臺灣博碩士論文加值系統

我們研究宏觀下被釐米大小的顆粒氣體(視為分子)所環繞的物體(公分大小)之布朗運動(Brownian motion)行為，當振動台給予系統垂直振動時，我們利用高速攝影技術來追蹤物體在水平面之位置、速度、方向及角速度。藉由這些影像，我們測量物體在水平面之兩個方向的移動及轉動的平均動能，令人驚訝的是，我們發現在誤差範圍下物體的移動自由度和轉動自由度的平均動能是相同的，即在非平衡態下的顆粒體系統中物體是遵守能量均分定律的。之後，我們做了二維分子動力學模擬，但是發現除非粒子的表面極度粗糙否則粒子無法有能量均分現象。

We study the Brownian motion of a macroscopic (centimeter size) granular object surrounded by a granular gas composed of millimeter size spheres acting as the molecules. While the system is vibrated vertically by an electromagnetic shaker, we use a fast camera to capture the temporal variations of the horizontal position and the orientation from above. From the captured image sequences, we manage to measure the translational and the rotational velocities in the horizontal plan. Suprisingly, we find that the average kinetic energies carried by the translational degrees of freedom and the rotational degree of freedom of the disk are the same within experimental uncertainty. Hence energy equipartition is found to valid even for nonequilibrium granular systems.
We find granular object obey Equipartition law. Then, we do the two dimentional molecular dynamic simulation and we find granular object won’t obey Equipartition law unless the granular object surface is extremely rough.

摘要 i
Abstract ii
目錄 iii
圖、表目錄 v
第一章 緒論 1
1-1 顆粒體 2
1-1-1 顆粒體介紹 2
1-1-2 顆粒氣體 3
1-1-3 顆粒氣體的能量均分問題 5
1-2 目標和計畫 7
1-2-1實驗方法 7
1-2-2論文大綱 8
第二章 實驗 10
2-1 實驗過程 11
2-1-1 實驗步驟 11
2-1-2 轉動慣量、碰撞系數、摩擦係數量測 12
2-2實驗數據分析 14
2-2-1 影像、數據分析 14
2-2-2 位置和速度變化 15
2-2-3 能量統計時間 16
2-3 顆粒布朗物體的和顆粒氣體的速度分布 17
2-4顆粒布朗物體的平均能量 18
2-4-1 顆粒布朗物體遵守能量均分 18
2-4-2 振動強度不影響能量均分 18
2-4-3 質量和大小不影響能量均分 19
2-5 實驗結論 19
第三章 分子動力學模擬 21
3-1系統設置 22
3-1-1 碰撞模型 22
3-1-2 溫度調節 24
3-1-3 蛙跳積分法 25
3-1-4 無因次單位 26
3-1-5 細節處理 27
3-1-6 系統測試 28
3-2模擬結果 29
3-2-1 顆粒布朗物體能量不均分 29
3-2-2 改變垂直碰撞係數 系統能量不均分 30
3-2-3 改變摩擦係數 對平均能量的影響 30
3-3模擬結論 30
第四章 討論與結論 32
參考資料 61
附錄:程式 63
gif2xyat 63
do.060302.s 66
m.c 68
m.h 77
m0.h 79
m1.h 81
m.in 85
do.m.s 85

[1]R. A. L. Jones, Soft Condensed Matter, Oxford University Press(2002).

[2]Stephen J. Blundell and Katherine M.Blundell, Thermal Physics, Oxford University Press(2010).

[3]賈魯強,黎璧賢,漫談顆粒體物理,物理雙月刊,23卷4期,503 (2001).

[4]T. Wang, K. To, Granular gas in a vibrating box, Chin. J. Phys. 45, 675 (2007).
[5]陳文楠, An experimental study of the motion of a quasi two-dimensional granular gas, 國立清華大學物理所碩士論文(2008).

[6]W. Chen, K. To, Unusual diffusion in a quasi-two-dimensional granular gas, Phys. Rev. E 80, 061305 (2009).

[7]K. To, Boltzmann distribution in a nonequilibrium steady state: Measuring local potential by granular Brownian particles, Phys. Rev. E 89, 062111 (2014).

[8]J. S. van Zon, F. C. MacKintosh, Velocity distributions in dilute granular systems, Phys. Rev. E 72, 051301 (2005).

[9]W. Losert, D.G.W. Cooper, J. Delour, A. Kudrolli, J.P. Gollub, Velocity statistics in excited granular media, Chaos 9, 682 (1999).

[10]F. Rouyer, N. Menon, Velocity fluctuations in a homogeneous 2D granular gas in steady state, Phys. Rev. Lett. 85, 3676 (2000).

[11]J.S. Olafsen, J.S. Urbach, Velocity distributions and density fluctuations in a granular gas, Phys. Rev. E 60, R2468 (1999).

[12]G.W. Baxter, J.S. Olafsen, The temperature of a vibrated granular gas, Granular Matter 9, 135 (2007).

[13]K. Feitosa, N. Menon, Breakdown of energy equipartition in a 2D binary vibrated granular gas, Phys. Rev. Lett. 88, 198301 (2002).

[14]K. Nichol, K.E. Daniels, Equipartition of rotational and translational energy in a dense granular gas, Phys. Rev. Lett. 108, 018001 (2012).

[15]S. McNamara, S. Luding, Energy nonequipartition in systems of inelastic, rough spheres, Phys. Rev. E 58, 2247 (1998).

[16]D. C. Rapaport, The Art of Molecular Dynamics Simulation - 2nd ed., Cambridge University Press, (2004).

[17]S. Luding, Granular materials under vibration: Simulations of rotating spheres, Phys. Rev. E 52, 4442 (1995).

[18]A. Barrat, E. Trizac, Molecular dynamics simulations of vibrated granular gases, Phys. Rev. E 66, 051303 (2002).

[19]O. Herbst, R. Cafiero, A. Zippelius, H.J. Herrmann, S. Luding, A driven two-dimensional granular gas with Coulomb friction, Phys. Fluids 17, 107102 (2005).

[20]A. Barrat, E. Trizac, Lack of energy equipartition in homogeneous heated binary granular mixtures, Granular Matter 4, 57 (2002).

[21]N.V. Brilliantov, T. Po¨schel, W. T. Kranz, A. Zippelius, Translations and Rotations Are Correlated in Granular Gases, Phys. Rev. Lett. 98, 128001 (2007).