The Multiscale Principle in phase transition: Geometric structure, thermodynamic entropy, and dynamical instability

Scaling and multi-scaling are two guiding principles for understanding nature or even commerce and industrial outcome. In the realm of critical phenomena, scaling laws describe the behavior of physical quantities near a continuous phase transition. This is a complex phenomena where systems exhibit critical fluctuations and long-range correlations, challenging classical formulations such as the fluctuation-dissipation theorem (FDT). In particular, near critical points, the divergence of the correlation length and the emergence of fractal geometry in spin clusters suggest the need for alternative frameworks. In this work, we explore the connection between critical exponents, fractal dimensions, and the thermodynamic formalism known as the Principium Luxuriae. By linking the thermodynamic fractal dimension D to the correlation length and entropy variations, we reinterpret energy dissipation in terms of multiscale processes. This approach allows us to reproduce effective scaling tendencies associated with the critical behavior of the 2d Ising model and provides new insight into how systems redistribute energy macroscopically during phase transitions. Furthermore, we demonstrate that the decrease in D near the critical temperature marks a transition toward macroscopic coherence, where the system begins to dissipate energy across its entire scale. This unified behavior, however, induces a heightened sensitivity to perturbations, where local fluctuations are transmitted globally through the fractal network. This mechanism establishes a direct bridge between geometric structure, thermodynamic entropy, and the onset of dynamical instability (chaos) as the system approaches criticality.