle\Delta \theta\rangle / \Omega, corresponds to the broadened peak at $f=1/⟨\mathrm{\Delta }t⟩$$f=1/⟨\mathrm{\Delta }t⟩$f=1//(:Delta t:)f=1 /\langle\Delta t\rangle in the power spectrum. The roll off at high frequencies above $f=1/⟨\tau ⟩$$f=1/⟨\tau ⟩$f=1//(:tau:)f=1 /\langle\tau\rangle is due to the finite width $\tau$$\tau$tau\tau of the individual avalanches. Its average power-law decay, close to $f^{-3}$$f^{-3}$f^(-3)f^{-3}, is explained by variations between rec-
图 1(b)显示了典型的时序图，图 1©显示了使用旋转鼓和球形玻璃珠获得的相应功率谱。使用开放式箱子和粗糙氧化铝颗粒的情况也获得了类似的结果。图 1(b)中的每个倒置尖峰几乎都对应于一次主要雪崩，几乎覆盖了整个系统的自由表面。雪崩从角度 ${\theta }_m\equiv {\theta }_r+\delta$${\theta }_m\equiv {\theta }_r+\delta$theta_(m)-=theta_(r)+delta\theta_{m} \equiv \theta_{r}+\delta （其中 $\delta$$\delta$delta\delta 通常是几度）开始，并在 ${\theta }_r$${\theta }_r$theta_(r)\theta_{r} 将堆回其亚稳态。雪崩之间的 $\mathrm{\Delta }t$$\mathrm{\Delta }t$Delta t\Delta t 间隔及其宽度 $\tau$$\tau$tau\tau 的分布如图 2 所示，在它们的平均值附近尖锐地峰值。雪崩的几乎均匀间隔 $⟨\mathrm{\Delta }t⟩$$⟨\mathrm{\Delta }t⟩$(:Delta t:)\langle\Delta t\rangle $=⟨\mathrm{\Delta }\theta ⟩/\mathrm{\Omega }$$=⟨\mathrm{\Delta }\theta ⟩/\mathrm{\Omega }$=(:Delta theta:)//Omega=\langle\Delta \theta\rangle / \Omega 对应于功率谱中 $f=1/⟨\mathrm{\Delta }t⟩$$f=1/⟨\mathrm{\Delta }t⟩$f=1//(:Delta t:)f=1 /\langle\Delta t\rangle 处的展宽峰值。高于 $f=1/⟨\tau ⟩$$f=1/⟨\tau ⟩$f=1//(:tau:)f=1 /\langle\tau\rangle 的高频衰减是由于单个雪崩的有限宽度 $\tau$$\tau$tau\tau 。其平均幂律衰减，接近 $f^{-3}$$f^{-3}$f^(-3)f^{-3} ，可由回

FIG. 2. (a) The distribution of intervals between avalanches, $P(\mathrm{\Delta }t)$$P(\mathrm{\Delta }t)$P(Delta t)P(\Delta t), and (b) the distribution of avalanche durations, $P(\tau )$$P(\tau )$P(tau)P(\tau), for a run using glass beads in the rotating drum ( $\mathrm{\Omega }={3.2}^{\circ }/\min$$\mathrm{\Omega }={3.2}^{\circ }/\min$Omega=3.2^(@)//min\Omega=3.2^{\circ} / \mathrm{min} ).
图 2。（a）使用旋转鼓中的玻璃珠进行一次运行时，滑坡间隔的分布， $P(\mathrm{\Delta }t)$$P(\mathrm{\Delta }t)$P(Delta t)P(\Delta t) ，以及（b）滑坡持续时间的分布， $P(\tau )$$P(\tau )$P(tau)P(\tau) 。
tangular and triangular pulse shapes of individual events. In the relevant frequency range, between $1/⟨\mathrm{\Delta }t⟩$$1/⟨\mathrm{\Delta }t⟩$1//(:Delta t:)1 /\langle\Delta t\rangle and $1/⟨\tau ⟩$$1/⟨\tau ⟩$1//(:tau:)1 /\langle\tau\rangle, the spectra appear frequency independent. This finding is in conflict with Bak, Tang, and Wiesenfeld who predicted power-law behavior, $f^{-\theta }$$f^{-\theta }$f^(-theta)f^{-\theta}, with $\theta \cong 1$$\theta \cong 1$theta~=1\theta \cong 1. Such behavior would have followed from a power-law distribution of widths $\tau$$\tau$tau\tau implying events over a wide range of time scales. Instead we find that the power spectrum is indicative of a linear superposition of global, system-spanning avalanches with a narrowly peaked distribution of time scales.
