TL;DR
This paper extends Einstein's classical Brownian motion to quantum particles in dissipative environments, deriving a quantum diffusion equation and exploring quantum tunneling and electron motion in metals.
Contribution
It introduces a quantum generalization of Einstein's law of Brownian motion and derives a thermo-quantum Smoluchowski diffusion equation for quantum systems.
Findings
Quantum tunneling at equilibrium described
Electron motion in metals analyzed
Quantum diffusion law formulated
Abstract
Einstein's Brownian motion of a quantum particle in a classical environment is studied via virial and equipartition theorems. The effect of continuous measurement in a strongly dissipative environment is accounted for and a quantum generalization of the classical Einstein law of Brownian motion is obtained. A thermo-quantum Smoluchowski diffusion equation is derived via a generalization of the Madelung quantum hydrodynamics. The latter is applied for description of the quantum tunneling at equilibrium and stationary states as well as of the motion of an electron in metals, i.e. the Smoluchowski-Poisson problem.
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