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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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The phase of a given substance depends on the pressure and temperature. Thus, plots of pressure versus temperature showing the phase in each region provide considerable insights into the thermal properties of substances. Such plots are known as phase diagrams. For instance, in the phase diagram for water (Figure 1), the solid curve boundaries between the phases indicate phase transitions (i.e., temperatures and pressures at which the phases coexist).
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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...
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We studied coupled oscillators with random interactions and thermal noise. The system shows a universal synchronization threshold, unaffected by noise or coupling randomness.

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Area of Science:

  • Complex Systems
  • Statistical Physics
  • Network Science

Background:

  • Understanding collective dynamics and synchronization in coupled oscillator systems is crucial for various scientific fields.
  • The influence of non-separable, random interactions and thermal noise on synchronization transitions remains an active area of research.

Purpose of the Study:

  • To investigate the collective dynamics of XY model-type oscillators with non-separable random interactions under thermal noise.
  • To derive and validate theoretical formulas for critical synchronization thresholds in both deterministic and stochastic regimes.
  • To explore the universality of synchronization phenomena in heterogeneous network structures.

Main Methods:

  • Analysis of collective dynamics using stability analysis for finite populations and mean-field theory with graphons for the thermodynamic limit.
  • Derivation of exact formulas for critical transition thresholds in deterministic (T=0) and stochastic (T>0) cases.
  • Extensive numerical simulations to support theoretical findings.

Main Results:

  • At T=0, a discontinuous, first-order-like phase transition from incoherence to full coherence is observed.
  • For T>0, the transition from incoherence to partial coherence becomes continuous, with a higher critical threshold.
  • The derived synchronization threshold is identical to that found in systems with uniform node strengths, suggesting universality.

Conclusions:

  • The study provides a rigorous theoretical framework for understanding synchronization in complex oscillator networks with random interactions.
  • Thermal noise alters the nature of the phase transition, making it continuous and increasing the synchronization threshold.
  • The observed universality of the synchronization threshold highlights fundamental principles governing collective behavior in diverse network topologies.