二、三維隨意堆積顆粒系統之 Voronoi 圖的幾何統計有許多實際的應用，吸引了物理、數學和生物學家的興趣。由於過去研究著重在硬圓盤與硬圓球，本研究將針對顆粒軟度的影響，探討面積（體積）、周長（表面積）和邊數（面數）等幾何量的統計分佈，並將結果與硬顆粒的情況比較。我們從（LAMMPS）分子動力學 (MD) 模擬、理論模型與實驗三方面齊頭並進。

我們在 MD 模擬軟顆粒的方式，是採用束縛粒子模型（Bonded-granule Model, BPM），透過 Lubachevsky–Stillinger（LS）演算法控制系統的 packing ratio，獲得穩定結構後，進行 Voronoi 劃分。由於二維顆粒的面積容易因變形而改變，我們留意控管使變化率小於10\%；至於三維的軟顆粒，體積變化則永遠低於5\%。
前人的理論只針對二維硬圓盤的 Voronoi 面積或邊數，以及三維的體積分佈。我們在此研究軟顆粒 Voronoi cell 的面積/體積、周長/表面積、邊數/面數分布，並和硬顆粒的情況做比對。針對上述幾何量，我們使用 $AIC$ 比較若干模型的優劣，發現對於軟顆粒，所有統計量皆以 Gamma 分布描述最佳；對於硬顆粒，除周長/表面積外，皆以 Gamma 分布最佳。周長/表面積在小 $\phi$ 時使用 generalized Gaussian model (origin) 最好，之後以 Gamma 分布描述較好，但在表面積在 $\phi>0.55$ 之後以 generalized Gaussian 分布描述顯著較好，推測這是因為系統進入 jammed state 的關係。
我們改良出一套基於巨正則系綜的統計理論模型，能同時預測二維硬圓盤的這兩個性質，並推廣到周長。該模型在小於 0.8 的範圍內皆具備一定解釋力。雖然理論模型原則上可推廣至三維以及軟顆粒的情況，但目前由於數值實作上的困難，本文中並未展示。
實驗部分，以前學長將水晶寶寶浸泡溶液，使其與溶液的折射率相同，以方便用雷射斷層掃描來決定內部結構。我們也嘗試使用 X-ray 斷層掃描，與直接（擺進冷凍室）固化樣本後，切割來比對斷層資訊。在論文中會說明，儘管我們花費了眾多人力與時間，這三種嘗試最終都無法成功的理由。

綜合而言，我們針對軟顆粒在隨意堆積下的 Voronoi diagram，先藉由 MD 模擬得到其統計幾何性質；接著為了彰顯我們具體統計理論模型的正確性，先示範能夠重現前人研究過的硬圓盤性質，再拓展至二維軟顆粒。這個模型的優勢是能夠針對二維顆粒系統，同時預測最多幾何統計性質；雖然我們的模型可以被推廣到三維，但是礙於目前無法寫出程式來對其進行數值分析，這一部分的著墨有待後人完成。

The geometric statistics of Voronoi diagrams in two- and three-dimensional randomly packed granular systems have numerous practical applications, attracting interest from physicists, mathematicians, and biologists. While previous studies have primarily focused on hard disks and hard spheres, our work investigates the influence of granule softness and extends the analysis to the statistical distributions of geometric quantities such as area (volume), perimeter (surface area), and number of polygon sides (polyhedron faces). This study integrates Molecular Dynamics (MD) simulations (via LAMMPS), theoretical modeling, and experimental efforts.

In our MD simulations of soft granules, we adopt the Bonded-granule Model (BPM) and use the Lubachevsky–Stillinger (LS) algorithm to control the packing ratio and generate stable configurations, followed by Voronoi tessellation. Since the area of 2D granules is more sensitive to deformation, we ensure that area variations remain below 10\%. For 3D soft granules, volume fluctuations are always kept below 5\%.
Previous theoretical work has mainly addressed the distributions of Voronoi area or number of sides for 2D hard disks, and the volume distribution for 3D hard spheres. In this work, we systematically analyze the distributions of area/volume, perimeter/surface area, and number of edges/faces for soft granules, and compare them with those of hard granules. We apply the Akaike Information Criterion (AIC) to evaluate various candidate distributions and find that for soft granules, all geometric quantities are best described by the Gamma distribution. For hard granules, Gamma remains the best fit for all but the perimeter/surface area. At low packing ratios, the perimeter/surface area distributions are better described by a generalized Gaussian (origin) model, while at higher packing ratios Gamma performs better. However, for 3D surface area beyond $\phi>0.55$, the generalized Gaussian model outperforms Gamma, likely due to the onset of jamming.
Theoretically, we develop an improved statistical model based on the grand canonical ensemble, which accurately predicts both the area and number of polygon sides distributions in 2D hard disk systems, and can be further extended to the perimeter. This model shows good predictive power up to $\phi \approx 0.8$. While the framework can in principle be generalized to 3D and to soft granules, numerical implementation remains challenging and is not presented in this work.
On the experimental front, previous students in our lab attempted to match the refractive index of hydrogel beads to that of the surrounding solution, enabling internal structure detection via laser tomography. We also explored X-ray tomography, as well as freezing and slicing the sample to retrieve cross-sectional data. In this thesis, we detail why, despite considerable effort and manpower, all three approaches ultimately failed.

In summary, we first obtain the statistical geometric properties of randomly packed soft granules via MD simulations, and then validate our theoretical model by showing its consistency with established results on hard disks, before extending it to soft granules. The key strength of our model lies in its ability to predict multiple geometric statistical quantities in 2D granular systems. Although the model is in principle extendable to 3D, its full realization remains a challenge for future research.