Related Experiment Video
Updated: May 18, 2026
06:26
Orientational Transition in a Liquid Crystal Triggered by the Thermodynamic Growth of Interfacial Wetting Sheets
Published on: May 15, 2017
Thermodynamic phase transitions for Pomeau-Manneville maps
1Centro de Matemática, Computação e Cognição, UFABC, 09210-170 Santo André, SP, Brazil. roberto.venegeroles@ufabc.edu.br
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 26, 2012
Summary
We explore phase transitions in Pomeau-Manneville intermittent maps using infinite ergodic theory. A distributional limit theorem helps calculate thermodynamic potentials and understand dynamic characteristics during instability phases.
Area of Science:
- Dynamical Systems and Chaos Theory
- Statistical Mechanics
- Information Theory
Background:
- Pomeau-Manneville maps exhibit intermittent behavior, a key characteristic of chaotic systems.
- Infinite ergodic theory provides tools to analyze systems with diverging measures, often found in complex dynamics.
- Understanding phase transitions is crucial for characterizing the behavior of dynamical systems.
Purpose of the Study:
- To investigate phase transitions in Pomeau-Manneville intermittent maps.
- To apply infinite ergodic theory and distributional limit theorems to these systems.
- To calculate thermodynamic potentials, topological pressure, and Rényi entropy.
Main Methods:
- Utilizing the framework of infinite ergodic theory.
- Applying a distributional limit theorem for analysis.
- Exact calculation of topological pressure and Rényi entropy.
Main Results:
- The distributional limit theorem serves as a powerful tool for calculating thermodynamic potentials.
- The theorem provides insights into dynamic characteristics across different instability phases.
- Exact values for topological pressure and Rényi entropy were determined.
- A connection was established between the distributional limit theorem and non-Gaussian fluctuations in algorithmic complexity.
Conclusions:
- Infinite ergodic theory offers a robust approach to studying phase transitions in intermittent maps.
- The distributional limit theorem is effective for both quantitative calculations and qualitative understanding of system dynamics.
- The findings link thermodynamic properties with information-theoretic concepts like algorithmic complexity.
Related Concept Videos
01:21Path Between Thermodynamics States
Consider the two thermodynamic processes involving an ideal gas that are represented by paths AC and ABC in Figure 1:
01:21Phase Transitions
A phase transition is the process in which a substance changes from one state of matter to another, like from a solid to a liquid, liquid to gas, or vice versa, at a specific temperature and under given pressure conditions. This change is spontaneous and is affected by alterations in temperature and pressure. These parameters impact the strength of the forces between molecules (intermolecular forces) in the substance.During a phase transition, both the initial and final phases of the substance...
02:31Phase 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 occupy...
01:28Phase Diagrams of Ternary Systems
Consider a ternary system, which is composed of three components: water (W), ethanoic acid (E), and trichloromethane (T). Here, Ethanoic acid (E) is fully miscible with both water (W) and trichloromethane (T), meaning it can mix entirely with either of them. However, water and trichloromethane have partial miscibility, meaning they can only mix to a certain extent, beyond which two separate phases will form.The phase diagram of a ternary system is represented as an equilateral triangle, where...
01:19Phase 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).
01:24Phase Diagram
A phase diagram is a graphical representation of the physical states of a substance under different conditions of temperature and pressure. It shows the boundaries between solid, liquid, and gas phases and the conditions at which these phases coexist in equilibrium. An area in a phase diagram represents a single phase, whereas lines or phase boundaries represent the equilibrium between two phases.In the phase diagram of water, the boundary line between the solid and liquid states illustrates...