tangular 和三角形的单个事件脉冲形状。在相关频率范围内，在 $1/⟨\mathrm{\Delta }t⟩$$1/⟨\mathrm{\Delta }t⟩$1//(:Delta t:)1 /\langle\Delta t\rangle 和 $1/⟨\tau ⟩$$1/⟨\tau ⟩$1//(:tau:)1 /\langle\tau\rangle 之间，光谱似乎与频率无关。这一发现与 Bak、Tang 和 Wiesenfeld 的预测相矛盾，他们预测了幂律行为， $f^{-\theta }$$f^{-\theta }$f^(-theta)f^{-\theta} ，具有 $\theta \cong 1$$\theta \cong 1$theta~=1\theta \cong 1 。这种行为将遵循宽度 $\tau$$\tau$tau\tau 的幂律分布，意味着跨越广泛时间尺度的事件。相反，我们发现功率谱表明全局、系统跨越的雪崩的线性叠加，具有时间尺度分布的窄峰。

This situation is a direct consequence of the existence of a nonzero value of $\delta$$\delta$delta\delta. The phenomenon was first explained as dilatation by Reynolds ${}^6$${}^6$^(6){ }^{6} in 1885. For particles to slide on the surface, ${\theta }_r$${\theta }_r$theta_(r)\theta_{r} (which corresponds to the metastable configuration after an avalanche) must be exceeded by an additional amount $\delta$$\delta$delta\delta to allow for clearance with the profile of the underlying layer. ${}^7$${}^7$^(7){ }^{7} As long as effects from the retaining walls of the container can be neglected we find $⟨{\theta }_r⟩={26}^{\circ }\left({39}^{\circ }\right)$$⟨{\theta }_r⟩={26}^{\circ }\left({39}^{\circ }\right)$(:theta_(r):)=26^(@)(39^(@))\left\langle\theta_{r}\right\rangle=26^{\circ}\left(39^{\circ}\right) and $⟨\delta ⟩={2.6}^{\circ }$$⟨\delta ⟩={2.6}^{\circ }$(:delta:)=2.6^(@)\langle\delta\rangle=2.6^{\circ} ( $5^{\circ }$$5^{\circ }$5^(@)5^{\circ} ) for glass beads (aluminum-oxide particles), independent of container or grain size. These values for $⟨\delta ⟩$$⟨\delta ⟩$(:delta:)\langle\delta\rangle agree with calculations by ${\mathrm{Bagnold}}^7$${\mathrm{Bagnold}}^7$Bagnold^(7)\mathrm{Bagnold}^{7} who also predicted the periodic occurrence of avalanches in the steady state. We believe that the observation of a wide spread of values for $\mathrm{\Delta }t$$\mathrm{\Delta }t$Delta t\Delta t by Evesque and Rajchenbach ${}^8$${}^8$^(8){ }^{8} is due to wall effects in a comparatively narrow system.
这种情况是存在非零值 $\delta$$\delta$delta\delta 的直接后果。这种现象最初由雷诺兹 ${}^6$${}^6$^(6){ }^{6} 在 1885 年解释为膨胀。为了使粒子在表面上滑动， ${\theta }_r$${\theta }_r$theta_(r)\theta_{r} （对应于雪崩后的亚稳态配置）必须超过额外的量 $\delta$$\delta$delta\delta ，以便与底层层的轮廓有间隙。 ${}^7$${}^7$^(7){ }^{7} 只要可以忽略容器侧壁的影响，我们就会找到 $⟨{\theta }_r⟩={26}^{\circ }\left({39}^{\circ }\right)$$⟨{\theta }_r⟩={26}^{\circ }\left({39}^{\circ }\right)$(:theta_(r):)=26^(@)(39^(@))\left\langle\theta_{r}\right\rangle=26^{\circ}\left(39^{\circ}\right) 和 $⟨\delta ⟩={2.6}^{\circ }$$⟨\delta ⟩={2.6}^{\circ }$(:delta:)=2.6^(@)\langle\delta\rangle=2.6^{\circ} （ $5^{\circ }$$5^{\circ }$5^(@)5^{\circ} ）对于玻璃珠（氧化铝颗粒），与容器或颗粒大小无关。这些值 $⟨\delta ⟩$$⟨\delta ⟩$(:delta:)\langle\delta\rangle 与 ${\mathrm{Bagnold}}^7$${\mathrm{Bagnold}}^7$Bagnold^(7)\mathrm{Bagnold}^{7} 的计算一致，他也预测了稳态中雪崩的周期性发生。我们认为 Evesque 和 Rajchenbach ${}^8$${}^8$^(8){ }^{8} 观察到的 $\mathrm{\Delta }t$$\mathrm{\Delta }t$Delta t\Delta t 值的广泛分布是由于相对较窄系统中壁效应造成的。

