Boltzmann to Lindblad: Classical and Quantum Approaches to Out-of-Equilibrium Statistical Mechanics

TL;DR

This paper develops a unified classical-quantum framework for out-of-equilibrium statistical mechanics, ensuring thermodynamic consistency and complete positivity in quantum models, with applications to nanoscale systems.

Contribution

It introduces a generalized Langevin and Klein-Kramers equation framework that bridges classical and quantum dynamics, ensuring thermodynamic laws and positivity in quantum master equations.

Findings

The framework admits Boltzmann distribution as stationary state.

Complete positivity requires friction and noise in both Hamiltonian equations.

Conditions for Lindblad-type reduction depend on friction operator properties.

Abstract

Open quantum systems play a central role in contemporary nanoscale technologies, including molecular electronics, quantum heat engines, quantum computation and information processing. A major theoretical challenge is to construct dynamical models that are simultaneously consistent with classical thermodynamics and complete positivity. In this work, we develop a framework that addresses this issue by extending classical stochastic dynamics to the quantum domain. We begin by formulating a generalized Langevin equation in which both friction and noise act symmetrically on the two Hamiltonian equations. From this, we derive a generalized Klein-Kramers equation expressed in terms of Poisson brackets, and we show that it admits the Boltzmann distribution as its stationary solution while satisfying the first and second laws of thermodynamics along individual trajectories. Applying canonical quantization to this classical framework yields two distinct quantum master equations, depending on whether the friction operators are taken to be Hermitian or non-Hermitian. By analyzing the dynamics of a harmonic oscillator, we determine the conditions under which these equations reduce to a Lindblad-type generator. Our results demonstrate that complete positivity is ensured only when friction and noise are included in both Hamiltonian equations, thus fully justifying the classical construction. Moreover, we find that the friction coefficients must satisfy the same positivity condition in both the Hermitian and non-Hermitian formulations, revealing a form of universality that transcends the specific operator representation. The formalism offers a versatile tool for deriving quantum versions of the thermodynamic laws and is directly applicable to a wide class of nonequilibrium nanoscale systems.