Wennan Chen 陳文楠 and Kiwing To 杜其永 Institute of Physics Academia Sinica 中央研究院物理所

1. Unusual diffusion in quasi-two-dimensional granular gas Wennan Chen 陳文楠and Kiwing To 杜其永Institute of Physics Academia Sinica中央研究院物理所 OUTLINE Introduction: granular physics Velocity distribution of granular gas Diffusion of quasi-2D granular gas Phenomenological two-state model Summary & Discussions

2. 「你們要給人、就必有給你們的並且用十足的升斗、連搖帶按、上尖下流的、倒在你們懷裡，因為你們用甚麼量器量給人、也必用甚麼量器量給你們。」「你們要給人、就必有給你們的並且用十足的升斗、連搖帶按、上尖下流的、倒在你們懷裡，因為你們用甚麼量器量給人、也必用甚麼量器量給你們。」 新約聖經、路加福音、第六章、第38節 Give, and it shall be given unto you; a good measure, pressed down, shaken together, and running over, they will give into your bosom. For with what measure you measure, it shall be measured to you in return. New Testament Bible, Luke, 6:38

3. Compaction demonstration Compaction demonstration video

4. Granular Solid Force distribution at granular pile bottom Vanel et. al., PRE 60, R5040, 1999 pile of sand 1.2 mm diameter 8 cm high 33o repose angle localized deposition uniform deposition

5. Granular Solid apparent weight Vanel, Duran, ‘97 cylinder radius (mm) 20 bead diameter (mm): 1.5, 2, 3 Where has the missing weight gone?

6. Granular Solid physical properties are history dependent ~d density increases due to rearrangement of beads in packing Potential energy of a grain ~ mgd with m~ 10-2 g, g=10m/s2, d~1mm ~ 10-7 J Thermodynamic temperature ~ kT with k=1.3810-21 J/K, T~300K ~ 410-21 J Temperature ~ 0 as compared to granular energy Thermodynamically non-equilibrium, meta-stable state does not relaxes spontaneously relaxes by external agitation (tapping or vibration)

7. dilute rapid flow external forcing friction dissipation immobile state Static and dynamicsdead and alive gas fluid solid dense, slow flow quasi-static meta-stable glassy region

8. Motivation Statistical properties of quasi-two-dimensional gas with inelastic inter-molecular interaction gas : a collection of small molecules moving around inter-molecular interaction: exchange of energy among molecules inelastic: interaction does not conserve energy statistical properties: velocity distribution temperature diffusion constant

9. energy change 0 hard sphere collision after collision before collision restitution coefficient

10. elastic collision average velocity mean flow = 1 velocity variance thermal speed temperature from equilibrium statistic mechanics reduced velocity Maxwell distribution velocity distribution change in velocity distribution due to collision Boltzmann Equation

11. inelastic collision average energy lost per collision < 1 average collision rate 2s v12 cooling rate Haff’s Law

12. inelastic hard spheres velocity distribution changes due to collision Boltzmann-Enskog Equation < 1 collision integral pair correlation function at contact adiabatic cooling approximation van Noije and Ernst, Gran. Matter 1, 57, 1998.

13. inelastic hard spheres < 1 << gain loss high energy tail exponential decay at high energy tail confirmed by direct Monte Carlo simulation Brey, et. al., PRE 59, 1256, 1999.

14. velocity distribution and temperature elastic collisions molecular gas in equilibrium state temperature inelastic collisions granular gas for c >> 1 non-equilibrium steady state granular temperature

15. mean square displacement MSD = log ( MSD ) slope = 1 log ( t ) Unusual diffusion in quasi-two-dimensional granular gas MSD = 6Dt diffusion constant in 2 dimension MSD = 4Dt in 1 dimension MSD = 2Dt

16. diameter, s mean free time typical speed, v number/area, r mean free path area swept per unit time = 2sn collision rate = 2rsn total molecule number in fixed area A Unusual diffusion in quasi-two-dimensional granular gas 2s diffusion constant [m2/s] v D inversely proportional to N usual diffusion

17. vibrating stage 12.5 mm 300 mm Unusual diffusion in quasi-two-dimensional granular gas granular gas : a collection of ‘molecules’ interact with each other with inelastic collision quasi-two-dimensional: two dimensional projection from three dimensional motion plastic ball diameter: 6 mm mass: 0.12 gm vibration frequency: 20 Hz amplitude: 1.84 mm camera resolution: 1024x1024 frame rate: 1000 fps

18. vibrating stage 12.5 mm 300 mm Unusual diffusion in quasi-two-dimensional granular gas get trajectory each particles measure MSD

19. MSD = 4Dt diffusive motion MSD = <(Dx)2> = <(vt)2> = <v2> t2 ballistic motion N =1000 slope=2

20. NTg 26 mJ 1116 mJ 500 1000 N =500 D=300 mm/s2 N =1000 D=100 mm/s2

21. 4D [mm2/s] diffusion increases with N for N < 1000 Tg [10-9 J] N

22. Langevin equation : ball dynamics motion in vertical direction motion in horizontal direction collision with top or bottom (type-1) collision with other balls (type-2) small velocity fluctuation in horizontal direction large velocity fluctuation in horizontal direction top and bottom act on the balls like a viscous fluid balls may gain kinetic energy in the horizontal direction due to inelastic collision

23. particle trajectory and speed high speed state low speed state

24. low speed high speed high speed Two state model ball excited to HSS after colliding with another ball ball in HSS relaxes to LSS in t2 due to collision with top and bottom 1800 500 200 50

25. type-2 collision : type-1 collision : with other balls with top/bottom large perturbation to horizontal velocity small perturbation to horizontal velocity effective temperature T2 effective temperature T1

26. enter HSS with speed v2 decay to LSS in t area swept = 2d v2t high speed state, T2 decay time, t type-2 collision type-1 collisions low speed state, T1 diffusion constant D1 diffusion in LSS with rate D1 , duration t1 area swept = 4pD1t1 total area swept:

27. type-1 collision rate, f mean speed in HSS, life-time of HSS, total time spend in LSS, end-to-end distance moved = area swept = area swept = time spend in HSS, f= ft total area swept in 1 second, =130 mm/s, t =100ms, D1 = 5.5 mm2/s type-1 collision rate = f fraction of ball in HSS argument break down when ft ~1, this happens when N = Nc = 945

28. =130 mm/s, t =100ms, D1 = 5.5 mm2/s, D2 =100 mm2/s T1 = 54 nJ, T2 =4400 nJ Two effect temperature baths collision with other ball collision with top or bottom temperature bath-2, T2 temperature bath-1, T1 effective granular temperature

29. Summary and discussions We studied diffusion in a quasi-two-dimensional granular gas (Q2DGS) composed of plastic balls confined in a vertically vibrating thin box. The motion of the particles in the Q2DGS was found to follow the Langevin equation with the top and bottom of the box acting on the balls like a viscous fluid. We found that both the granular temperature and the diffusion constant increased with the number of balls N in the box for small N. Based on the velocity distributions and the two different kinds of horizontal motions observed in the experiments, we proposed a simple two-state model to explain the unusual diffusion behavior. Thank you for your attention !

31. Motivation Can we understand the statistical properties of granular gas using simple kinetic theory ? kinetic theory: each molecule move according to Newton’s law statistical properties: velocity distribution temperature diffusion constant