Since ${\theta }_m$${\theta }_m$theta_(m)\theta_{m} is inherently unstable, the hysteresis, $\delta$$\delta$delta\delta, is easily reduced to zero by the application of mechanical vibrations to the system. By adjusting the intensity of the vibrations one can span the transition from solidlike to liquidlike behavior of the granular material in a very controlled way. We induced vibrations by a speaker mechanically coupled to the system. This configuration allowed us in a convenient way to address whether $1/f$$1/f$1//f1 / f fluctuations in the steady-state particle flow for $\mathrm{\Omega }$$\mathrm{\Omega }$Omega\Omega $=$$=$== const could be recovered if $\delta \to 0$$\delta \to 0$delta rarr0\delta \rightarrow 0. It also allowed us to investigate, for $\mathrm{\Omega }=0$$\mathrm{\Omega }=0$Omega=0\Omega=0, the time dependence of $\theta (t)$$\theta (t)$theta(t)\theta(t) as the average angle of the free surface relaxes under the influence of the vibrations.
由于 ${\theta }_m$${\theta }_m$theta_(m)\theta_{m} 本质上是不可稳定的，通过向系统施加机械振动，可以将滞后 $\delta$$\delta$delta\delta 轻易地降低到零。通过调整振动的强度，可以非常可控地跨越颗粒材料从固体到液体行为的转变。我们通过将扬声器机械地耦合到系统中来诱导振动。这种配置使我们能够方便地探讨，当 $\delta \to 0$$\delta \to 0$delta rarr0\delta \rightarrow 0 时，对于 $\mathrm{\Omega }$$\mathrm{\Omega }$Omega\Omega $=$$=$== 恒定的情况，是否可以恢复到稳态粒子流的 $1/f$$1/f$1//f1 / f 波动。这还使我们能够研究，对于 $\mathrm{\Omega }=0$$\mathrm{\Omega }=0$Omega=0\Omega=0 ，随着自由表面的平均角度在振动的影响下松弛， $\theta (t)$$\theta (t)$theta(t)\theta(t) 的时间依赖性。

Figure 3 shows the variation of the power spectrum as the vibration intensity in the drum is increased. The spectra show that increasing intensity leads to a broader distribution of time scales, yet no universal power-law behavior emerges. Instead the fall off in the curve broadens smoothly as the intensity is increased and at the highest vibration intensity the spectrum can be approximated by $f^{-0.8}$$f^{-0.8}$f^(-0.8)f^{-0.8} over a limited range. If a finite value of $\delta$$\delta$delta\delta were the main reason for the absence of universal power-law dependence, that dependence should have emerged for vibration intensities even smaller than that corresponding to the top curve in Fig. 3. Clearly the introduction of noise does not bring the system closer to a critical state. The transition that occurs appears to be more similar to a first-order transition than to a critical point.
图 3 显示了鼓中振动强度增加时功率谱的变化。谱图显示，强度的增加导致时间尺度分布变宽，但没有出现普遍的幂律行为。相反，随着强度的增加，曲线的衰减部分平滑变宽，在最高振动强度下，谱可以在一定范围内近似为 $f^{-0.8}$$f^{-0.8}$f^(-0.8)f^{-0.8} 。如果 $\delta$$\delta$delta\delta 的有限值是普遍幂律依赖性缺失的主要原因，那么这种依赖性应该已经出现在比图 3 中顶部曲线对应的振动强度更小的振动强度中。显然，引入噪声并没有使系统更接近临界状态。发生的转变似乎更类似于一级转变，而不是临界点。

We have studied the nature of the particle flow by in-
我们研究了粒子流的性质，通过..

FIG. 3. The power spectra for avalanches of aluminumoxide particles in a rotating drum $(\mathrm{\Omega }={1.3}^{\circ }/\min )$$\left(\mathrm{\Omega }={1.3}^{\circ }/\min \right)$(Omega=1.3^(@)//min)\left(\Omega=1.3^{\circ} / \mathrm{min}\right) for different vibration intensities (increasing from top to bottom) parametrized by the steady-state angle, ${\theta }_{ss}$${\theta }_{ss}$theta_(ss)\theta_{s s}. Dashed line shows a $1/f$$1/f$1//f1 / f power spectrum for comparison. Similar results were found for spherical beads.
