A formalism for the construction of quantum friction equations

Joachim Ankerhold

Stefano Giordano, Giuseppe Florio, Giuseppe Puglisi +2 more

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.

B. A. Arbuzov

Stig Stenholm

C E Smith

Methods are presented for extending a promising formalism that incorporates dissipative forces into Quantum Mechanics. Hamiltonians and Lagrangians are constructed that have additional force and frictional terms while still being consistent with the rules of the formalism. The simple systems of a damped particle in a force field and a quadratically damped oscillator are discussed.

Laurent Bruneau