The document presents a simulation of diffusion using agent-based modeling (ABM) implemented in JavaScript. It outlines the principles of Fick's law to model diffusion, discusses numerical approaches for solving related equations, and illustrates how agents represent particles with specific behaviors. Furthermore, it provides a framework for implementing the model in a web browser, alongside discussions on collaboration and suggestions for improvement.

1. 17th SEACOMP 2019
3rd PIT-FMB 2019
8-10 August 2019, Badung, Indonesia 1
Diffusion Simulation with JS:
Writing Your Own ABM Code
S. Viridi1
, F. Haryanto1
, A. Sutresno2
, I. Arif1
1
Department of Physics, Institut Teknologi Bandung, Bandung 40132 Indonesia
2
Department of Physics, Universitas Kristen Satya Wacana, Salatiga 50711, Indonesia
v201890808_2

2. This slide is part of
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Outline
• Introduction
• Agent-based model (ABM)
• JavaScript (JS) implementation
• Discussion

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Introduction

5. Fick’s law
• Original form
• If container cross-section Q is not function of
x
which is the well-known form, Fick’s 2nd law
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( ) ( )
2
2
,,
x
txC
D
t
txC
∂
∂
−=
∂
∂
( ) ( ) ( )








∂
∂
+
∂
∂
−=
∂
∂
x
txC
dx
dQ
Qx
txC
D
t
txC ,1,,
2
2
A. Fick, "Über Diffusion", Annalen der Physik [Ann. Phys. (Leipzig)], vol. 170, no. 1, pp. 59-86, 1855, url
https://doi.org/10.1002/andp.18551700105

6. Modification
• In order to accomodate finite-time limit, non-
Gaussian distribution, Burnett coefficients,
previous equation will be generalized into
which is reduced to Fick’s 2nd law if N = 1 and
t′ = 0, where φ2 is distribution change kernel
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W. E. Alley, B. J. Alder, "Modification of Fick's Law", Physical Review Letters [Phys. Rev. Lett.] vol. 43, no. 10, pp. 653-
656, Sep 1979, url https://doi.org/10.1103/PhysRevLett.43.653
( ) ( ) ( )
n
nN
n
t
n
x
txC
tttd
D
D
t
txC
2
2
1 0
2
2 ,,
∂
′∂
′−′−=
∂
∂
∑ ∫=
ϕ

7. Numerical approach: FD
• With finite difference method, partial
differen-tial equation related to diffusion can
be solved
• And the Fick’s 2nd law will be
with j is for time and i is for position
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J. C. Wu, N. A. Peppas, "Numerical Simulation of Anomalous Penetrant Diffusion in Polymers", Journal of Applied
Polymer Science [J. Appl. Polym. Sci.], vol. 49, no. 10, pp. 1845-1856, Sep 1993, url
https://doi.org/10.1002/app.1993.070491015
( ) ( ) j
i
j
i
j
i
j
i
j
i
j
i CCC
x
tD
CC
x
tD
C +−
∆
∆
−−
∆
∆
−= −+
+
1212
1

8. Illustration of FD approach
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Agent-based model

10. Agents as particles
• An agent will represent a particle
• Various behaviours can be defined, even the
not physical ones
• The freedom of this model is an advantage
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11. Previous work
• ABM has been implemented using matrix
representation to simulate diffusion of gas
and fluid
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S. Viridi, Z. F. T. W. Lestari, T. D. K. Wungu, Suprijadi, "Simple Grid-Based Model of Sandpile Avalanche Due to
Heating Process", International Workshop on Computational Science, 30 June 2018, Kanazawa University, Japan, url
https://doi.org/10.17605/osf.io/nker6

12. Previous work: Initial condition
• System grid (square with blue line) with object
grid (black rectangle)
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13. • p← = –1, p→ = +1
• p↑ = +1, p↓ = –1
Previous work: Gas phase
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t = 0.01 t = 0.04 t = 0.2

14. • p← = –1, p→ = +1
• p↑ = +0.5, p↓ = –1
Previous work: Fluid phase
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t = 0.01 t = 0.1 t = 0.2

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JS Implementation

16. World matrix W
• Matrix that represents static boundary
condition
• Value of 0: free space, where agents can move
into
• Value of 1: occupied space, where agent are
not allowed to move into
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17. Agent matrix A
• Matrix that represents position of all agents
• It will have value from 2 to N + 1, where N is
types of agents
• Each type is related to specific diffusion
coefficient D, which indicates how active one
type can be
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18. Algorithm
• Step 1 t = tbeg
• Step 2 Choose not empty element in A
• Step 3 Generate new position using D
• Step 4 Move agent if destination value is 0
• Step 5 Repeat Step 2 for all agents
• Step 6 t = t + Δt
• Step 7 If t < tend Repeat Step 2
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19. JavaScript implementation
• url https://bit.do/workshop-biophysics
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23. JS implementation (cont.)
• In a web browser, use CTRL + U to see the
code
• You can modify the code and implement in
your own way
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Discussion

26. Discussion
• Any suggestion?
• Any idea for collaboration?
• Source is avalable at
https://github.com/dudung/butiran.js
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Thank you