Exploring Order–Disorder Transitions Using a Two-State Master Equation

Abstract

In this study, we explore the order–disorder transition in the dynamics of a straightforward master equation that describes the evolution of a probability distribution between two states, p1 and p2 (with p1+p2=1). We focus on (1) the behavior of entropy S, (2) the distance D from the uniform distribution (p1=p2=1/2), and (3) the free energy F. To facilitate understanding, we introduce two price-ratios: ηS=dS/dtdF/dt and ηD=dD/dtdF/dt. They respectively define the energetic costs of modifying (1) S and (2) D. Our findings indicate that both energy costs diverge to plus and minus infinity as the system approaches the uniform distribution, marking a critical transition point where the master equation temporarily loses its physical meaning. Following this divergence, the system stabilizes itself into a new well-behaved regime, reaching finite values that signify a new steady state. This two-regime behavior showcases the intricate dynamics of simple probabilistic systems and offers valuable insights into the relationships between entropy, distance in probability space, and free energy within the framework of statistical mechanics, making it a useful case study that highlights the underlying principles of the system’s evolution and equilibrium. Our discussion revolves about the order–disorder contrast that is important in various scientific disciplines, including physics, chemistry, and material science, and even in broader contexts like philosophy and social sciences.

Keywords

• master equation
• order
• disorder
• entropy
• disequilibrium