https://htsod.github.io/posts/ Recent content in Studies in Complex Systems on ButterCat https://htsod.github.io/%3Clink%20or%20path%20of%20image%20for%20opengraph,%20twitter-cards%3E https://htsod.github.io/%3Clink%20or%20path%20of%20image%20for%20opengraph,%20twitter-cards%3E Hugo -- 0.123.7 en Mon, 27 Jan 2025 00:00:00 +0000 https://htsod.github.io/posts/percolation_network/ Tue, 07 Jan 2025 00:00:00 +0000 https://htsod.github.io/posts/percolation_network/ Site and bond percolation models exhibit a phase transition at a shared critical point, where both demonstrate self-similarity and scale invariance—hallmarks of continuous phase transitions. Using the Renormalization Group (RG) method in ( Citation: Sethna, 2020 Sethna,  J.  (2020).   Entropy, Order Parameters, and Complexity (2).   Clarendon Press. ) , which proposed a scaling procedure from the assumption of self-similar, we derive the scaling exponents for the percolation universality class. While this top-down approach offers pedagogical simplicity, it lacks the physical intuition provided by the Ginzburg-Landau framework. https://htsod.github.io/posts/small_world/ Sun, 06 Oct 2024 00:00:00 +0000 https://htsod.github.io/posts/small_world/ In this blog, we start with a qualitative overview of the small-world effect, also known as the “six degrees of separation,” followed by a quantitative analysis using computational methods, and finally, a description of the phenomenon through the framework of phase transitions and critical phenomena. From the surface level to deeper insights, I aim to explore the mechanics underlying this phenomenon and how this understanding can be extended to other large-scale phenomena. https://htsod.github.io/posts/maxwell_eqns_derive/ Mon, 19 Jun 2023 00:00:00 +0000 https://htsod.github.io/posts/maxwell_eqns_derive/ Can the order of physics discovery be differernt? From Zee Einstein’s Gravity in a nutshell, a hypothetical situation is proposed: a far away civilization where life was evolved from modular planet happen to understand the constancy of light before understanding electromagnetism. A brilliant physicist then could derived E.M. and gravity from the correct conception of space and time. The derivation was casted in a simplisitc manner: By placing the potential term either inside or outside the action square root, we obtain the familiar interactions of gravity or electromagnetism, with major concepts left unexplained. https://htsod.github.io/posts/fourier/ Wed, 27 Mar 2024 00:00:00 +0000 https://htsod.github.io/posts/fourier/ Group theory is powerful in analysis problem with symmetry. This blog aims at giving an example on how Fourier Theorem could be derived on the ground of group theory. https://htsod.github.io/posts/wave_particle/ Sun, 24 Mar 2024 00:00:00 +0000 https://htsod.github.io/posts/wave_particle/ Both the Schrödinger wave equation and the Heisenberg matrix method are fundamental approaches to quantum mechanics, each offering a unique perspective. To unify them, we recognize that while they approach the problem differently, they ultimately describe the same quantum phenomena. https://htsod.github.io/posts/ising_model/ Wed, 21 Jun 2023 00:00:00 +0000 https://htsod.github.io/posts/ising_model/ This blogh reprodcues the transfer matrix solution to one dimensional Ising model and the numerical solution to the two dimensional Ising model. https://htsod.github.io/posts/rg_method/ Sun, 11 Jun 2023 00:00:00 +0000 https://htsod.github.io/posts/rg_method/ Renormalization Group Approach in Dynamical System The renormalization group (RG) method is an approximation technique initially developed for solving strongly interacting many-body problems in quantum field theory, where perturbative solutions deviate from the actual solutions. The fundamental concept of the renormalization group approach is to eliminate irrelevant degrees of freedom in a physical system while preserving its essential characteristics ( Citation: P. Kopietz, 2010 P. Kopietz,  F.  (2010).   Introduction to the Functional Renormalization Group (1).