  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.
