自從上個世紀五零年代左右, Enskog在假設碰撞只牽涉到兩個分子的前提下給出了波茲曼方程的近似解. 這是第一次科學家能夠對氣體分子擴散有一個普適與成功的描述. 除了Rosenfeld在1977年的先驅工作, 擴散理論上的進展停滯了將近40年之久. 其中的困難點主要是在於液體分子間的距離只比固體的多百分之十左右, 所以液體分子的擴散機制是迥異於氣體的(碰撞). 液體分子大部分的時間都被附近鄰居困在原地作微小的隨機震盪. 只有當其附近的鄰居在某個時刻出現較為鬆散的結構時, 該液體分子才可以離開原地而擴散. 在氣體中, 因為分子距離較遠, 因此碰撞的擴散機制較與分子間作用力無關. 然而, 在液體中分子的距離是如此的靠近, 因此形形色色的作用力, 也對描述液體結構造成了困難. 因此對液體分子的擴散一直缺乏一個普適性的描述.
1996年, Dzugutov基於上述機制, 提出了一個普適的, 經驗性的指數定律, 可以很好的解釋不同液體分子的擴散. 2004年Samanta等人, 更從Mode coupling theory出發得到了更完全的結果. 然而由於同時量測分子擴散與結構上的困難, 長久以來都沒有實驗能夠驗證這些關係.
在這篇論文裡, 我們建造了一個新的二維氣墊床顆粒系統. 在這個系統裡的顆粒能夠浮在一個通氣平板上. 依靠改變顆粒間的作用力, 密度, 以及風速, 這些鈕扣狀顆粒可以形成稀薄液體(模擬氣體), 稠密液體(液體), 與晶體. 這些顆粒的速度分佈與真實液體相同, 都是高斯分佈. 藉由此分佈, 我們定義了這些顆粒的”溫度”. 這個速度分佈只有在顆粒密度足夠高(依舊稀薄)的條件下才會發生, 所以我們推論其來源並不是氣墊床的氣體, 而是經由多體互相交互作用而來. 我們同時也量測了液體的結構, 擴散微觀運動與擴散行為. 最後將所量測到的擴散關係與至少三個上述的理論比較. 結果相符合的程度令人驚訝, 特別是這些理論是針對三維真實液體而不是二維顆粒流體. 這篇論文也闡述了這個新的系統的核心組成, 與這個系統的基本特性.

In this thesis, a new 2D air table granular system has been built and demonstrated its advantages on "simulating" such transportation and micro-dynamics of real liquids.
The main components of this air table system are a pinhole plate and a vertical wind tunnel.
The detail of how we made such pinhole plates and the design of the vertical wind tunnel are also given in this article.
Besides of the pinhole plate and the uniform airflow, the flexibility on the control of interactions and the shapes of particles, compared to traditional granular systems, also makes this system to be suitable on studying the topics in related condensed matter.
We found the statistics of these far from equilibrium granules is very close to the Boltzmann statistics and reveals Gaussian velocity distribution functions (GVDF).
The origin of the GVDF, unlike to previous works, is from many-body dynamics, which makes these granular liquids very similar to real liquid molecules.
We also found that the reduced dimensionless diffusion constant $D^*$ and excess entropy $S^*_2$ follow two distinct scaling laws D*~e^{S*2} for dense liquids and D*~ e^{3S^*_2} for dilute ones.
The scaling for dense liquids is very similar to the laws for 3D real liquids proposed previously [Nature (London) 381, 137 (1996); Phys. Rev. Lett. 92, 145901 (2004)].
In the dilute regime, a power law [J. Phys. Condens. Matter 11, 5415 (1999)] also fits our data reasonably.
In our system, particles experience low air drag dissipation and interact with each other through embedded magnets.
These near-conservative many-body interactions together with Boltzmann statistics, we believe, are responsible for the satisfied correspondence between our results to those laws for 3D real liquids.
The dominance of cage relaxations in dense liquids also leads to the two different scaling laws for dense and dilute regimes.
The phases diagrams with number densities and "temperature" are investigated and the structures are identified by the pair correlation function.
A detail description on microscopic stringlike trajectories is also given in this work.
The stringlike trajectories are the trace of 7-fold defects accompanied by rotated (caged) order patches on both sides.

[1] S. Chapman and T. G. Cowling. The Mathematical Theory of Non-uniform
Gases. Cambridge University Press, Cambridge, 3 edition, 1990.
[2] Mikhail Dzugutov. A universal scaling law for atomic diffusion in con-
densed matter. Nature (London), 381(6578):137{139, May 1996.
[3] John D. Weeks, David Chandler, and Hans C. Andersen. Role of repulsive
forces in determining the equilibrium structure of simple liquids. The
Journal of Chemical Physics, 54(12):5237{5247, 1971.