图 3. 不同振动强度下（从上到下依次增加）以稳态角度 ${\theta }_{ss}$${\theta }_{ss}$theta_(ss)\theta_{s s} 为参数的旋转鼓中氧化铝颗粒雪崩的功率谱 $(\mathrm{\Omega }={1.3}^{\circ }/\min )$$\left(\mathrm{\Omega }={1.3}^{\circ }/\min \right)$(Omega=1.3^(@)//min)\left(\Omega=1.3^{\circ} / \mathrm{min}\right) 。虚线显示用于比较的功率谱。对于球形珠子也得到了类似的结果。
vestigating the dependence of the average slope on vibration intensity and rotation speed $\mathrm{\Omega }$$\mathrm{\Omega }$Omega\Omega. The data were taken both in the semicircular drum and in a fully circular drum with no opening for particles to leave [Fig. 1(a)]. This second configuration had radius 15 cm and width 10 cm and allowed continuous rotation without refilling. The steady state of the system at constant speed $\mathrm{\Omega }$$\mathrm{\Omega }$Omega\Omega and constant nonzero vibration intensity is characterized by an angle ${\theta }_{ss}$${\theta }_{ss}$theta_(ss)\theta_{s s}. Switching off the vibrations thus is a way to prepare the system in a supercritical state, $\theta >{\theta }_{ss}$$\theta >{\theta }_{ss}$theta > theta_(ss)\theta>\theta_{s s}, and one can then observe the relaxation to the steady state as the vibrations are turned back on. Such relaxation is shown in Fig. 4 for the case $\mathrm{\Omega }=0$$\mathrm{\Omega }=0$Omega=0\Omega=0 (stationary drum) and initial condition $\theta (t=0)={\theta }_r$$\theta (t=0)={\theta }_r$theta(t=0)=theta_(r)\theta(t=0)=\theta_{r}. The model of self-organized criticality predicts ${}^{4,9}$${}^{4,9}$^(4,9){ }^{4,9} a power-law dependence on $t$$t$tt while our data cannot be fitted by a power law with reasonable parameters over any wide interval of time, even for small intensity. Instead the data for high vibration intensity are consistent with a $\log (t)$$\log (t)$log(t)\log (t) dependence over many decades. For smaller intensity the curves follow a $\log (t)$$\log (t)$log(t)\log (t) behavior over several decades but turn over to a slower decay rate at very large times.
研究平均斜率对振动强度和旋转速度的依赖性。数据是在半圆形鼓和没有开口让颗粒离开的完整圆形鼓中采集的[图 1(a)]。第二个配置的半径为 15 厘米，宽度为 10 厘米，允许连续旋转而不需要重新填充。在恒定速度 $\mathrm{\Omega }$$\mathrm{\Omega }$Omega\Omega 和恒定非零振动强度下，系统的稳态特征为一个角度 ${\theta }_{ss}$${\theta }_{ss}$theta_(ss)\theta_{s s} 。关闭振动因此是一种将系统准备成超临界状态 $\theta >{\theta }_{ss}$$\theta >{\theta }_{ss}$theta > theta_(ss)\theta>\theta_{s s} 的方法，然后可以观察到在振动重新开启时系统的松弛到稳态。这种松弛在图 4 中对于 $\mathrm{\Omega }=0$$\mathrm{\Omega }=0$Omega=0\Omega=0 （静止鼓）和初始条件 $\theta (t=0)={\theta }_r$$\theta (t=0)={\theta }_r$theta(t=0)=theta_(r)\theta(t=0)=\theta_{r} 的情况进行了展示。自组织临界性模型预测 ${}^{4,9}$${}^{4,9}$^(4,9){ }^{4,9} 对 $t$$t$tt 的幂律依赖性，而我们的数据在任何合理参数的宽时间间隔内都无法用幂律拟合，即使是对于小强度。相反，高振动强度的数据与多个十进制数的 $\log (t)$$\log (t)$log(t)\log (t) 依赖性一致。 对于较小的强度，曲线在几十年的时间跨度内遵循 $\log (t)$$\log (t)$log(t)\log (t) 行为，但在非常大的时间尺度上转变为较慢的衰减率。

To make such dependence plausible, consider a simple model based on the notion that the vibration intensity plays the role of an effective temperature. In analogy with electrical conduction we relate the particle flow $j$$j$jj to an applied field $E$$E$EE by a conductivity $\sigma$$\sigma$sigma\sigma whose $T$$T$TT dependence is governed by an effective rate of escape from random trapping sites. For the rotating drum we replace $j$$j$jj by $d\theta /dt$$d\theta /dt$d theta//dtd \theta / d t and $E$$E$EE by $\theta$$\theta$theta\theta. The motion of particles is impeded by the barriers posed by neighboring beads and the corresponding random potential will be a complicated function of the local configurations. Here, however, we
