How Do Surfaces Grow?


Le Chen

le.chen@auburn.edu
Department of Mathematics and Statistics
Auburn University

Auburn Summer Science Institute (AU-SSI)
Auburn, AL, June 5, 2024

NSF

DMS-Probability: No. 2246850
(2023-2026)

Simons

No. 959981
(2022-2027)

Two graduate students who have been working on this project throughout

Math 7820/30: Applied Stochastic Processes (2023/24):

Mau

Mauricio Montes

mauricio.montes@auburn.edu

Ian

Ian Ruau

ian.ruau@auburn.edu

Sand pile at the beach vs Snow pile in winter

Contrasting Sand and Snow Piles

Image created by OpenAI's DALL-E

Simulation by Tetris-Ballistic package

Central Limit Theorem (CLT)

Galton Box

Video from by Wikipeidia

Video created by 3Blue1Brown

Video created by 3Blue1Brown

Video created by 3Blue1Brown

Mean and standard deviation

Video created by 3Blue1Brown

General ideas of CLT

Video created by 3Blue1Brown

Video created by 3Blue1Brown

Assumptions for CLT

Video created by 3Blue1Brown

Simulation by Tetris-Ballistic package

Nonsticky

Sticky

Growing interfaces in a thin film

[4] K. A. Takeuchi, M. Sano, T. Sasamoto, and H. Spohn. Growing interfaces uncover universal fluctuations behind scale invariance. Sci. Rep., 1(1):1--5, 2011. [ bib ]

Other examples

Paper inking

[1] A.-L. Barabási and H. E. Stanley. Fractal concepts in surface growth. Cambridge University Press, Cambridge, 1995. [ bib | DOI | http ]

Paper burning

[5] J. Zhang, Y.-C. Zhang, P. Alstrøm, and M. Levinsen. Modeling forest fire by a paper-burning experiment, a realization of the interface growth mechanism. Phys. A: Stat. Mech. Appl., 189(3):383--389, 1992. [ bib | DOI | http ]

Snow pile in one dimension

[1] A.-L. Barabási and H. E. Stanley. Fractal concepts in surface growth. Cambridge University Press, Cambridge, 1995. [ bib | DOI | http ]

Tetris Pieces?

Are there any non KPZ types?
But when it is sticky...

Any crossover of Gaussian and KPZ?

5% nonsticky + 95% sticky

Log-Log Plot 05%

50% nonsticky + 50% sticky

Log-Log Plot 50%

90% nonsticky + 10% sticky

Log-Log Plot 90%

95% nonsticky + 5% sticky

Log-Log Plot 95%

98% nonsticky + 2% sticky

Original 98%

99% nonsticky + 1% sticky

Log-Log Plot 99%

Install

PyPI: Tetris Ballistic

Visit on PyPI

Source code

GitHub: Simulations on Some Surface Growth Models

Visit on GitHub

Diffusion-limited aggregation (DLA)

[5] T. A. Witten and L. M. Sander. Diffusion-limited aggregation, a kinetic critical phenomenon. Phys. Rev. Lett., 47:1400--1403, 1981-11. [ bib | DOI | http ]

Simulation by markstock/dla-nd

Thank you!