[4] Yoel Forterre Bruno Andreotti and Olivier Pouliquen. Granular Media:
Between Fluid and Solid. Cambridge University Press, Cambridge, 1st
edition, 2013.
[5] Rongxin Huang, Isaac Chavez, Katja M. Taute, Branimir Lukic, Sylvia
Jeney, Mark G. Raizen, and Ernst-Ludwig Florin. Direct observation of
the full transition from ballistic to diffusive brownian motion in a liquid.
Nat Phys, 7(7):576{580, July 2011.
[6] Simon Kheifets, Akarsh Simha, Kevin Melin, Tongcang Li, and Mark G.
Raizen. Observation of brownian motion in liquids at short times: Instan-
taneous velocity and memory loss. Science, 343(6178):1493{1496, 2014.
[7] G. Baxter and J. Olafsen. Experimental evidence for molecular chaos in
granular gases. Phys. Rev. Lett., 99:028001, Jul 2007.
55
[8] David Bray, Michael Swift, and P. King. Velocity statistics in dissipative,
dense granular media. Phys. Rev. E, 75:062301, Jun 2007. Theory.
[9] Z.A. Daya, E. Ben-Naim, and R.E. Ecke. Experimental characterization
of vibrated granular rings. The European Physical Journal E, 21(1):1{10,
2006.
[10] K. Kohlstedt, A. Snezhko, M. Sapozhnikov, I. Aranson, J. Olafsen, and
E. Ben-Naim. Velocity distributions of granular gases with drag and with
long-range interactions. Phys. Rev. Lett., 95:068001, Aug 2005.
[11] W. Losert, D. G. W. Cooper, J. Delour, A. Kudrolli, and J. P. Gollub.
Velocity statistics in excited granular media. Chaos: An Interdisciplinary
Journal of Nonlinear Science, 9(3):682{690, 1999.
[12] J. Olafsen and J. Urbach. Velocity distributions and density
uctuations
in a granular gas. Phys. Rev. E, 60:R2468{R2471, Sep 1999.
[13] G. W. Baxter and J. S. Olafsen. Kinetics: Gaussian statistics in granular
gases. Nature (London), 425(6959):680{680, October 2003.
[14] Michael Hay, Richard Workman, and Srinivas Manne. Two-dimensional
condensed phases from particles with tunable interactions. Phys. Rev. E,
67:012401, Jan 2003.
[15] X. Zheng and R. Grieve. Melting behavior of single two-dimensional crys-
tal. Phys. Rev. B, 73:064205, Feb 2006.
[16] J. Schockmel, E. Mersch, N. Vandewalle, and G. Lumay. Melting of a
conned monolayer of magnetized beads. Phys. Rev. E, 87:062201, Jun
2013.
56
[17] J.-C. Tsai, G. Voth, and J. Gollub. Internal granular dynamics, shear-
induced crystallization, and compaction steps. Phys. Rev. Lett., 91:064301,
Aug 2003.
[18] J. Olafsen and J. Urbach. Clustering, order, and collapse in a driven
granular monolayer. Phys. Rev. Lett., 81:4369{4372, Nov 1998.
[19] G. Straburger and I. Rehberg. Crystallization in a horizontally vibrated
monolayer of spheres. Phys. Rev. E, 62:2517{2520, Aug 2000.
[20] J. Olafsen and J. Urbach. Two-dimensional melting far from equilibrium
in a granular monolayer. Phys. Rev. Lett., 95:098002, Aug 2005.
[21] Kiwing To. Boltzmann distribution in a nonequilibrium steady state:
Measuring local potential by granular brownian particles. Phys. Rev. E,
89:062111, Jun 2014.
[22] W.-T. Lin, Y.-C. Sun, C.-C. Chang, Y.-C. Lin, C.-W. Peng, W.-T. Juan,
and J.-C. Tsai. Ratcheting and transitions: Short granular chain in a
gradient of vibration. Phys. Rev. Lett., 112:058001, Feb 2014.
[23] J Lema^tre, A Gervois, H Peerhossaini, D Bideau, and J-P Troadec. An
air table designed to study two-dimensional disc packings: preliminary
tests and rst results. J. Phys. D, 23(11):1396, 1990.
[24] R. P. Ojha, P.-A. Lemieux, P. K. Dixon, A. J. Liu, and D. J. Durian.
Statistical mechanics of a gas-
uidized particle. Nature, 427(6974):521{
523, February 2004.
[25] James Puckett, Frederic Lechenault, and Karen Daniels. Local origins of
volume fraction
uctuations in dense granular materials. Phys. Rev. E,
83:041301, Apr 2011.
