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Third Law of Thermodynamics02:38

Third Law of Thermodynamics

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A pure, perfectly crystalline solid possessing no kinetic energy (that is, at a temperature of absolute zero, 0 K) may be described by a single microstate, as its purity, perfect crystallinity,and complete lack of motion means there is but one possible location for each identical atom or molecule comprising the crystal (W = 1). According to the Boltzmann equation, the entropy of this system is zero.
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Phase Transitions02:31

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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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Entropy and the Second Law of Thermodynamics

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The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
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Phase Transitions: Melting and Freezing02:39

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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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Phase transitions play an important theoretical and practical role in the study of heat flow. In melting or fusion, a solid turns into a liquid; the opposite process is freezing. In evaporation, a liquid turns into a gas; the opposite process is condensation.
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Thermodynamic quantum phase transition by the structural-stability-based catastrophe theory.

Jiu Hui Wu1, Jiamin Niu1, Hong Lin Liu1

  • 1School of Mechanical Engineering, Xi'an Jiaotong University & State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi'an 710049, China.

Iscience
|April 25, 2025
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Summary
This summary is machine-generated.

This study quantitatively investigates thermodynamic quantum phase transitions using catastrophe theory. The approach accurately models systems like liquid helium, validating its potential for quantum many-body problems.

Keywords:
Natural sciencesPhysicsQuantum theoryThermodynamics

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Area of Science:

  • Thermodynamics
  • Quantum Mechanics
  • Statistical Mechanics

Background:

  • Understanding quantum phase transitions is crucial for condensed matter physics.
  • Existing models often struggle to bridge quantum and macroscopic scales.
  • Catastrophe theory offers a novel framework for analyzing abrupt system changes.

Purpose of the Study:

  • To quantitatively investigate thermodynamic quantum phase transitions.
  • To develop a model applicable from quantum to macroscopic scales.
  • To validate the model using experimental data from liquid helium.

Main Methods:

  • Utilizing structural-stability-based catastrophe theory.
  • Adopting the cusp catastrophe model for average free energy.
  • Applying dimensionless analysis to derive quantum state equations.
  • Calculating ensemble free energy, partition function, and specific heat capacity.

Main Results:

  • A general quantum state equation for pressure was derived.
  • Exact expressions for ensemble free energy, partition function, and specific heat were obtained.
  • The model accurately predicted the superfluid phase transition of liquid helium.

Conclusions:

  • The catastrophe theory-based approach provides a robust method for quantitatively analyzing quantum phase transitions.
  • This framework successfully bridges quantum and macroscopic scales.
  • The validated theory offers a new tool for exploring quantum many-body problems.