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## Quantum out of equilibrium dynamics

Besides their experimental relevance and possible technological application,

the behavior of quantum systems out of equilibrium is a topic of fundamental

interest, the understanding of which is still at an early stage.

Some techniques of classical dissipative systems have been generalized to the quantum case but important questions remain to be answered such as which would be the meaning of the effective temperature appearing in fluctuation-dissipation relations out of equilibrium.

The time-dependent (second and higher order) correlation functions and the (linear and non-linear) responses of systems evolving close to equilibrium satisfy a number of model independent relations. In classical dynamics these fluctuation-dissipation theorems are Ward identities of a symmetry of the, generically non-Markovian, dynamic generating functional. The quantum version is work in progress. We studied the effect of a drive, realized in the form of the coupling to

two fermion reservoirs at different chemical potentials, on the phase transition and coarsening dynamics of coupled quantum rotors. In experimental systems the coupling to an environment is usually unavoidable. We studied the effect of a non-Ohmic classical bath on the critical (Langevin) dynamics of the classical —phi-4, O(N) model in the large N limit by using the dynamic renormalization-group. We showed that although the critical temperature and critical dimensions

are not altered by the bath, the dynamic critical exponent is, for sufficiently sub-Ohmic baths. As a step towards understanding impurity motion in one dimensional ultra cold quantum liquids, we derived the non equilibrium correlation function of a quantum Brownian particle with general Gaussian

non-factorizing initial conditions. We studied the possible emergence of thermal behavior long after the quench of a closed quantum system by studying fluctuation-dissipation relations in which parameters replace the equilibrium temperature. We analyzed these relations in the stationary regime after a quench in the transverse field of the quantum Ising chain.

The lack of thermalization to a Gibbs ensemble became apparent with this approach.