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Related Concept Videos

Phase Transitions01:21

Phase 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...
Phase Transitions02:31

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 occupy...
Transition State Theory01:25

Transition State Theory

Transition-state theory, also known as activated-complex theory, provides a molecular-level explanation of reaction rates in both gas-phase and solution-phase reactions. It extends earlier kinetic models by considering the formation of a short-lived, high-energy configuration during a reaction.The progress of a chemical reaction can be represented using a reaction profile, which plots potential energy against the reaction coordinate. As two reactant molecules approach one another, their...
Scale-Up Processes01:14

Scale-Up Processes

The scale-up of microbial fermentation processes is essential in industrial biotechnology, allowing the transition from laboratory-scale experiments to commercial-scale production while aiming to maintain product yield and quality. This process requires meticulous adjustment of equipment design, process parameters, and contamination control strategies to accommodate increasing culture volumes.At the laboratory scale, cultures are typically maintained in 1 to 10-liter glass or autoclavable...
Entropy Changes Accompanying Specific Processes01:21

Entropy Changes Accompanying Specific Processes

Entropy, a measure of disorder in a system, changes during phase transitions like freezing or boiling. At the transition temperature Ttrs, where two phases are in equilibrium, the phase transition is a reversible process. The entropy change can be calculated from a substance's enthalpy of transition using the equation ΔStrs = ΔtrsH /Ttrs.When a perfect gas expands isothermally from one volume to another, entropy increases logarithmically with volume. Conversely, isothermal compression results...
Transformations of Functions II01:29

Transformations of Functions II

Transformations in mathematics alter the position or orientation of a function’s graph while preserving its fundamental shape. One important type of transformation is the horizontal shift, which involves modifying the input variable within a function’s equation. This operation affects where outputs occur along the horizontal axis but does not alter the function’s overall structure.A horizontal shift is achieved by replacing the input variable x with either x + c or x - c, where c is a constant.

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Related Experiment Video

Updated: May 24, 2026

A Method for Investigating Age-related Differences in the Functional Connectivity of Cognitive Control Networks Associated with Dimensional Change Card Sort Performance
09:01

A Method for Investigating Age-related Differences in the Functional Connectivity of Cognitive Control Networks Associated with Dimensional Change Card Sort Performance

Published on: May 7, 2014

Phase transition without global ordering in a hierarchical scale-free network.

Takehisa Hasegawa1, Masataka Sato, Koji Nemoto

  • 1Graduate School of Information Sciences, Tohoku University, 6-3-09, Aramaki-Aza-Aoba, Sendai 980-8579, Japan. hasegawa@m.tohoku.ac.jp

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 10, 2012
PubMed
Summary

Site-bond percolation on a decorated (2,2)-flower network reveals critical phases. Transitions occur between these critical states, characterized by fractal exponents, not traditional order parameters.

Related Experiment Videos

Last Updated: May 24, 2026

A Method for Investigating Age-related Differences in the Functional Connectivity of Cognitive Control Networks Associated with Dimensional Change Card Sort Performance
09:01

A Method for Investigating Age-related Differences in the Functional Connectivity of Cognitive Control Networks Associated with Dimensional Change Card Sort Performance

Published on: May 7, 2014

Area of Science:

  • Statistical Physics
  • Network Science
  • Complex Systems

Background:

  • Percolation theory investigates the behavior of connected components in random networks.
  • Scale-free networks exhibit unique properties due to their heterogeneous degree distributions.
  • Hierarchical networks possess self-similar structures across different scales.

Purpose of the Study:

  • To analyze site-bond percolation on a decorated hierarchical scale-free network, specifically the (2,2)-flower graph.
  • To investigate the influence of site occupation probability on the network's phase diagram.
  • To characterize novel critical phases and transitions beyond standard percolating and non-percolating states.

Main Methods:

  • Renormalization group technique applied to the decorated (2,2)-flower network.
  • Analysis of the phase diagram as a function of the fraction of occupied sites.
  • Characterization of transitions using fractal exponents instead of traditional order parameters.

Main Results:

  • The phase diagram is highly sensitive to the fraction of occupied sites.
  • Unexpectedly, low site occupation probabilities lead exclusively to critical phases, not percolating or non-percolating ones.
  • Transitions between critical phases exist, marked by jumps in the fractal exponent, even with a zero order parameter.
  • A critical threshold for unoccupied sites exists, beyond which a single critical phase dominates.

Conclusions:

  • The decorated (2,2)-flower network exhibits complex phase behavior under site-bond percolation.
  • Fractal exponents serve as key indicators for transitions between distinct critical states.
  • The system demonstrates a unique phase transition mechanism driven by criticality rather than long-range order.