Statistical mechanics and the quantum friction

M. Mijatović, A. D. Jannussis, A. Streclas

Morton D. Kostin

I.R. Svin'in

Lev N Kantorovich

Using the nonequilibrium statistical operator method, we suggest a new general method of treating dynamics of a combined system consisting of interacting classical and quantum parts. The method is illustrated on the tip dynamics in the noncontact atomic force microscopy (NC-AFM) where a macroscopic tip interacts with a quantum microscopic system (the surface and the nanotip). The derived general equation of motion for the tip and the Fokker-Planck equation, applicable even at low temperatures, contain memory effects and a friction term which should (at least partially) be responsible for the observed energy dissipation in NC-AFM experiments.

Michele Cini

Stefano Giordano, Giuseppe Florio, Giuseppe Puglisi +2 more

Abstract Open quantum systems play a central role in current nanoscale technologies, such as 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 with the requirement of complete positivity of quantum evolution. In this work we develop a framework that addresses this issue by systematically 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 fulfilling 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, fully justifying the classical construction. Moreover, we find that the friction coefficients must adhere to 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 developed here presents a thermodynamically consistent and completely positive quantum extension of classical stochastic mechanics. It offers a versatile tool for deriving quantum versions of thermodynamic laws and is directly applicable to a broad class of non-equilibrium nanoscale systems of current theoretical and technological interest.