Dimensionless fluctuations balance applied to statistics and quantum physics

Area of Science:

• Theoretical Physics
• Statistical Mechanics
• Quantum Mechanics

Background:

• Partial Differential Equations (PDEs) are fundamental in describing physical phenomena.
• Deriving distribution functions from PDEs often requires complex analytical or numerical methods.
• Existing methods may not seamlessly integrate quantum effects or statistical distributions.

Purpose of the Study:

• Introduce a novel Dimensionless Fluctuation Balance (DFB) methodology.
• Demonstrate DFB's capability to derive distribution functions as solutions to PDEs.
• Validate DFB across classical and quantum statistical mechanics.

Main Methods:

• The Dimensionless Fluctuation Balance (DFB) method was developed.
• DFB was applied to derive the Boltzmann PDE and its solutions.
• DFB was extended to incorporate quantum effects using Heisenberg uncertainty relations.

Main Results:

• DFB successfully obtained the Boltzmann PDE and distributions for Boltzmann, Planck, Fermi-Dirac, and Bose-Einstein gases.
• A PDE for Boltzmann's entropy law was derived from thermal and entropy energies using DFB.
• DFB yielded a Schrödinger-type PDE for free particles, consistent with Hamiltonian formalism.

Conclusions:

• The DFB method provides a unified approach to solving PDEs for various distributions.
• DFB offers a pathway to connect classical statistical mechanics with quantum mechanics.
• The DFB methodology shows significant potential for applications in diverse fields like materials modeling and theoretical physics.