Microstate Sequence Theory of Phase Transition: Theory Construction and Application on 3-Dimensional Ising Model

Abstract

The concepts of microstates and statistical ensembles form a fundamental starting point for various statistical physics theories that address thermodynamic and phase transition behaviors of correlated many-body systems. In this paper, we propose microstate sequence (MSS) theory built on a novel idea of arranging all microstates of a discrete thermodynamic system into a sequence with monotonically increasing property of key parameters and strict “smooth structure variation” property. Because of the properties, it obtains better analytical ability to express the derivation with the essential parameter change (in the cubic Ising model, the parameter is the dimensionality) at any micro-structure to figure out the qualitative issues like the relationship between phase transition order and dimensionality. With this idea in mind, the microstate sequence (MSS) of the Ising model in arbitrary dimension is constructed through a nontrivial iteration method based on a series of number-theoretic transformation tricks. After obtaining the complete form of the MSS for the Ising model, we provide a concise proof of the second-order phase transition nature for the Ising model in all n $>$ 2 dimensions starting from the well-known exact result for the two-dimensional Ising model, as a test of the qualitative issue of MSS theory. Finally, we discuss the MSS theory in other lattice models like the Potts model and temperature derivation model to explore the correlations of number theory and phase trajectory in an extended range of discrete thermodynamic systems.