为了使这种依赖性显得合理，考虑一个基于振动强度起有效温度作用的简单模型。类似于电导，我们通过一个导电率 $\sigma$$\sigma$sigma\sigma 将粒子流 $j$$j$jj 与施加的场 $E$$E$EE 联系起来，其 $T$$T$TT 的依赖性受从随机捕获位点的有效逃逸速率控制。对于旋转鼓，我们将 $j$$j$jj 替换为 $d\theta /dt$$d\theta /dt$d theta//dtd \theta / d t ，将 $E$$E$EE 替换为 $\theta$$\theta$theta\theta 。粒子的运动受到相邻珠子形成的障碍的阻碍，相应的随机势将是一个复杂的局部配置函数。然而，在这里，然而，

FIG. 4. The relaxation of $\theta$$\theta$theta\theta in a stationary drum with glass beads. Vibration intensities increase from top to bottom. Straight lines indicate $\log (t)$$\log (t)$log(t)\log (t) behavior.
are interested in the average effective barrier height $U$$U$UU as the angle $\theta$$\theta$theta\theta is varied. Expanding to first order around the experimental starting value $\theta (t=0)={\theta }_r$$\theta (t=0)={\theta }_r$theta(t=0)=theta_(r)\theta(t=0)=\theta_{r} we have $U\cong U_0+U_1({\theta }_r-\theta )$$U\cong U_0+U_1\left({\theta }_r-\theta \right)$U~=U_(0)+U_(1)(theta_(r)-theta)U \cong U_{0}+U_{1}\left(\theta_{r}-\theta\right). Since we know that spontaneous flow sets in at ${\theta }_m$${\theta }_m$theta_(m)\theta_{m} we require $U\left({\theta }_m\right)=0$$U\left({\theta }_m\right)=0$U(theta_(m))=0U\left(\theta_{m}\right)=0 which gives $\delta \equiv {\theta }_m-{\theta }_r=U_0/U_1$$\delta \equiv {\theta }_m-{\theta }_r=U_0/U_1$delta-=theta_(m)-theta_(r)=U_(0)//U_(1)\delta \equiv \theta_{m}-\theta_{r}=U_{0} / U_{1}. Assuming now an effective temperature $T_{\text{eff}}$$T_{\text{eff }}$T_("eff ")T_{\text {eff }} due to the mechanical vibrations, we obtain a rate of escape over the barrier exponentially dependent on the ratio $U/k{T}_{\text{eff}}$$U/k{T}_{\text{eff }}$U//kT_("eff ")U / k T_{\text {eff }}. This leads to

where $A\equiv A_0\exp (-U_0/k{T}_{\text{eff}})$$A\equiv A_0\exp \left(-U_0/k{T}_{\text{eff }}\right)$A-=A_(0)exp(-U_(0)//kT_("eff "))A \equiv A_{0} \exp \left(-U_{0} / k T_{\text {eff }}\right) and $\beta \equiv U_1/k{T}_{\text{eff}}$$\beta \equiv U_1/k{T}_{\text{eff }}$beta-=U_(1)//kT_("eff ")\beta \equiv U_{1} / k T_{\text {eff }} are independent of $\theta$$\theta$theta\theta. The equation can be solved analytically for $t(\theta )$$t(\theta )$t(theta)t(\theta) in terms of the exponential integral function ${}^{10}$${}^{10}$^(10){ }^{10} $E_1(\beta \theta )$$E_1(\beta \theta )$E_(1)(beta theta)E_{1}(\beta \theta). For $\beta \theta \gg 1$$\beta \theta \gg 1$beta theta≫1\beta \theta \gg 1 and, up to logarithmic corrections in $\theta /{\theta }_r$$\theta /{\theta }_r$theta//theta_(r)\theta / \theta_{r}, the solution is well approximated by $\theta ={\theta }_r-(1/\beta )\ln (\beta A{\theta }_{r}t+1)$$\theta ={\theta }_r-(1/\beta )\ln \left(\beta A{\theta }_{r}t+1\right)$theta=theta_(r)-(1//beta)ln(beta Atheta_(r)t+1)\theta=\theta_{r}-(1 / \beta) \ln \left(\beta A \theta_{r} t+1\right). This reproduces the logarithmic dependence on $t$$t$tt seen in Fig. 4 for times larger than $t_0\equiv 1/\beta A{\theta }_r$$t_0\equiv 1/\beta A{\theta }_r$t_(0)-=1//beta Atheta_(r)t_{0} \equiv 1 / \beta A \theta_{r} and also gives a good fit for shorter times.
