Scaling invariance: a bridge between geometry, dynamics and criticality

Abstract

Scale invariance is a central organizing principle in physics, underlying phenomena that range from critical behaviour in statistical mechanics to transport and chaos in nonlinear dynamical systems. Here we present a unified and physically motivated exploration of scaling concepts, emphasizing how invariance under rescaling transformations emerges across systems of increasing dynamical complexity. Rather than adopting a purely abstract approach, we combine simple geometrical constructions, analytical arguments, and prototypical dynamical models to build physical intuition. We begin with elementary, easily reproducible examples governed by a single control parameter, showing how power-law behaviour naturally arises when characteristic scales are absent. We then extend the discussion to nonlinear dynamical systems exhibiting local bifurcations, where two scaling variables control the relaxation toward stationary states. In this context, scaling invariance manifests through critical exponents, crossover phenomena, and critical slowing down, allowing systems of different dimensionality to be grouped into universality classes. Finally, we address continuous phase transitions in chaotic dynamical systems, including transitions from integrability to non-integrability and from bounded to unbounded diffusion. By drawing on concepts traditionally associated with statistical mechanics, such as order parameters, susceptibilities, symmetry breaking, elementary excitations, and topological defects, we show how these transitions can be interpreted within a coherent scaling framework. Taken together, the examples discussed here demonstrate that scaling invariance provides a unifying language for understanding structure, transport, and criticality in nonlinear systems, bridging deterministic dynamics and nonequilibrium statistical physics in a transparent and physically intuitive manner.

Figures (22)

• Figure 1: (a) Plot of $l/l_0$ versus $m/m_0$ for the paper boat folding experiment. A power-law fit yields a slope $0.51(1)\approx 1/2$, corresponding to a scaling exponent $a=-2$. (b) Illustation of the paper boat folding and the crumpled paper. Error bars in (a) represent the uncertainty associated with each measurement.
• Figure 2: Plot of $m/m_0$ versus $r/r_0$ for the crumpled paper balls. A power-law fit yields a slope $D_f = 2.47(3)$.Error bars correspond to the standard deviation computed from 20 measurements.
• Figure 3: Orbit diagram of the mapping (\ref{['c9_eq1']}) for two different values of $\gamma$: (a) $\gamma=1$ and (b) $\gamma=2$.
• Figure 4: (a,c) Plot of $x~vs.~n$ at the bifurcation point $R=1$, for $\gamma=1$ and $\gamma=3/2$, respectively, and for different initial conditions $x_0$. (b,d) Collapse of the curves in (a,c) onto universal plot under the transformations $x \rightarrow x / x_0^{\alpha}$ and $n \rightarrow n / x_0^z$.
• Figure 5: (a) Plot of the crossover iteration number $n_x$ with the initial condition $x_0$ for $\gamma=1$ and $\gamma=2$. Power-law fits yield $z=-1.0002(3)$ and $z=-2.001(2)$, respectively. (b) Plot of the relaxation time $\tau$ as a function of $\mu = R - R_c$ for $\gamma = 1$ and $\gamma = 3/2$. A power-law fit yields $\delta = -1$, independent of $\gamma$.