A theory of nonequilibrium steady states in quantum chaotic systems

TL;DR

This paper develops a theoretical framework for nonequilibrium steady states (NESS) in quantum chaotic systems, linking the structure of density matrices to eigenstate properties and providing numerical validation.

Contribution

It introduces a universal structure of density matrices in quantum chaos that predicts the emergence of NESS and connects to the eigenstate thermalization hypothesis.

Findings

Density matrix elements behave as Laplace-distributed random variables.

Variance of off-diagonal elements scales as 1/(E_α - E_β)^2 near zero energy difference.

Numerical evidence supports the proposed universal structure in chaotic models.

Abstract

Nonequilibrium steady state (NESS) is a quasistationary state, in which exist currents that continuously produce entropy, but the local observables are stationary everywhere. We propose a theory of NESS under the framework of quantum chaos. In an isolated quantum system, there exist some initial states for which the thermodynamic limit and the long-time limit are noncommutative. The density matrix $\hat \rho$ of these states displays a universal structure. Suppose that $\alpha$ and $\beta$ are different eigenstates of the Hamiltonian with energies $E_\alpha$ and $E_\beta$, respectively. $<\alpha|\hat \rho|\beta>$ behaves as a random number which approximately follows the Laplace distribution with zero mean. In thermodynamic limit, the variance of $<\alpha|\hat \rho|\beta>$ is a smooth function of $\left|E_\alpha-E_\beta\right|$, scaling as $1/(E_\alpha-E_\beta)^2$ in the limit $\left|E_\alpha-E_\beta\right|\to 0$. If and only if this scaling law is obeyed, the initial state evolves into NESS in the long time limit. We present numerical evidence of our hypothesis in a few chaotic models. Furthermore, we find that our hypothesis implies the eigenstate thermalization hypothesis (ETH) in a bipartite system.