On the modeling of irreversibility by relaxator Liouville dynamics

TL;DR

The work presents a general framework, the relaxator Liouville dynamics, for deriving irreversible macroscopic behavior from reversible microscopic quantum mechanics by projecting onto relevant degrees of freedom and tracing out an environment. This yields a frequency-dependent Liouvillian $L(\omega)=L_{\cal P}+L_{\cal P\cal Q} G_{\cal Q}(\omega) L_{\cal Q\cal P}$ with a non-Hermitian dissipator $\Gamma(\omega)$ and a spectral shift $\Delta H(\omega)$, ensuring a unique stationary state and a Pauli master equation in equilibrium. It develops kinetic equations for one-particle densities, generalizes linear response to open systems, and shows how Markov semigroup dynamics emerge in the appropriate weak-coupling, secular limit, while retaining memory effects and initial correlations in the general case. The framework explains two routes to irreversibility—environment-induced lifetimes and dense spectra leading to decay of correlations—and provides a versatile tool for modeling irreversible dynamics in macroscopic quantum systems with clear conditions for its applicability.

Abstract

A general approach to modeling irreversibility starting from microscopic reversibility is presented. The time $t_s$ up to which relevant degrees of freedom of a system are tracked is extremely much shorter than the spectral resolution time $t_e$ that would be necessary to resolve the spectrum of all degrees of freedom involved. A relaxator that breaks reversibility condenses in the Liouville operator of the relevant degrees of freedom. The irrelevant degrees of freedom act as an environment to the system. The irreversible relaxator Liouville equation contains memory effects and initial correlations of all degrees of freedom. Stationary states turn out to be generically unique and independent of the initial conditions and exceptions are due to degeneracies. Equilibrium states lie in the relaxator's kernel yielding a stationary Pauli master equation. Kinetic equations for oneparticle densities are constructed as special cases of relaxator Liouville dynamics. Kubo's linear response theory is generalized to relaxator Liouville dynamics and related to irreversibility within the system. In a weak coupling approximation between system and environment the relaxator can be reduced to environmental correlations and bilinear system operators. Markov approximation turns the relaxator Liouville dynamics into a semi-group dynamics.

Figures (3)

• Figure 1: Visualizing relaxator Liouville dynamics in ${\cal P}$-space by direct processes and virtual processes via hopping and propagation in ${\cal Q}$-space.
• Figure 2: Visualizing the contribution of virtual processes by initial correlations to the opens sytems's initial state. The correlation part ${\Delta \rho}^{\rm corr}_0$ of the initial state within the complementary ${\cal Q}$-space propagates there and hops to the ${\cal P}$-space to contribute to the system's initial state in ${\cal P}$-space with ${\Delta \rho}^{\rm corr}_0(z)$.
• Figure 3: Sketch of spectral properties of (a) the isolated system dynamic, (b) the complementary dynamic and (c) of the relaxator Liouville dynamic.