The Connection between the Stochastic Schr\"{o}dinger Equation and Boltzmann Equation

TL;DR

This paper explores the theoretical link between the stochastic Schrödinger equation and the Boltzmann equation in describing heavy quark transport in quark-gluon plasma, confirming their consistency through analytical and numerical methods.

Contribution

It establishes the connection between the stochastic Schrödinger equation and the Boltzmann equation within the Keldysh Green's function formalism, demonstrating their equivalence in the weak coupling limit.

Findings

SSE is consistent with the Boltzmann equation in the weak coupling limit

Analytical comparison of Green's functions confirms the connection

Numerical calculations support the theoretical results

Abstract

The heavy quarks present in the quark-gluon plasma (QGP) can act as a probe of relativistic heavy ion collisions as they retain the memory of their interaction history. In a previous study, a stochastic Schr\"{o}dinger equation (SSE) has been applied to describe the transport process of heavy quarks, where an external field with random phases is used to simulate the thermal medium. In this work, we study the connection between the SSE and the Boltzmann equation (BE) approach in the Keldysh Green's function formalism. By comparing the Green's function of the heavy quark from the SSE and the Keldysh Green's functions leading to the Boltzmann equation, we demonstrate that the SSE is consistent with the Boltzmann equation in the weak coupling limit. We subsequently confirm their consistency through numerical calculations.