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A reversible chemical reaction represents a chemical process that proceeds in both forward (left to right) and reverse (right to left) directions. When the rates of the forward and reverse reactions are equal, the concentrations of the reactant and product species remain constant over time and the system is at equilibrium. A special double arrow is used to emphasize the reversible nature of the reaction. The relative concentrations of reactants and products in equilibrium systems vary greatly;...
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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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Some solids can transition directly into the gaseous state, bypassing the liquid state, via a process known as sublimation. At room temperature and standard pressure, a piece of dry ice (solid CO2) sublimes, appearing to gradually disappear without ever forming any liquid. Snow and ice sublimate at temperatures below the melting point of water, a slow process that may be accelerated by winds and the reduced atmospheric pressures at high altitudes. When solid iodine is warmed, the solid sublimes...
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Phase Diagram01:19

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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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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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In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
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Dynamical Phase Transitions in Nonequilibrium Networks.

Jiazhen Liu1, Nathaniel M Aden1, Debasish Sarker1

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|October 31, 2025
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This summary is machine-generated.

Dynamical phase transitions (DPTs) are critical changes in system behavior. This study introduces a minimal network model showing that nonlinear interactions cause DPTs, with network degree diverging at a critical time.

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Area of Science:

  • Complex systems
  • Network science
  • Statistical physics

Background:

  • Dynamical phase transitions (DPTs) describe critical changes in system behavior at finite times, extending beyond equilibrium physics.
  • While studied in quantum systems, DPTs are less explored in classical and complex systems like social or financial networks.
  • Empirical observations show abrupt dynamical changes in complex systems, necessitating a theoretical framework.

Purpose of the Study:

  • To present a minimal theoretical model for nonequilibrium networks.
  • To demonstrate the emergence of DPTs from nonlinear interactions within networks.
  • To provide a theoretical foundation for understanding emergent nonequilibrium criticality.

Main Methods:

  • Development of a minimal model for nonequilibrium networks.
  • Analytical investigation of network dynamics and critical phenomena.
  • Analysis of network properties such as degree and clustering coefficients.

Main Results:

  • Nonlinear interactions among network edges naturally induce DPTs.
  • Network degree diverges at a finite critical time, exhibiting universal hyperbolic scaling.
  • Key network properties show critical scaling as the system approaches criticality.

Conclusions:

  • The study establishes a theoretical basis for DPTs in classical complex systems.
  • Findings align with empirical observations of abrupt dynamical changes in real-world networks.
  • The model provides insights into emergent nonequilibrium criticality across diverse systems.