A theory of nonequilibrium steady states in quantum chaotic systems

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.

10 pages, 4 figures

References in corpus (2)

• Thermalization and its mechanism for generic isolated quantum systems
• Quantum many-body systems out of equilibrium

Cited by in corpus (3)

• An Extension of ETH to Non-Equilibrium Steady States
• Coherent transport through a resonant level coupled to random-matrix leads
• Density matrix of chaotic quantum systems