Fluctuation theorem is tested for case of 2-d Brazil nut effect through entropy definition using concept of contactopy

1. Fluctuation Theorem Application on 2-D
Granular Materials Configurations
S. Viridi, S. N. Khotimah, Novitrian, Widayani
Nuclear Physics and Biophysics Research Division,
Institut Teknologi Bandung, Bandung 40132, Indonesia
Solo, Indonesia,
16 September 2014
1
2nd International Conference on Advanced
Materials Science and Technology
(ICAMST)

2. Outline
• Fluctuation theorem
• Brazil nut effect
• 2-d artificial configuration
• Observation
• Summary
Solo, Indonesia,
16 September 2014
2nd International Conference on Advanced
Materials Science and Technology
(ICAMST)
2

3. Fluctuation theorem
• One of the FTs states that (Seifert, 2005)
p - D
s = -D
D
which holds for finite length of time δt for
entropy change Δs
• Constraints: steady state of constant force and
constant temperature
Solo, Indonesia,
16 September 2014
2nd International Conference on Advanced
Materials Science and Technology
(ICAMST)
3
( )
( )
e s
p s

4. Fluctuation theorem (cont.)
• Second law of thermodynamics predicts that
the entropy of an isolated system should tend
to increase until it reaches equilibrium
• This theorem tells us that reverse process,
where entropy decrease, could have high
probability to happen
• It violates the law (in small scale during finite
length of time)
Solo, Indonesia,
16 September 2014
2nd International Conference on Advanced
Materials Science and Technology
(ICAMST)
4

5. Brazil nut effect (BNE)
• Mixture of two different size of granular
materials under vibration will produce
seggreation  larger grains on top, smaller
grains on the bottom
• It holds also for only single intruder
• It is observed in two- and three-dimension
Solo, Indonesia,
16 September 2014
2nd International Conference on Advanced
Materials Science and Technology
(ICAMST)
5

6. Brazil nut effect (BNE) (cont.)
• It has also the reverse BNE (RBNE) as reported
(Hong et al., 2001)
Solo, Indonesia,
16 September 2014
2nd International Conference on Advanced
Materials Science and Technology
(ICAMST)
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7. Brazil nut effect (BNE) (cont.)
• There is a crossover condition
BNE: ●, RBNE: ▪
Solo, Indonesia,
16 September 2014
2nd International Conference on Advanced
Materials Science and Technology
(ICAMST)
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8. 2-d artificial configuration
• It is based on real observation
• Configuration are set artificially (Viridi et al.,
2014)
• Time series of intruder and granular bed can
be analyzed
Solo, Indonesia,
16 September 2014
2nd International Conference on Advanced
Materials Science and Technology
(ICAMST)
8

9. 2-d artificial configuration (cont.)
Solo, Indonesia,
16 September 2014
2nd International Conference on Advanced
Materials Science and Technology
(ICAMST)
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10. Observation
• Number of contacts between grains are
observed
• It is named as contactopy (Herrmann, 1993)
Solo, Indonesia,
16 September 2014
2nd International Conference on Advanced
Materials Science and Technology
(ICAMST)
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11. Observation (cont.)
• Potential energy
N
g i U t m g r t
• Contact energy
c ij U t t
Solo, Indonesia,
16 September 2014
g sign x
2nd International Conference on Advanced
Materials Science and Technology
(ICAMST)
11
( ) = ×
( ) å=
i
1
 
( ) åå [ ( )]
= >
=
N
i
N
j
1 1

12. Observation (cont.)
• Overlap between grains (Schäfer et al., 1996)
x max 0, 1
• Temperature T = mgDα
• Entropy
Solo, Indonesia,
16 September 2014
Ds = - 1 +d -
2nd International Conference on Advanced
Materials Science and Technology
(ICAMST)
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( ) ( ) ( ) ( )
þ ý ü
î í ì
 
t = D + D - r t - r t ij i j i j
2
[U(t t) U(t)]
T

13. Observation (cont.)
• Center of mass posi-tion
for mA/mB = 1
• Intruder movement
seems to violate
MPEP
• Change of COM for
the whole system
agrees with MPEP
Solo, Indonesia,
16 September 2014
2nd International Conference on Advanced
Materials Science and Technology
(ICAMST)
13

14. Observation (cont.)
• For mA/mB = 1, γ = 1, and α = 8
Solo, Indonesia,
16 September 2014
2nd International Conference on Advanced
Materials Science and Technology
(ICAMST)
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15. Summary
• BNE obey second law of thermodynamics
• RBNE has small probability to occur
• FT does not violate second law of
thermodynamics for chosen parameters
(mA/mB = 1, γ = 1, and α = 8)
Solo, Indonesia,
16 September 2014
2nd International Conference on Advanced
Materials Science and Technology
(ICAMST)
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16. THANK YOU
Solo, Indonesia,
16 September 2014
2nd International Conference on Advanced
Materials Science and Technology
(ICAMST)
16