$A\equiv A_0\exp (-U_0/k{T}_{\text{eff}})$$A\equiv A_0\exp \left(-U_0/k{T}_{\text{eff }}\right)$A-=A_(0)exp(-U_(0)//kT_("eff "))A \equiv A_{0} \exp \left(-U_{0} / k T_{\text {eff }}\right) 和 $\beta \equiv U_1/k{T}_{\text{eff}}$$\beta \equiv U_1/k{T}_{\text{eff }}$beta-=U_(1)//kT_("eff ")\beta \equiv U_{1} / k T_{\text {eff }} 与 $\theta$$\theta$theta\theta 独立。该方程可以针对 $t(\theta )$$t(\theta )$t(theta)t(\theta) 通过指数积分函数 ${}^{10}$${}^{10}$^(10){ }^{10} $E_1(\beta \theta )$$E_1(\beta \theta )$E_(1)(beta theta)E_{1}(\beta \theta) 进行解析求解。对于 $\beta \theta \gg 1$$\beta \theta \gg 1$beta theta≫1\beta \theta \gg 1 ，以及到对数修正的 $\theta /{\theta }_r$$\theta /{\theta }_r$theta//theta_(r)\theta / \theta_{r} ，解可以用 $\theta ={\theta }_r-(1/\beta )\ln (\beta A{\theta }_{r}t+1)$$\theta ={\theta }_r-(1/\beta )\ln \left(\beta A{\theta }_{r}t+1\right)$theta=theta_(r)-(1//beta)ln(beta Atheta_(r)t+1)\theta=\theta_{r}-(1 / \beta) \ln \left(\beta A \theta_{r} t+1\right) 良好地近似。这再现了在图 4 中大于 $t_0\equiv 1/\beta A{\theta }_r$$t_0\equiv 1/\beta A{\theta }_r$t_(0)-=1//beta Atheta_(r)t_{0} \equiv 1 / \beta A \theta_{r} 的时间上观察到的对数依赖性，同时也为较短时间提供了良好的拟合。

For small vibration intensities the assumption that mechanical vibrations mimic an effective temperature fails when $k{T}_{\text{eff}}$$k{T}_{\text{eff }}$kT_("eff ")k T_{\text {eff }} becomes less than $U$$U$UU. In contrast to thermal fluctuations we expect the mechanical energy distribution to be cut off sharply above a finite value corresponding to the maximum vibration intensity. Particles therefore get stuck when $\beta ({\theta }_r-\theta )<c$$\beta \left({\theta }_r-\theta \right)<c$beta(theta_(r)-theta) < c\beta\left(\theta_{r}-\theta\right)<c, where $c$$c$cc is some constant of order unity, and the logarithmic decay seen in Fig. 4 flattens out. Also at very long times it becomes difficult to determine the angle $\theta$$\theta$theta\theta precisely since the profile of the free surface starts to deviate from a straight line. $\theta$$\theta$theta\theta in Fig. 4 is therefore the average angle,
对于小振动强度，当 $k{T}_{\text{eff}}$$k{T}_{\text{eff }}$kT_("eff ")k T_{\text {eff }} 小于 $U$$U$UU 时，机械振动模拟有效温度的假设失效。与热波动相反，我们预计机械能量分布将在对应最大振动强度的有限值以上急剧截止。因此，当 $\beta ({\theta }_r-\theta )<c$$\beta \left({\theta }_r-\theta \right)<c$beta(theta_(r)-theta) < c\beta\left(\theta_{r}-\theta\right)<c 时，粒子会卡住，其中 $c$$c$cc 是约等于 1 的常数，图 4 中看到的对数衰减会变得平坦。此外，在非常长的时间内，由于自由表面的轮廓开始偏离直线，很难精确确定角度 $\theta$$\theta$theta\theta 。因此，图 4 中的 $\theta$$\theta$theta\theta 是平均角度。