57
[26] P Melby, F Vega Reyes, A Prevost, R Robertson, P Kumar, D A Egolf,
and J S Urbach. The dynamics of thin vibrated granular layers. Journal
of Physics: Condensed Matter, 17(24):S2689, 2005.
[27] Andreas Gotzendorfer, Jennifer Kreft, Christof Kruelle, and Ingo Rehberg.
Sublimation of a vibrated granular monolayer: Coexistence of gas and
solid. Phys. Rev. Lett., 95:135704, Sep 2005.
[28] J. Atwell and J. Olafsen. Anisotropic dynamics in a shaken granular dimer
gas experiment. Phys. Rev. E, 71:062301, Jun 2005.
[29] P. Reis, R. Ingale, and M. Shattuck. Crystallization of a quasi-two-
dimensional granular
uid. Phys. Rev. Lett., 96:258001, Jun 2006.
[30] Andrea J. Liu and Sidney R. Nagel. Nonlinear dynamics: Jamming is not
just cool any more. Nature, 396(6706):21{22, November 1998.
[31] See Soft Matter Volume 6 and the articles within.
[32] Kiwing To, Pik-Yin Lai, and H. Pak. Jamming of granular
ow in a
two-dimensional hopper. Phys. Rev. Lett., 86:71{74, Jan 2001.
[33] Malte Schmick and Mario Markus. Gaussian distributions of rotational
velocities in a granular medium. Phys. Rev. E, 78:010302, Jul 2008.
[34] Alexis Burdeau and Pascal Viot. Quasi-gaussian velocity distribution of
a vibrated granular bilayer system. Phys. Rev. E, 79:061306, Jun 2009.
Theory, simulation.
[35] E. Ben-Naim, Z. Daya, P. Vorobieff, and R. Ecke. Knots and random
walks in vibrated granular chains. Phys. Rev. Lett., 86:1414{1417, Feb
2001.
58
[36] Kevin Safford, Yacov Kantor, Mehran Kardar, and Arshad Kudrolli.
Structure and dynamics of vibrated granular chains: Comparison to equi-
librium polymers. Phys. Rev. E, 79:061304, Jun 2009.
[37] Pei-Ren Jeng, Kuan Hua Chen, Gwo-jen Hwang, Chenhsin Lien, Kiwing
To, and Y. C. Chou. Collapse kinetics of vibrated granular chains. The
Journal of Chemical Physics, 135(24):{, 2011.
[38] Pei-Ren Jeng, KuanHua Chen, Gwo-jen Hwang, Ethan Y. Cho, Chenhsin
Lien, Kiwing To, and Y. C. Chou. Entropic force on granular chains self-
extracting from one-dimensional connement. The Journal of Chemical
Physics, 140(2):{, 2014.
[39] Matt Harrington, Joost Weijs, and Wolfgang Losert. Suppression and
emergence of granular segregation under cyclic shear. Phys. Rev. Lett.,
111:078001, Aug 2013.
[40] Mitch Mailman, Matt Harrington, Michelle Girvan, and Wolfgang Losert.
Consequences of anomalous diffusion in disordered systems under cyclic
forcing. Phys. Rev. Lett., 112:228001, Jun 2014.
[41] James G. Puckett and Karen E. Daniels. Equilibrating temperaturelike
variables in jammed granular subsystems. Phys. Rev. Lett., 110:058001,
Jan 2013.
[42] Irene Ippolito, Chrystele Annic, Jacques Lema^tre, Luc Oger, and Daniel
Bideau. Granular temperature: Experimental analysis. Phys. Rev. E,
52:2072{2075, Aug 1995.
[43] A. Abate and D. Durian. Partition of energy for air-
uidized grains. Phys.
Rev. E, 72:031305, Sep 2005.
[44] A. Abate and D. Durian. Approach to jamming in an air-
uidized granular
bed. Phys. Rev. E, 74:031308, Sep 2006.
59
[45] F. Lechenault and Karen E. Daniels. Equilibration of granular subsystems.
Soft Matter, 6:3074{3081, 2010.
[46] M. E. Beverland, L. J. Daniels, and D. J. Durian. Air-
uidized balls in
a background of smaller beads. Journal of Statistical Mechanics: Theory
and Experiment, page P03027, 2011.
[47] Kiri Nichol and Karen Daniels. Equipartition of rotational and transla-
tional energy in a dense granular gas. Phys. Rev. Lett., 108:018001, Jan
2012.
[48] E.G.D. Cohen. Fifty years of kinetic theory. Physica A: Statistical Me-
chanics and its Applications, 194:229{257, March 1993.
[49] J M Kosterlitz and D J Thouless. Ordering, metastability and phase
transitions in two-dimensional systems. Journal of Physics C: Solid State
Physics, 6(7):1181, 1973.
