局部激發在一個垂直振動顆粒層

Localized excitations in a vertically vibrated granular layer 

Paul B. Umbanhowar*, Francisco Melot & Harry L. Swinney

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研究背景與目標

• 繁體中文：在生物、化學及物理系統中，二維圖案的形成常透過平面波的非線性相互作用來描述。

• English: The formation of two-dimensional patterns in biological, chemical, and physical systems is often described by the nonlinear interaction of plane waves.

• 繁體中文：另一種方法將圖案視為局部化物件的集合，類似於原子組成晶體的過程。

• English: An alternative approach views patterns as ensembles of interacting localized objects, analogous to the assembly of crystals from atoms.

• 繁體中文：在巨觀圖案形成系統中，局部化激發可類比為“原子”的角色，儘管它們實際上並不存在。

• English: For macroscopic pattern-forming systems, spatially localized excitations can play an analogous role to 'atoms', although they do not physically exist.

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一維與二維局部化態的研究現狀

• 繁體中文：一維局部化態常見於水中的孤立波及光纖等系統，並可形成簡單的圖案結構。

• English: One-dimensional localized states are observed in many systems, such as solitary waves in water and optical fibers, and can organize into simple patterns.

• 繁體中文：然而，二維局部化態的例子稀少，且多數不穩定或無法形成簡單的圖案結構。

• English: However, few examples of two-dimensional localized states are known, and these are often unstable or fail to form simple patterns.

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研究成果與發現

• 繁體中文：本研究發現穩定的二維局部化激發，稱為「振盪子」（oscillons），出現在垂直振動的砂層中。

• English: This study reports the observation of stable, two-dimensional localized excitations, termed 'oscillons', in a vibrating layer of sand.

• 繁體中文：「振盪子」傾向於組成分子及晶體結構。

• English: The 'oscillons' have a propensity to assemble into 'molecular' and 'crystalline' structures.

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意義與推論

• 繁體中文：實驗結果表明，滯後性與耗散性在振盪子的形成中具有關鍵作用，並可能出現在連續介質中。

• English: The experimental results indicate a crucial, cooperative role for hysteresis and dissipation in the formation of oscillons, and suggest similar behavior may occur in continuous media.

• 繁體中文：類似的局部化激發也出現在模型微分方程中，支持這一發現。

• English: Similar localized excitations observed in model differential equations further support this finding.

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研究背景與顆粒材料特性

• 繁體中文：振盪子（oscillons）出現在顆粒材料中，這是一種高耗散性介質，由透過接觸力相互作用的宏觀顆粒組成。

• English: Oscillons are found in granular materials, a highly dissipative medium composed of macroscopic grains that interact through contact forces.

• 繁體中文：儘管對顆粒材料的研究興趣濃厚，但它們仍然是尚未被充分理解的系統。

• English: Despite substantial interest, granular materials remain poorly understood systems.

• 繁體中文：維持顆粒材料運動需要能量來源。

• English: To maintain motion in granular materials, an energy source is required.

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顆粒材料的動力學現象

• 繁體中文：實驗表明，垂直振動板供能可導致多種現象，包括堆積形成、對流、尺寸分離、氣泡化，以及次諧波駐波的生成。

• English: Experiments with vertically vibrated plates show phenomena such as heap formation, convection, size segregation, bubbling, and subharmonic standing waves.

• 繁體中文：除次諧波駐波外，其他現象主要由周圍氣體和/或側壁摩擦引起。

• English: All but the subharmonic standing waves are caused by the surrounding gas and/or sidewall friction.

• 繁體中文：顆粒駐波完全由粒子間的碰撞協作行為引發，可呈現條紋、方形或六邊形，具體形狀取決於振動頻率與振幅。

• English: Granular standing waves arise solely from cooperative particle collisions and can form stripes, squares, or hexagons depending on the vibration frequency and amplitude.

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振盪子的形成條件與實驗設置

• 繁體中文：振盪子形成於顆粒駐波的相同條件下。

• English: Oscillons form under the same conditions required for granular standing waves.

• 繁體中文：實驗中，顆粒層位於一個垂直驅動的圓柱容器底部，該容器處於真空環境中。

• English: In the experiment, a granular layer is placed at the bottom of a vertically driven cylindrical container in a vacuum environment.

• 繁體中文：振動以位移函數表示為 $z = A \sin(2\pi f t)$，其中 $A$ 是位移振幅，$f$ 是驅動頻率。

• English: The vibration displacement is given by $z = A \sin(2\pi f t)$, where $A$ is the displacement amplitude and $f$ is the drive frequency.

