Growing fluctuation of quantum weak invariant and dissipation

TL;DR

This paper investigates the time evolution of fluctuations in weak invariants within nonequilibrium quantum thermodynamics, revealing that such fluctuations do not decrease over time under completely positive maps and linking growth rates to dissipation.

Contribution

It provides a detailed analysis of fluctuation dynamics of weak invariants and connects their growth to dissipation in Lindblad dynamics, with practical examples and a general thermodynamic relation.

Findings

Fluctuations of weak invariants do not diminish over time under completely positive maps.

Growth rate of fluctuations in Lindblad equations is related to the dissipator.

Derived a general relation between specific heat and temperature near equilibrium.

Abstract

The concept of weak invariants has recently been introduced in the context of conserved quantities in finite-time processes in nonequilibrium quantum thermodynamics. A weak invariant itself has a time-dependent spectrum, but its expectation value remains constant under time evolution defined by a relevant master equation. Although its expectation value is thus conserved by definition, its fluctuation is not. Here, time evolution of such a fluctuation is studied. It is shown that if the subdynamics is given by a completely positive map, then the fluctuation of the associated weak invariant does not decrease in time. It is also shown, in the case of the Lindblad equation, how the growth rate of the fluctuation is connected to the dissipator. As examples, the harmonic oscillator with a time-dependent frequency and the spin in a varying external magnetic field are discussed, and the fluctuations of their Hamiltonians as the weak invariants are analyzed. Furthermore, a general relation is presented for the specific heat and temperature of any subsystem near equilibrium following the slow Markovian isoenergetic process.