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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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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...
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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...
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Cooperative allosteric transitions can occur in multimeric proteins, where each subunit of the protein has its own ligand-binding site. When a ligand binds to any of these subunits, it triggers a conformational change that affects the binding sites in the other subunits; this can change the affinity of the other sites for their respective ligands. The ability of the protein to change the shape of its binding site is attributed to the presence of a mix of flexible and stable segments in the...
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Phase transitions in Pareto optimal complex networks.

Luís F Seoane1,2, Ricard Solé1,2,3

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Network organization in complex systems is shaped by optimization processes. Phase transitions in these networks depend on the functions being optimized, not just structural changes.

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Area of Science:

  • Complex systems science
  • Network theory
  • Statistical physics

Background:

  • Complex systems are organized by interaction networks, represented as graphs.
  • Network topology arises from various mechanisms, including optimization under constraints.
  • Spatial networks, like power grids or neural wiring, often involve optimization.

Purpose of the Study:

  • To investigate network designs emerging from Pareto optimization processes.
  • To analyze how simultaneous constraints influence network organization.
  • To identify the drivers of phase transitions in optimized networks.

Main Methods:

  • Studied network designs resulting from Pareto optimization.
  • Analyzed three variations of an optimization problem.
  • Identified distinct network phases and their associated topological arrangements.

Main Results:

  • Observed phase transitions of different kinds in network structures.
  • Found that distinct phases correlate with specific connection arrangements.
  • Demonstrated that the functions under optimization, not topological changes, determine phase transitions.

Conclusions:

  • Phase transitions in optimized networks are driven by the interplay between the system and its constraints.
  • The functions being optimized play a crucial role in shaping network organization and transitions.
  • Network structure is a result of optimization balancing multiple, simultaneous objectives.