Generalized quantum Fokker-Planck, diffusion, and Smoluchowski equations with true probability distribution functions

Area of Science:

• Quantum mechanics
• Statistical physics
• Condensed matter theory

Background:

• Traditional quantum Brownian motion models often use quasiprobability distributions like Wigner functions, which can become problematic (singular or negative) in the full quantum regime.
• Existing methods for describing quantum Brownian motion face challenges with mathematical rigor and applicability under certain conditions.

Purpose of the Study:

• To develop a simplified, non-Markovian theory for quantum Brownian motion.
• To utilize true probability distribution functions for a more robust theoretical framework.
• To establish a method amenable to classical analytical techniques.

Main Methods:

• The approach is based on an initial coherent state representation of bath oscillators and an equilibrium canonical distribution.
• A generalized quantum Langevin equation in c-numbers is derived.
• The derived equations are analyzed using classical non-Markovian dynamics theory.

Main Results:

• The generalized quantum Langevin equation allows for theoretical analysis using classical non-Markovian dynamics.
• The derived Fokker-Planck, diffusion, and Smoluchowski equations are exact quantum analogs of their classical counterparts.
• The theory is valid for arbitrary temperatures and friction, with the Smoluchowski equation applicable in the overdamped limit.

Conclusions:

• This work presents a straightforward and valid extension of classical non-Markovian dynamics to quantum Brownian motion.
• The use of true probability distributions overcomes limitations of quasiprobability functions in quantum regimes.
• The developed theory provides a powerful tool for studying quantum systems without relying on path integral techniques.