• 繁體中文：控制參數為驅動頻率 $f$ 和無量綱加速度振幅 $r = \frac{4\pi^2 A f^2}{g}$（$g$ 為重力加速度）。

• English: The control parameters are the drive frequency $f$ and the dimensionless acceleration amplitude $r = \frac{4\pi^2 A f^2}{g}$ (where $g$ is the acceleration due to gravity).

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振盪子的特性

• 繁體中文：振盪子是一種小型、圓形對稱的激發，振盪頻率為 $f/2$。

• English: An oscillon is a small, circularly symmetric excitation oscillating at frequency $f/2$.

• 繁體中文：在容器的一個週期內，振盪子是一個峰值；在下一個週期則變成一個凹陷（坑洞）。

• English: During one cycle of the container, the oscillon is a peak; in the next cycle, it becomes a crater.

• 繁體中文：由於振盪子是次諧波，因此峰值和坑洞可以共存。

• English: Because oscillons are subharmonic, peaks and craters can coexist.

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振盪子的行為特徵

• 繁體中文：在每隔一個週期觀察顆粒層時，坑洞始終保持為坑洞，峰值也始終保持為峰值。

• English: Observing the layer on alternate cycles shows that craters remain craters and peaks remain peaks.

• 繁體中文：振盪子的直徑依賴於 $r$，但通常為 30 顆粒徑，約為容器直徑的 4%。

• English: The diameter of oscillons depends on $r$ but is typically 30 particle diameters (about 4% of the container diameter).

• 繁體中文：振盪子在層的平面內以均等概率形成，壽命長，通常可持續超過 $5 \times 10^5$ 次容器振動。

• English: Oscillons form with equal probability at all locations within the plane and are long-lived, often persisting for more than $5 \times 10^5$ container oscillations.

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振盪子的移動與形態

• 繁體中文：振盪子本質上不會傳播，但會隨機緩慢漂移，大約需要 $10^3$ 週期移動一個自身直徑的距離。

• English: Oscillons do not propagate but drift slowly and randomly, taking about $10^3$ cycles to move a distance equal to their diameter.

• 繁體中文：在振動週期的一個時刻，振盪子呈現為被抬升物質環繞的圓形凹陷（坑洞）。

• English: At one point in the cycle, the oscillon appears as a circular crater surrounded by a raised ring of displaced material.

• 繁體中文：在下一週期，振盪子變為一個高出層厚度約四倍的峰值。

• English: In the next cycle, the oscillon becomes a peak extending to about four times the layer depth.

• 繁體中文：無論是坑洞還是峰值，振盪子均由其附近區域的物質位移形成。

• English: In both phases, the oscillon is formed by material displaced from its immediate vicinity.

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圖例描述（FIG. 1）

振盪子的表現與視角

• 繁體中文：圖 1a 顯示兩個相位相反的振盪子。

• English: Figure 1a shows two oscillons with opposite phases.

• 繁體中文：圖 1b 和 1c 顯示單個振盪子的俯視圖，時間相差 $1/f$。

• English: Figures 1b and 1c show the top view of a single oscillon at times differing by $1/f$.

• 繁體中文：圖 1d 和 1e 為相應的側視圖（振動頻率 $f = 26 \, \text{Hz}$，加速度參數 $r = 2.45$，層厚度為 17 顆粒徑）。

• English: Figures 1d and 1e show corresponding side views ($f = 26 \, \text{Hz}$, $r = 2.45$, layer depth of 17 particle diameters).

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顆粒特性與實驗設置

• 繁體中文：顆粒為直徑 0.15–0.18 毫米的青銅球，彼此緊密接觸，可見於圖 1b–1e。

• English: The particles are 0.15–0.18 mm diameter bronze spheres in close contact, as shown in Figures 1b–1e.

• 繁體中文：圖中展示了容器內 7 毫米 × 7 毫米的區域，容器直徑為 127 毫米。

• English: The images show 7 mm × 7 mm regions of the 127 mm diameter container.

• 繁體中文：容器內光源從側面照射，光脈衝與驅動信號同步，相位一致。高振幅區域為亮色，低振幅區域為暗色。

• English: The images are side-illuminated, with light strobed in phase with the drive signal. Regions with large amplitudes are bright, while regions with small amplitudes are dark.

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真空條件

• 繁體中文：容器內壓力降至 0.1 torr，此壓力下殘餘氣體的影響可忽略不計。
• English: The container is evacuated to 0.1 torr, at which the effects of the remaining gas are negligible.

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圖例描述（FIG. 2）

穩定區域與相圖特性

• 繁體中文：圖 2 顯示不同狀態的穩定區域，作為頻率 $f$ 和加速度參數 $r$ 的函數，並區分 $r$ 增加與減少的過程。

• English: Figure 2 shows the stability regions for different states as a function of frequency $f$ and acceleration parameter $r$, distinguishing between increasing $r$ and decreasing $r$.

