Analytic solution of the domain-wall nonequilibrium stationary state

This article reviews recent experimental and theoretical progress on many-body phenomena in dilute, ultracold gases. Its focus are effects beyond standard weak-coupling descriptions, like the Mott-Hubbard-transition in optical lattices, strongly interacting gases in one and two dimensions or lowest Landau level physics in quasi two-dimensional gases in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near Feshbach resonances in the BCS-BEC crossover.

This colloquium gives an overview of recent theoretical and experimental progress in the area of nonequilibrium dynamics of isolated quantum systems. We particularly focus on quantum quenches: the temporal evolution following a sudden or slow change of the coupling constants of the system Hamiltonian. We discuss several aspects of the slow dynamics in driven systems and emphasize the universality of such dynamics in gapless systems with specific focus on dynamics near continuous quantum phase transitions. We also review recent progress on understanding thermalization in closed systems through the eigenstate thermalization hypothesis and discuss relaxation in integrable systems. Finally we overview key experiments probing quantum dynamics in cold atom systems and put them in the context of our current theoretical understanding.

Scaling level-spacing distribution functions in the ``bulk of the spectrum'' in random matrix models of \(N\times N\) hermitian matrices and then going to the limit \(N\to\infty\), leads to the Fredholm determinant of the sine kernel \(\sin\pi(x-y)/\pi (x-y)\). Similarly a scaling limit at the ``edge of the spectrum'' leads to the Airy kernel \([{\rm Ai}(x) {\rm Ai}'(y) -{\rm Ai}'(x) {\rm Ai}(y)]/(x-y)\). In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, M{\^o}ri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlev{\'e} transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for general \(n\), of the probability that an interval contains precisely \(n\) eigenvalues.