Search research articles

ABOUT JoVE

Overview Leadership Blog JoVE Help Center

AUTHORS

Publishing Process Editorial Board Scope & Policies Peer Review FAQ Submit

LIBRARIANS

Testimonials Subscriptions Access Resources Library Advisory Board FAQ

RESEARCH

JoVE Journal Methods Collections JoVE Encyclopedia of Experiments Archive

EDUCATION

JoVE Core JoVE Business JoVE Science Education JoVE Lab Manual Faculty Resource Center Faculty Site

Terms & Conditions of Use

Related Concept Videos

Phase Transitions

Phase Transitions

Whether solid, liquid, or gas, a substance's state depends on the order and arrangement of its particles (atoms, molecules, or ions). Particles in the solid pack closely together, generally in a pattern. The particles vibrate about their fixed positions but do not move or squeeze past their neighbors. In liquids, although the particles are closely spaced, they are randomly arranged. The position of the particles are not fixed—that is, they are free to move past their neighbors to...

Phase Transitions: Melting and Freezing

Phase Transitions: Melting and Freezing

Heating a crystalline solid increases the average energy of its atoms, molecules, or ions, and the solid gets hotter. At some point, the added energy becomes large enough to partially overcome the forces holding the molecules or ions of the solid in their fixed positions, and the solid begins the process of transitioning to the liquid state or melting. At this point, the temperature of the solid stops rising, despite the continual input of heat, and it remains constant until all of the solid is...

Phase Transitions: Vaporization and Condensation

Phase Transitions: Vaporization and Condensation

The physical form of a substance changes on changing its temperature. For example, raising the temperature of a liquid causes the liquid to vaporize (convert into vapor). The process is called vaporization—a surface phenomenon. Vaporization occurs when the thermal motion of the molecules overcome the intermolecular forces, and the molecules (at the surface) escape into the gaseous state. When a liquid vaporizes in a closed container, gas molecules cannot escape. As these gas phase...

Scaling

Scaling

In designing and analyzing filters, resonant circuits, or circuit analysis at large, working with standard element values like 1 ohm, 1 henry, or 1 farad can be convenient before scaling these values to more realistic figures. This approach is widely utilized by not employing realistic element values in numerous examples and problems; it simplifies mastering circuit analysis through convenient component values. The complexity of calculations is thereby reduced, with the understanding that...

Phase Diagram

Phase Diagram

The phase of a given substance depends on the pressure and temperature. Thus, plots of pressure versus temperature showing the phase in each region provide considerable insights into the thermal properties of substances. Such plots are known as phase diagrams. For instance, in the phase diagram for water (Figure 1), the solid curve boundaries between the phases indicate phase transitions (i.e., temperatures and pressures at which the phases coexist).

Phase Transitions: Sublimation and Deposition

Phase Transitions: Sublimation and Deposition

Some solids can transition directly into the gaseous state, bypassing the liquid state, via a process known as sublimation. At room temperature and standard pressure, a piece of dry ice (solid CO2) sublimes, appearing to gradually disappear without ever forming any liquid. Snow and ice sublimate at temperatures below the melting point of water, a slow process that may be accelerated by winds and the reduced atmospheric pressures at high altitudes. When solid iodine is warmed, the solid sublimes...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by

Same author

Hierarchical fragmentation of regular islands in a discontinuous nontwist map.

Chaos (Woodbury, N.Y.)·2026

Same author

Discrete dynamical systems with scaling and inversion symmetries.

Chaos (Woodbury, N.Y.)·2026

Same author

Analysis of invariant spanning curves in oval billiards: A numerical approach based on Slater's theorem.

Chaos (Woodbury, N.Y.)·2025

Same author

Finding critical exponents and parameter space for a family of dissipative two-dimensional mappings.

Chaos (Woodbury, N.Y.)·2024

Same author

Shrinking shrimp-shaped domains and multistability in the dissipative asymmetric kicked rotor map.

Chaos (Woodbury, N.Y.)·2024

Same author

An investigation of escape and scaling properties of a billiard system.

Chaos (Woodbury, N.Y.)·2024

Search research articles

Related Experiment Video

Updated: Sep 10, 2025

Phase Behavior of Charged Vesicles Under Symmetric and Asymmetric Solution Conditions Monitored with Fluorescence Microscopy

Phase Behavior of Charged Vesicles Under Symmetric and Asymmetric Solution Conditions Monitored with Fluorescence Microscopy

Published on: October 24, 2017

Scaling Invariance: A Gateway to Phase Transitions.

Edson Denis Leonel¹

¹Departamento de Física, Universidade Estadual Paulista (UNESP), Av. 24A, 1515, São Paulo 13506-900, SP, Brazil.

Entropy (Basel, Switzerland)

|August 28, 2025

Summary

This study reveals scaling invariance in dynamical systems transitioning from regularity to chaos. This behavior mirrors a continuous phase transition, evidenced by diverging susceptibility near the critical point.

Keywords:

chaos chaotic diffusion critical exponents phase transition

More Related Videos

Orientational Transition in a Liquid Crystal Triggered by the Thermodynamic Growth of Interfacial Wetting Sheets

Orientational Transition in a Liquid Crystal Triggered by the Thermodynamic Growth of Interfacial Wetting Sheets

Published on: May 15, 2017

Experimental Manipulation of Body Size to Estimate Morphological Scaling Relationships in Drosophila

Experimental Manipulation of Body Size to Estimate Morphological Scaling Relationships in Drosophila

Published on: October 1, 2011

Related Experiment Videos

Last Updated: Sep 10, 2025

Phase Behavior of Charged Vesicles Under Symmetric and Asymmetric Solution Conditions Monitored with Fluorescence Microscopy

Phase Behavior of Charged Vesicles Under Symmetric and Asymmetric Solution Conditions Monitored with Fluorescence Microscopy

Published on: October 24, 2017

Orientational Transition in a Liquid Crystal Triggered by the Thermodynamic Growth of Interfacial Wetting Sheets

Orientational Transition in a Liquid Crystal Triggered by the Thermodynamic Growth of Interfacial Wetting Sheets

Published on: May 15, 2017

Experimental Manipulation of Body Size to Estimate Morphological Scaling Relationships in Drosophila

Experimental Manipulation of Body Size to Estimate Morphological Scaling Relationships in Drosophila

Published on: October 1, 2011

Area of Science:

Nonlinear dynamics
Statistical mechanics
Chaos theory

Background:

Dynamical systems can transition between ordered (regular) and disordered (chaotic) states.
Hamiltonian systems, characterized by action and angle variables, are often used to model such transitions.
Phase transitions in physical systems involve critical phenomena and scaling laws.

Purpose of the Study:

To investigate scaling invariance in a specific class of 2D nonlinear area-preserving mappings.
To characterize the transition from regular to chaotic behavior in these systems.
To determine if this transition exhibits properties of a continuous phase transition.

Main Methods:

Utilized a two-dimensional nonlinear mapping preserving area in phase space.
Analyzed the behavior of action and angle variables under variation of a control parameter.
Calculated the average squared action and susceptibility within the chaotic regime.

Main Results:

Observed scaling invariance in the average squared action in the chaotic region.
Identified an order parameter that approaches zero at the transition point.
Demonstrated that susceptibility diverges as the order parameter approaches zero.

Conclusions:

The transition from regularity to chaos in these systems exhibits characteristics of a second-order or continuous phase transition.
Scaling invariance provides evidence for universality in the transition to chaos.
The diverging susceptibility further supports the analogy to phase transitions in statistical mechanics.