Extended Weyl-Wigner phase-space framework for nonlinear systems: Typical and modified prey-predator-like dynamics.

Area of Science:

• Quantum Mechanics
• Statistical Mechanics
• Phase-Space Formalism

Background:

• The Weyl-Wigner formalism provides a quantum mechanical description in phase space.
• Extending this formalism to more complex Hamiltonians is crucial for understanding quantum systems.
• Classical and quantum dynamics often exhibit deviations, particularly in non-stationary scenarios.

Purpose of the Study:

• To revisit and extend phase-space Weyl-Wigner quantum mechanics for Hamiltonians of the form H(q,p)=K(p)+V(q).
• To identify and quantify deviations from classical and stationary profiles using Wigner functions and currents.
• To develop a novel algorithm for treating quantum modifications in phase space.

Main Methods:

• Application of the extended phase-space Weyl-Wigner formalism to Hamiltonians with K(p) replacing p^2.
• Analysis of Wigner functions and Wigner currents for Gaussian and gamma/Laplacian distribution ensembles.
• Specialization of general results to specific Hamiltonians exhibiting nonlinear dynamics.

Main Results:

• Deviations from classical and stationary profiles were successfully identified.
• The procedure accurately accounts for quantum fluctuations compared to classical phase-space patterns.
• Nonlinear dynamics were revealed for specific Hamiltonians, including quantized prey-predator-like scenarios.

Conclusions:

• The extended phase-space Weyl-Wigner formalism effectively captures quantum fluctuations and nonlinear dynamics.
• The proposed method offers a novel algorithm for analyzing quantum modifications via Wigner currents.
• The framework is applicable to diverse systems, including those with statistical constraints.