[50] David Nelson and B. Halperin. Dislocation-mediated melting in two di-
mensions. Phys. Rev. B, 19:2457{2484, Mar 1979.
[51] A. Young. Melting and the vector coulomb gas in two dimensions. Phys.
Rev. B, 19:1855{1866, Feb 1979.
[52] Katherine Strandburg. Two-dimensional melting. Rev. Mod. Phys.,
60:161{207, Jan 1988.
[53] David R. Nelson. Defects and Geometry in Condensed Matter Physics.
Cambridge University Press, Cambridge, 1st edition, 2002.
[54] Francisco Ramos, Cristobal Lopez, Emilio Hernandez-Garca, and Miguel
Mu~noz. Crystallization and melting of bacteria colonies and brownian
bugs. Phys. Rev. E, 77:021102, Feb 2008.
60
[55] Yaakov Rosenfeld. Relation between the transport coefficients and the
internal entropy of simple systems. Phys. Rev. A, 15:2545{2549, Jun 1977.
[56] Andras Baranyai and Denis J. Evans. Direct entropy calculation from
computer simulation of liquids. Phys. Rev. A, 40:3817{3822, Oct 1989.
[57] Yaakov Rosenfeld. A quasi-universal scaling law for atomic transport in
simple
uids. J. Phys. Condens. Matter, 11(28):5415, 1999.
[58] Alok Samanta, Sk. M. Ali, and Swapan K. Ghosh. Universal scaling laws
of diffusion in a binary
uid mixture. Phys. Rev. Lett., 87:245901, Nov
2001.
[59] Alok Samanta, Sk. M. Ali, and Swapan K. Ghosh. New universal scaling
laws of diffusion and kolmogorov-sinai entropy in simple liquids. Phys.
Rev. Lett., 92:145901, Apr 2004.
[60] A. Meyer, S. Stuber, D. Holland-Moritz, O. Heinen, and T. Unruh. De-
termination of self-diffusion coefficients by quasielastic neutron scattering
measurements of levitated ni droplets. Phys. Rev. B, 77:092201, Mar 2008.
[61] A. Meyer. Self-diffusion in liquid copper as seen by quasielastic neutron
scattering. Phys. Rev. B, 81:012102, Jan 2010.
[62] J. Brillo, A. I. Pommrich, and A. Meyer. Relation between self-diffusion
and viscosity in dense liquids: New experimental results from electrostatic
levitation. Phys. Rev. Lett., 107:165902, Oct 2011.
[63] Xiaoguang Ma, Wei Chen, Ziren Wang, Yuan Peng, Yilong Han, and
Penger Tong. Test of the universal scaling law of diffusion in colloidal
monolayers. Phys. Rev. Lett., 110:078302, Feb 2013.
[64] Natsuki Hashitsume Ryogo Kubo, Morikazu Toda. Statistical Physics II:
Nonequilibrium Statistical Mechanics. Springer, Berlin, 1st edition, 1978.
61
[65] Jean-Pierre Hansen and Ian R. McDonald. Theory of Simple Liquids.
Academic Press, San Diego, 2nd edition, 1990.
[66] J. B. Barlow, W. H. Rae, Jr., and A. Pope. Low-speed wind tunnel testing.
John Wiley & Sons Inc., New York, 3 edition, 1993.
[67] Daan Frenkel and Berend Smit. Understanding Molecular Simulation:
From Algorithms to Applications. Academic Press, San Diego, 2nd edition,
2001.
[68] Chi Yang, Chong-Wai Io, and Lin I. Cooperative-motion-induced struc-
tural evolution in dusty-plasma liquids with microheterogeneity: Rup-
ture, rotation, healing, and growth of ordered domains. Phys. Rev. Lett.,
109:225003, Nov 2012.
[69] P. Reis, R. Ingale, and M. Shattuck. Caging dynamics in a granular
uid.
Phys. Rev. Lett., 98:188301, Apr 2007.
[70] Eric R Weeks and D.A Weitz. Subdiffusion and the cage effect studied
near the colloidal glass transition. Chem. Phys., 284:361 { 367, 2002.
[71] Wei Chen, Susheng Tan, Zhoushen Huang, Tai-Kai Ng, Warren T. Ford,
and Penger Tong. Measured long-ranged attractive interaction between
charged polystyrene latex spheres at a water-air interface. Phys. Rev. E,
74:021406, Aug 2006.
[72] Sven H. Behrens and David G. Grier. Pair interaction of charged colloidal
spheres near a charged wall. Phys. Rev. E, 64:050401, Oct 2001.
[73] David Chandler. Introduction to Modern Statistical Mechanics. Oxford
University Press, New York, 1st edition, 1987.
[74] The plot is reconstructed from the solid line in Fig. 1 from [59].