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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: 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...

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 molecules...

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 Diagrams

Phase Diagrams

A phase diagram combines plots of pressure versus temperature for the liquid-gas, solid-liquid, and solid-gas phase-transition equilibria of a substance. These diagrams indicate the physical states that exist under specific conditions of pressure and temperature and also provide the pressure dependence of the phase-transition temperatures (melting points, sublimation points, boiling points). Regions or areas labeled solid, liquid, and gas represent single phases, while lines or curves represent...

Properties of Transition Metals

Properties of Transition Metals

Transition metals are defined as those elements that have partially filled d orbitals. As shown in Figure 1, the d-block elements in groups 3–12 are transition elements. The f-block elements, also called inner transition metals (the lanthanides and actinides), also meet this criterion because the d orbital is partially occupied before the f orbitals.

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Determination of the Gas-phase Acidities of Oligopeptides

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Published on: June 24, 2013

Third-Order Phase Transition: Random Matrices and Screened Coulomb Gas with Hard Walls.

Fabio Deelan Cunden¹, Paolo Facchi^2,3, Marilena Ligabò⁴

¹1School of Mathematics and Statistics, University College Dublin, Dublin 4, Ireland.

Journal of Statistical Physics

|July 6, 2019

Summary

A phase transition occurs in a confined gas when its volume changes, driven by repulsive interactions. This study proves a third-order phase transition in random matrix theory and Yukawa gases, identifying electrostatic pressure as the key order parameter.

Keywords:

Coulomb and Yukaw gases Extreme value statistics Large deviations Phase transitions Potential theory Random matrices log-gases

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Area of Science:

Statistical Mechanics
Mathematical Physics
Condensed Matter Physics

Background:

Understanding phase transitions in systems with constraints is crucial for statistical mechanics.
Confined gases with repulsive interactions exhibit complex behaviors under volume variations.
Random matrix theory and Yukawa gases serve as key models for studying such phenomena.

Purpose of the Study:

To rigorously prove the existence of a third-order phase transition in specific gas models under volume constraints.
To derive an exact formula for the free energy of these constrained systems.
To identify the order parameter governing the observed phase transition.

Main Methods:

Analysis of eigenvalues for one-cut, off-critical random matrices (log-gas) with hard walls.
Application of methods to a d-dimensional gas with Yukawa interaction in a confining potential.
Derivation of exact free energy formulas for constrained systems.

Main Results:

A third-order phase transition is proven for both random matrix eigenvalues and Yukawa gases when volume is constrained.
An exact formula for the free energy is derived, explicitly showing a jump in its third derivative.
The 'electrostatic pressure' is identified as the order parameter for the 'pulled' and 'pushed' phases.

Conclusions:

The study confirms a third-order phase transition in constrained d-dimensional gases, applicable to both random matrix models and Yukawa gases.
The derived free energy formula provides a precise mathematical description of the transition.
The electrostatic pressure is established as a fundamental quantity characterizing this phase transition.