Derivation of Fokker-Planck equation from Schrodinger dynamics

TL;DR

This paper derives the Fokker-Planck equation from quantum Schrödinger dynamics, providing a quantum mechanical foundation and discussing its path integral representation, linking thermodynamic and quantum entropy.

Contribution

It presents a novel derivation of the Fokker-Planck equation purely from quantum mechanics, connecting classical stochastic processes with quantum entropy.

Findings

Fokker-Planck equation derived from Schrödinger dynamics

Path integral representation discussed in context of derivation

Quantum entropy corresponds to coarse-grained thermodynamic entropy

Abstract

The Fokker_Planck equation can be derived in a consistent manner through a microscopic approach based on a unified scheme of classical and quantum mechanics. Here we shall derive it through a purely quantum mechanical approach based on the reversible Schrodinger dynamics. We also give a brief discussion of the path integral representation of the Fokker_Planck equation in light of our derivation. We conclude that, because of the use of the representation of eigenstates of the time-independent Hamiltonian in our derivation, the thermodynamical entropy in this case must correspond to a coarse-graining of the quantum entropy.