Studies in Complex Systems

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....

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....

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....

Fourier Methods could be derived entirely from Group theory! As the title suggests, the entire concept of the Fourier transform can be derived if we understand some basic group theory. We start by introducing the group \(Z_{N}\) and its irreducible represenations. Using the orthogonality theorem, we will then derive the discrete Fourier transform (DFT) and the Fourier transform (FT). Fourier analysis studies the periodicities of functions. Any continuous and differentiable function can be broken down into a linear combination of its frequency components, which is the foundation of Fourier series....

The Wave and Particle Dispute The contradiction between classical physics and microscopic phenomena is one of the most fascinating episodes in the history of science, reshaping our fundamental understanding of waves and particles. To grasp the weirdness of wave-particle duality, let’s start with a simple analogy. Imagine shooting bullets at a wall with two equally spaced gaps. As expected, the bullets passing through each gap will behave independently, forming two distinct patterns on a measurement panel behind the wall....

The Ising Model: Capturing Nearest Neighbor Interactions in Nature Any model is a simplification of reality. Real-world systems are often complex, high-dimensional entities with intricate couplings, making them impossible to model or calculate in full detail. Fortunately, not all degrees of freedom are equally relevant for predicting the qualitative behavior of a system. The Ising model simplifies reality by focusing on lattice neighbor interactions and external fields. It’s straightforward once we see the equation....

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)....