Notes

Quantum Mechanics

Projective Geometry Description and Applications for Quantum Entanglement

(Slide Version)

Algebraic Topology Aspects of Feynman Path Integral

Topology and Geometry in Statistical Mechanics

Sympcletic Geometry and Classical Mechanics

The Genetic Algorithms

Preliminary Study

Here are some small-scale research projects I've completed. During these projects, I observed some intriguing phenomena that I think warrant further exploration in the future.

Special Case of 5-Body Central Configuration

This study examines central configurations within the framework of the 5-body problem, with a specific focus on systems where four of the masses lie on a common circle. This investigation does not assume that the four co-circular masses form a central configuration on their own. I explore potential placements of a fifth body that would transform the entire 5-body system into a central configuration. This approach broadens the scope of understanding by considering configurations that deviate from standard symmetry or balance.
Furthermore, the study delves into specific geometric arrangements of the co-circular four bodies, identifying cases where their particular shapes or distributions inherently prevent the inclusion of a fifth body that could satisfy the central configuration conditions. By combining analytical methods with geometric considerations, this work aims to provide deeper insights into the constraints and possibilities of central configurations in multi-body gravitational systems. These findings contribute to the broader understanding of celestial mechanics and the intricate interplay of forces in systems with nontrivial geometric constraints.


Leveraging Neural Network on Detect Phases and Phase Transitions in the Ising Model

The classical Ising model is one of the simplest yet most important models in statistical mechanics, renowned for its phase transition phenomena. In dimensions \(d\geq 2\), the Ising model displays two distinct phases: ordered and disordered, with a temperature-dependent phase transition described by an order parameter. Exact solutions for the Ising model are challenging, making numerical methods crucial for studying it. Machine learning has become a valuable tool in many-body quantum mechanics and theoretical condensed matter physics. My objective is to apply neural networks to tackle the complexities of this model. In this project, I use a Monte Carlo method, to sample the system’s configurations, generating a dataset across different phases of the Ising model. This dataset will then train a neural network to identify and characterize phase properties, enhancing our understanding of phase transitions in the Ising model.


Numerical Simulation of Simplify Stellar Wind Model

The magnetohydrodynamics (MHD) equation model is considered one of the most straightforward frameworks for describing the behavior of plasma, combining the principles of magnetism and fluid dynamics. Although this model offers a foundational understanding, the complex nature of plasma introduces significant challenges that extend beyond the capabilities of the MHD equations alone. To address these challenges and gain deeper insights into plasma behavior, we construct a detailed stellar wind model. By examining this model, we can observe and analyze a range of phenomena related to plasma dynamics. This approach allows us to investigate and better understand some of the more intriguing and complex behaviors exhibited by plasma, offering valuable perspectives on its properties and interactions in various contexts.


Extended SIR Models with Monte Carlo Simulations

To enhance the completeness of the SIS model, additional variables can be included, such as in the SEIR model, which introduces an "exposed" variable (E). However, adding more variables increases the model's complexity, making the differential equations more challenging to solve. To manage this complexity, we use the Monte Carlo method to simulate the equations. This approach simplifies the simulation of multiple differential equations and provides clear results.


Neural Network and the Renormalization Group

The renormalization group theory is an influential tool used to study phase transitions, providing deep insights into how systems behave at different scales. On the other hand, neural networks represent a sophisticated theoretical framework within artificial intelligence, designed to model and understand complex patterns and relationships in data. Recent developments have highlighted an intriguing connection between these two areas: neural networks can be interpreted as a form of renormalization process. This perspective suggests that the mechanisms underlying neural networks bear a resemblance to the principles of renormalization, where the process of simplifying and scaling data in neural networks parallels the renormalization techniques used in phase transition analysis. To explore this connection further, we analyze and discuss various numerical results that illustrate the similarities between neural networks and renormalization processes. This examination sheds light on how concepts from one domain can enrich our understanding of the other.