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Phase Transitions: Melting and Freezing02:39

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...
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Phase Transitions: Vaporization and Condensation02:39

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Updated: Jun 23, 2026

Confocal Imaging of Confined Quiescent and Flowing Colloid-polymer Mixtures
10:56

Confocal Imaging of Confined Quiescent and Flowing Colloid-polymer Mixtures

Published on: May 20, 2014

Phase transitions in diluted negative-weight percolation models.

L Apolo1, O Melchert, A K Hartmann

  • 1City College of the City University of New York, New York, New York 10031, USA. lapolo00@ccny.cuny.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 28, 2009
PubMed
Summary
This summary is machine-generated.

We studied negative-weight loops on diluted lattice graphs. Dilution by removing edges changes the universality class of the percolation transition, unlike zero-weight dilution.

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Area of Science:

  • Statistical physics
  • Graph theory
  • Computational mathematics

Background:

  • Investigating geometric properties of loops on 2D lattice graphs with positive and negative edge weights.
  • Exploring the appearance of spanning loops with a total negative weight.
  • Analyzing a percolation problem fundamentally different from conventional percolation.

Purpose of the Study:

  • To understand how dilution affects the percolation transition in lattice graphs with weighted edges.
  • To examine two types of dilution: edges with zero weight and absent edges.
  • To determine the critical properties and phase diagram of these diluted systems.

Main Methods:

  • Numerical simulations using exact combinatorial optimization techniques.
  • Graph transformations and matching algorithms.
  • Finite-size scaling analysis to determine critical properties and phase diagrams.

Main Results:

  • The first type of dilution (zero-weight edges) does not alter the universality class compared to the undiluted case.
  • The second type of dilution (absent edges) leads to a change in the universality class.
  • Phase diagrams and critical properties of the phase boundary were determined.

Conclusions:

  • Dilution significantly impacts the universality class of percolation transitions in weighted lattice graphs.
  • The nature of dilution (zero-weight vs. absent edges) dictates the universality class.
  • This research provides insights into complex percolation phenomena in diluted graph systems.