Consensus Dynamics in Statistical and Quantum Mechanics Using Mathematical and Computational Simulations

In statistical mechanics, gases or liquids consist of many particles whose collective behavior leads to macroscopic phenomena such as temperature or pressure. Specifically in plasma physics, charged particles interact and evolve through collective electromagnetic fields. Consensus dynamics can be observed when the particles’ velocities or positions converge towards a common configuration influenced by collective interactions.

Consensus dynamics in network or graph theory plays a significant role in quantum physics.  This project presented various examples using a numerical method and applications to understand the structure of randum interactions of particles with initial conditions. Through network theory and its applications, this project studied various frameworks for understanding and solving real-world problems related to connectivity, optimization, and the analysis of complex systems by simplifying the complex procedures and concepts occurring in physical systems. The geometrical properties of various designed networks were assessed to perform the consensus dynamics in multiagent systems using mathematical and computational simulations. After calculating the Adjacency matrix, Diagonal matrix, and Laplacian matrix of each network, the eigenvalues of each system were found to present a model for opinion formation(consensus) among individuals of systems within a domain.

The outcome of consensus dynamics or influence propagation showed that it is not solely dependent on the active interaction of particles but also on the network's structural properties and geometrical dependencies. For example, in a network with strong clustering (nodes tend to be connected to other nearby nodes), consensus within clusters was formed more quickly. MATLAB and Python programming were used to perform the computer simulation and calculations.