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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...
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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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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...
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Recent Progress on Hybrid Percolation Transitions.

Young Sul Cho1, Byungnam Kahng2

  • 1Department of Physics, Jeonbuk National University, Jeonju 54896, Republic of Korea.

Entropy (Basel, Switzerland)
|January 28, 2026
PubMed
Summary
This summary is machine-generated.

Hybrid percolation transitions (HPTs) combine discontinuous order parameter jumps with continuous critical behavior. This review classifies HPTs into global suppression-induced and cascading failure-induced types, revealing universal scaling relations.

Keywords:
continuous percolation transitiondiscontinuous percolation transitionexplosive percolation transitionhybrid percolation transitionhybrid phase transition

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Area of Science:

  • Complex systems science
  • Network theory
  • Statistical physics

Background:

  • Percolation theory explains giant cluster formation in networks above a critical average degree.
  • Hybrid percolation transitions (HPTs) exhibit both discontinuous order parameter jumps and continuous critical behavior simultaneously.
  • HPTs have been observed across diverse scientific systems.

Purpose of the Study:

  • To review and classify representative examples of hybrid percolation transitions.
  • To categorize HPTs into two main classes: global suppression-induced and cascading failure-induced.
  • To identify and outline universal scaling relations common to both HPT categories.

Main Methods:

  • Review of existing literature on hybrid percolation transitions.
  • Classification of HPTs based on the underlying inducing mechanism (global suppression vs. cascading failure).
  • Analysis of critical behavior manifestation in cluster size distributions (global suppression) and avalanche size distributions (cascading failure).

Main Results:

  • HPTs are classified into two distinct categories: global suppression-induced and cascading failure-induced.
  • In global suppression-induced HPTs, critical behavior is observed in cluster size distributions.
  • In cascading failure-induced HPTs, critical behavior is observed in avalanche size distributions.
  • Universal scaling relations governing both types of HPTs have been identified.

Conclusions:

  • Hybrid percolation transitions represent a unique class of phase transitions in complex systems.
  • The classification provides a framework for understanding HPTs based on their inducing mechanisms.
  • The identified universal scaling relations offer insights into the fundamental nature of these transitions across different systems.