A theory of nonequilibrium steady states in quantum chaotic systems

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 $1/\left| E_\alpha - E_\beta\right|^2$ 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.