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Temperature Oscillations in a Compartmetalized Bidisperse Granular Gas C. K. Chan 陳志強 Institute of Physics, Academia Sinica, Dept of Physics,National Central University, Taiwan

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Collaborators May Hou, Institute of Physics, CAS 厚美英 P. Y. Lai, National Central University 黎璧賢

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Content What is a clock? What is special about a granular clock? Unstable Evaporation/Condensation Two temperature in a bi-disperse system Model for bidisperse oscillation Summary

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What is a clock ? Periodic motion sun, moon, pendulum etc … Periodic Reaction BZ reaction, enzyme circadian rhythm Periodic Collective behavior suprachiasmatic nuclei, sinoatrial node, comparmentalized granular gases, etc…

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BZ reaction From S. Mueller

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Granular Oscillation

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Second Law no clock? Belousov-Zhabotinsky reaction A B A B; Why not: A B Two-compartment granular Clock

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Molecular gases

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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

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

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

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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

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Heaping

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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)

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What happens for a binary mixture? What are the steady state? How many granular temperatures ?

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Oscillation of millet ( 小米, N=4000) and mung beans ( 绿豆, N=400) F = 20Hz. Amp = 2mm

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soda lime glass 138 small spheres diameter : 2 mm 27 large spheres diameter 4 mm box height:7.7 cmx0.73cmx5 cm

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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 a=6 mm, f =20 Hz. Times: a=0, b=3.1, c=58.3, d=66.2, e=103.2 s.

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Granular Oscillations in compartmentalized bidisperse granular gas 2.6cmx5.4cmx13.3cm barrier at1.5 cm Steel glass balls Radius = 0.5 mm N = 960 f = 60 Hz

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Phase Diagram

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Model of two temperatures 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)

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Model Objectives Quantitative description A model to understand the quantitative data

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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) T B 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:

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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:

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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)

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

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V>Vc V<Vc V<V f Numerical solution

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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???)

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

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Oscillation period

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Phase diagram

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

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

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Summary Dissipation is density dependent “Maxwell demon” Different collision dissipations in binary system existence of two “granular temperatures” Non-homogeneous temperature with homogenous energy input both spatially and temporally Granular steady state + compartment oscillations

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Thermophoresis or Janus ?

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A worm in a temperature bath

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