1. Granular Clock & Temperature Oscillations in a bidisperse Granular Gas Pik-Yin Lai ( 黎璧賢 ) Dept. of Physics & Center for Complex Systems, National Central University, Chung-Li,Taiwan Collaborators: C. K. Chan( 陳志強 ), Institute of Physics Academia Sinica May Hou ( 厚美英 ), IOP, Chinese Academy of Sciences

2. Granular materials( 顆粒體 ) refer to collections of a large number of discrete solid components. 日常生活中所易見的穀物、土石、砂、乃至公 路上的車流、輸送帶上的物流等 Granular materials have properties betwixt-and -between solids and fluids (flow). Basic physics is NOT understood Complex and non-linear medium

3. Grains Everywhere Food: almost everything we eat,: rice, cereal, peas... Engineering: Powder mechanics, soil mechanics Construction: Rocks, bricks, sand.. Agriculture: transport, storage & manipulation of seeds, grains & foodstuffs Pharmaceutical: pills & powder processing Transportation: shock absorption packing Industries: Mixing & segregation of grains & powder by shaking or rotation Geological: desert, landslides, earthquake dynamics 1 trillion US$/year

4. Discrete & Macroscopic Hardcore interactions & Dissipative Zero temperature: mgd/kT ~ 10 Breakdown of hydrodynamics Friction is important Dynamically Driven Inhomogeneous static stresses Complex Many-body systems Mixing & Segregation Pattern formation Complex Flow Physical aspects of Grains 12 Rev. Mod. Phys. 68, 1259 (1996) 71, S374 (1999) 71, 435 (1999) 物理雙月刊 23, 503 (2001)

5. Far from equilibrium Dissipation: inelastic collisions Energy input: vibrating bed (bottom collision) Dissipation rate ~ input rate  steady state Heap formation Granular gas Grains under excitations: vibrating bed

6. Properties of Granular Gases Particles in “random” motion and collisions “similar” to molecular gases But … Inelastic Collisions / Highly dissipative Energy input from vibration table Far from thermal equilibrium  Brazil Nut Effect, Clustering, Maxwell’s demon

7. Molecular gases

8. monodisperse granular gas in compartments: Maxwell’s Demon Eggers, PRL, 83 5322 (1999) v

9. Clustering Granular gas in Compartmentalized chamber under vertical vibration D. Lohse’s group

10. Maxwell’s Demon is possible in granular system Steady state: input energy rate = kinetic energy loss rate due to inelastic collisions N v kinetic temp Evaporation-condensation Unstable ! Bottom plate velocity (input) Dissipation (output) u Evaporation condensation characteristic

11. Flux model n h 1-n large V stable; as V decrease  bifurcation  ! uniform  cluster to 1 side is always a fixed point Eggers, PRL, 83 5322 (1999)

12. What happens for a binary mixture?

13. Granular Oscillations in compartmentalized bidisperse granular gas NA grain A NB grain B co =NA/NB

14. Phase Diagram

15. Objectives Quantitative description A model to understand the quantitative data

16. Effects of compartments + bidispersity: Granular Clock Markus et al, Phys. Rev. E, 74, 04301 (2006) Big and small grains. Explained by Reverse Brazil Nuts effects

17. Binary mixture in a single compartment A B inelastic collision is asymmetric: A can get K.E. from B (B heats up A & A slows down B) TB is lowered by the presence of A grains Change of K.E. of A grain due to A-B inelastic collision: Dissipation rate of A grain due to A-B inelastic collision:

18. Binary mixture in a single compartment A B inelastic collision is asymmetric: suppose A gets K.E. from B (B heats up A & A slows down B) TB is lowered by the presence of A grains Balancing input energy rate from vibrating plate with total dissipation due to collision:

19. binary mixture of A & B grains in 2 compartments Very large V, A & B are uniform in L & R, As V is lowered, at some point only A is free to exchange:  clustering instability of A T BR gets higher, then B evaporates to L Enough B jumped to L to heat up As, T AL increases  A evaporates from L to R A oscillates ! (B heats up A & A slows down B)

20. Flux Model for binary mixture of A & B grains in 2 compartments L R PRL, 100, 068001 (2008) J. Phys. Soc. Jpn. 78, 041001 (2009)

21. Theoretical result for p & q Balancing input energy and dissipation due to inelastic collisions:

22. p(c) & q(c) can be calculated theoretically

23. is always a fixed point, stable for V>Vc For V<Vc, Hopf bifurcation  oscillation L R

24. V>Vc V<Vc V<V f Numerical solution

25. Model Results V>Vc, A & B evenly distributed in 2 chambers Supercritical Hopf bifurcation near V c V<Vc, limit cycle. Granular clock for A & B. Amplitude  (v-v c ) 0.5 [Hopf] Period  ~ (v- v f ) -  (numerical solution of Flux model) V < V f, clustering into one chamber Saddle-node bifurcation at V f (to be proved rigorously)

26. Vc-V (cm/s) Oscillation amplitude: exptal data Numerical soln. of Flux model

27. Oscillation period

28. Phase diagram

29. Analytic results Fix point (0,0) loses stability at v c

30. Supercritical Hopf bifurcation at vc Theorem: supercritical Hopf if verified Expanding near (0,0):

31. Analytic result for phase boundary Fix point (N A /2,N B /2) loses stability at v c Vc calculated from

32. Analytic result for emergent frequency at vc Hopf bifurcation at v c : Larger c (more A), longer time to heat up for evaporation  smaller freq.

33. Saddle-node bifurcation at vf Phase boundary of vf: New stable fixed point emerges: V < Vf, clustering

34. Other interesting cases: Tri-dispersed grains : A, B,C 3-dim nonlinear dynamical system  complex dynamics, Chaos…

35. Other interesting cases: Bi-dispersed grains in M-compartments: 2(M-1)-dim nonlinear dynamical system  complex dynamics,…… 3 12

36. Summary Evaporation /Condensation in granular compartmentalized gas is unstable when dissipations become important  “Maxwell demon” Temperature difference is generated spontaneously. Each grain type has difference temperature in a bi-disperse vibrating grain mixture because of asymmetric properties of collisions (mass, size,…) [even for single compartment] Binary mixtures can generate oscillatory temperature differences in the two compartments Oscillations: Hopf bifurcation at vc Clustering: saddle-node bifurcation at vf Our model is confirmed by experiments. Systems with rich and complex dynamics