An Endogenous Quantum-Classical Crossover Temperature in the van der Waals Fluid: Quantumness as an Emergent Behavior

Area of Science:

• Thermodynamics
• Statistical Mechanics
• Quantum Mechanics

Background:

• Traditional quantum behavior onset in gases relies on external criteria, like comparing the thermal de Broglie wavelength to mean intermolecular separation (nλT3∼1).
• This criterion is not inherent in classical statistical mechanics frameworks such as ideal-gas or van der Waals partition functions.

Purpose of the Study:

• To demonstrate that a grand-canonical treatment of the van der Waals fluid can naturally generate an interaction-corrected crossover temperature, T3.
• To show that this T3 is determined by particle mass, density, and van der Waals interaction parameters (a,b).
• To establish T3 as a self-consistent condition within the grand-canonical van der Waals description, indicating the breakdown of classical theory.

Main Methods:

• Utilized a grand-canonical ensemble treatment for the van der Waals fluid.
• Derived a self-consistent condition to determine the interaction-corrected crossover temperature T3.
• Analyzed the implications of T3 for the validity of classical statistical assumptions.

Main Results:

• A crossover temperature T3(a,b,m,n) is naturally generated, incorporating particle mass, density, and van der Waals interaction parameters.
• The interaction-induced correction to T3 is obtained without direct use of Bose-Einstein or Fermi-Dirac distributions.
• Below T3, classical statistical assumptions become self-inconsistent, signaling the onset of the quantum-degenerate regime.

Conclusions:

• The derived T3 provides an interaction-corrected boundary for the validity of the classical van der Waals description.
• Intermolecular interactions significantly modify the crossover to the quantum-degenerate regime.
• This work offers a new perspective on the limits of applicability of classical van der Waals theory in the context of quantum effects.