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An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

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Quantum Smoluchowski equation for driven systems.

Raoul Dillenschneider¹, Eric Lutz

¹Department of Physics, University of Augsburg, D-86135 Augsburg, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|November 13, 2009

Summary

This summary is machine-generated.

We derived a quantum Smoluchowski equation for a driven harmonic oscillator coupled to a heat bath. This equation describes the probability distribution in position space under high friction and temperature conditions.

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Area of Science:

Quantum mechanics
Statistical mechanics
Condensed matter physics

Background:

The behavior of quantum systems interacting with their environment is crucial for understanding phenomena like decoherence.
The quantum harmonic oscillator is a fundamental model in quantum mechanics, often used to study dissipation and noise.

Purpose of the Study:

To derive a semiclassical equation for the Wigner phase space distribution of a driven quantum harmonic oscillator strongly coupled to a heat bath.
To obtain the quantum Smoluchowski equation for the probability distribution in position space in the high friction and high temperature limit.
To determine the validity and discuss special cases of the derived quantum Smoluchowski equation.

Main Methods:

Starting with the exact quantum Langevin equation.
Employing a Green's function approach to obtain the semiclassical Wigner equation.
Applying Brinkman's method in the high friction and high temperature limit to derive the quantum Smoluchowski equation.

Main Results:

A semiclassical equation for the Wigner phase space distribution was determined.
The quantum Smoluchowski equation was successfully derived for the probability distribution in position space.
The range of validity for the quantum Smoluchowski equation was established.

Conclusions:

The quantum Smoluchowski equation provides a simplified description for the dynamics of a driven quantum harmonic oscillator under specific conditions.
The derived equation is applicable in the high friction and high temperature regimes.
The study also explored the specific case of a Brownian parametric oscillator.