• 繁體中文：使用方形符號表示 $r$ 增加，使用三角形和圓形符號表示 $r$ 減少。

• English: Squares represent increasing $r$, while triangles and circles represent decreasing $r$.

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平層與圖案化轉換的滯後現象

• 繁體中文：從平層轉換為方形或條紋圖案具有滯後性，但條紋的滯後效應較小。
• English: Transitions from the flat layer to squares and stripes are hysteretic, with stripes exhibiting much smaller hysteresis.

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振盪子與顆粒層厚度的影響

• 繁體中文：振盪子出現在顆粒層厚度超過 13 顆粒徑的情況下，其出現的頻率範圍隨層厚度增加而擴大。

• English: Oscillons are observed in layers greater than 13 particle diameters thick, with their frequency range increasing as the layer depth increases.

• 繁體中文：對於更薄的顆粒層，相圖類似，但不包括振盪子區域。

• English: For thinner layers, the phase diagram is similar but without the oscillon region.

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振盪子的穩定結構與排列

• 繁體中文：除了雙極結構外，還存在其他穩定的峰值與坑洞組合，其中構成振盪子的配位數超過一。

• English: In addition to dipoles, other stable combinations of peaks and craters exist, where the constituent oscillons have coordination numbers greater than one.

• 繁體中文：圖 3c、3d 顯示了內部振盪子具有配位數為二的鏈狀結構，這些鏈條呈直線排列，表明即使未直接接觸，振盪子之間仍存在相互作用，可能由鏈條中的連接點傳遞。

• English: Figures 3c and 3d show a chain where the internal oscillons have a coordination number of two. The chains are straight, indicating that bound oscillons influence each other even when not in direct contact, possibly mediated by the intervening links.

• 繁體中文：三角形四聚結構（tetramer）也會形成，其中央振盪子的配位數為三（見圖 3e、3f）。

• English: Triangular tetramer structures also form, with the central oscillon having a coordination number of three (see Figs. 3e and 3f).

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配位數限制與圖案分解

• 繁體中文：未觀察到配位數為四或更高的孤立複合結構。

• English: Isolated compound structures with coordination numbers of four or higher are not observed.

• 繁體中文：當 $r$ 稍微降低到方形圖案穩定性邊界以下時，方形圖案會在數百個週期內被平層入侵。在此過程中，小型鏈狀結構和獨立振盪子從晶格中分離並最終消失（見圖 3g）。

• English: When $r$ is slightly decreased below the lower stability boundary for squares, the flat layer invades the square pattern over hundreds of cycles. During this process, small chains and individual oscillons separate from the lattice and then disappear (see Fig. 3g).

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振盪子排列的類比

• 繁體中文：在這一區域內，穩定的平面方形圖案可能由峰值與坑洞組成，其排列類似於離子晶體。
• English: In this region, stable planar square patterns are likely composed of peaks and craters arranged similarly to an ionic crystal.

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振盪子與平面圖案的關聯

• 繁體中文：進一步的證據顯示，平面圖案在某些情況下可被視為振盪子的集合，這是通過在振頻 $f/2$ 添加次諧強迫項的實驗所證明的。

• English: Additional evidence shows that planar patterns can, in some cases, be considered as collections of oscillons, as demonstrated by experiments with an added subharmonic forcing term at frequency $f/2$.

• 繁體中文：時間平移對稱性 $t \rightarrow - t + 1/f$ 被打破，平面六邊形圖案在之前出現方形和條紋圖案的區域形成（見圖 4）。

• English: The time translation symmetry $t \rightarrow - t + 1/f$ is broken, and planar hexagonal patterns form where squares and stripes appeared previously (see Fig. 4).

• 繁體中文：當 $\gamma$ 減小進入振盪子區域時，六邊形圖案會變成由相同相位的振盪子組成的六邊形晶體。這些振盪子不會結合，因為它們之間有弱排斥作用，這與單頻振盪子相同。

• English: When $\gamma$ is decreased into the oscillon region, the hexagonal pattern becomes a hexagonal crystal of oscillons with identical phase. These oscillons do not bind together, as they are weakly repelling, similar to single-frequency oscillons with the same phase.

• 繁體中文：從這一狀態增加 $\gamma$ 會使振盪子擴展，直到恢復蜂窩狀的細胞圖案。

• English: Increasing $\gamma$ from this state makes the oscillons expand until the cellular honeycomb pattern is recovered.

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振盪子形成與互動的解釋

• 繁體中文：目前沒有適用於顆粒介質的運動方程，可以對振盪子的形成和相互作用提供定量解釋。然而，基於滯後和耗散在局部結構形成中的角色的定性解釋提供了很多洞見。

• English: There is no equation of motion for granular media that can provide a quantitative explanation of the formation and mutual interaction of oscillons. However, a qualitative explanation based on the roles of hysteresis and dissipation in localized structure formation provides much insight.

• 繁體中文：為了使振盪子存在，主要的分叉必須具有滯後性，以便波和平層能在相同的參數值下共存。在我們的實驗中，低頻時滯後性較大，高頻時較小。

• English: For oscillons to exist, the primary bifurcation must be hysteretic so that waves and the flat layer can coexist at the same parameter values. In our experiments, hysteresis is large for low $f$ and small for high $f$.

• 繁體中文：然而，振盪子僅出現在滯後性隨著頻率增大而減少的頻率範圍內（見圖 2）。我們認為，對於低頻，耗散太小，無法穩定振盪子。

• English: Oscillons, however, appear only in a range of $f$ where the hysteresis is decreasing with increasing frequency (see Fig. 2). We suggest that dissipation, which keeps structures localized, is too small to stabilize oscillons for low $f$.

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耗散作用的角色與振盪子形成的實驗分析

• 繁體中文：為了探討耗散作用的可能角色，我們進行了使用不同直徑 $D$ 顆粒的實驗。結果顯示，振盪子形成的頻率與 $D^{-1/2}$ 成正比。

• English: To investigate the possible role of dissipation, we conducted experiments using particles with different diameters $D$. The frequency at the center of the range in which oscillons occur was found to be proportional to $D^{-1/2}$.

• 繁體中文：這一比例可以輕易理解：當顆粒的相對動能與將顆粒抬升一個直徑所需的勢能之比 $v^2 / gD$ 很大時（其中 $v$ 是相對於相鄰顆粒的垂直速度），顆粒的水平流動性高，耗散較小。

• English: This scaling can be easily understood: for large ratios of the relative kinetic energy of a particle to the potential energy needed to raise the particle by one diameter, that is $v^2 / gD$ (where $v$ is the vertical velocity relative to a neighboring particle), the horizontal mobility is high and the dissipation is small.

• 繁體中文：假設 $\gamma$ 是常數，並且假設 $v$ 與最大容器速度成正比（使得 $v \propto f^{-1}$），那麼在低頻 $f$ 下，$v^2 / gD \gg 1$，耗散較小；而在高頻 $f$ 下，$v^2 / gD \ll 1$，耗散較大。

• English: We assume that $\gamma$ is constant and that $v$ is proportional to the maximum container velocity (so that $v \propto f^{-1}$); then at low $f$, $v^2 / gD \gg 1$ and the dissipation is small, while at high $f$, $v^2 / gD \ll 1$ and the dissipation is large.

• 繁體中文：觀察到的中心頻率與顆粒直徑 $D$ 之間的比例符合 $v^2 / gD$ 或等效的常數耗散值。因此，振盪子只在耗散與滯後作用較大的狹窄頻率範圍內形成。

• English: The observed scaling of the center frequency for oscillon formation with $D$ is consistent with a constant value of $v^2 / gD$ or, equivalently, a constant dissipation. Thus, oscillons form only in the narrow range of $f$ where both dissipation and hysteresis are large.

• 繁體中文：對於垂直振動的流體（法拉第系統），在低粘度下會出現方形圖案，而在高粘度下則會出現條紋圖案。這一過渡與我們的實驗觀察結果一致，即隨著 $f$ 增加進入振盪子區域。

• English: Support for the dependence of dissipation on $f$ also comes from the observation that in vertically vibrated fluids (the Faraday system), square patterns appear at low viscosity while stripes occur at high viscosity; this is the same transition observed in our experiment as $f$ is increased through the oscillon region.

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數值模擬與振盪子形成的理論分析

• 繁體中文：具有滯後和耗散作用的模型系統的數值模擬也顯示了局部的二維結構。

• English: Numerical simulations of model systems with hysteresis and dissipation also show localized two-dimensional structures.

• 繁體中文：成對的局部激發可以形成類似於我們的振盪子雙極結構的綁定態：相鄰局部激發的中心之間的距離略大於一個半徑；個別解決方案在綁定後保持其身份；而配對結構是穩定的。

• English: Pairs of localized excitations can form bound states similar to our oscillon dipoles: the centers of adjacent localized excitations are separated by slightly more than one radius; individual solutions maintain their identity after binding; and paired structures are stable.

• 繁體中文：我們的實驗和這些模擬表明，滯後和耗散在二維局部結構形成中的關鍵合作作用，並暗示具有這些特性的其他物理系統中可能也會發生類似行為。

• English: Our experiments and these simulations indicate the crucial, cooperative role of hysteresis and dissipation in the formation of two-dimensional localized structures, and suggest that similar behavior may also occur in other physical systems with these properties.

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