We present an effective evolution equation for a coarse-grained distribution function of a long-rangeinteracting system preserving the symplectic structure of the noncollisional Boltzmann, or Vlasov, equation. First, we derive a general form of such an equation based on symmetry considerations only. Then we explicitly derive the equation for one-dimensional systems, finding that it has the form predicted on general grounds. Finally, we use this equation to predict the dependence of the damping times on the coarse-graining scale and numerically check it for some one-dimensional models, including the Hamiltonian mean-field model, a scalar field with quartic interaction, a 1-d self-gravitating system, and a self-gravitating ring.

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In N -body systems with long-range interactions mean-field effects dominate over binary interactions (collisions), so that relaxation to thermal equilibrium occurs on time scales that grow with N , diverging in the N → ∞ limit. However, a much faster and completely non-collisional relaxation process, referred to as violent relaxation, sets in when starting from generic initial conditions: collective oscillations (referred to as virial oscillations) develop and damp out on timescales not depending on the system's size. After the damping of such oscillations the system is found in a quasi-stationary state that may be very far from a thermal one, and that survives until the slow relaxation driven by two-body interactions becomes effective, that is, virtually forever when the system is very large. During violent relaxation the distribution function obeys the collisionless Boltzmann (or Vlasov) equation, that, being invariant under time reversal, does not "naturally" describe a relaxation process. Indeed, the dynamics is moved to smaller and smaller scales in phase space as time goes on, so that observables that do not depend on small-scale details appear as relaxed after a short time.Here we propose an approximation scheme to describe the collisionless relaxation process, based on the introduction of suitable moments of the distribution function, and apply it to a simple toy model, the Hamiltonian Mean Field (HMF) model. To the leading order, virial oscillations are equivalent to the motion of a particle in a one-dimensional potential. Inserting higher-order contributions in an effective way, inspired by the Caldeira-Leggett model of quantum dissipation, we derive a dissipative equation describing the damping of the oscillations, including a renormalization of the effective potential and yielding predictions for collective properties of the system after the damping in very good agreement with numerical simulations. Here we restrict ourselves to "cold" initial conditions, i.e., where the velocities of all the particles are set to zero: generic initial conditions will be considered in a forthcoming paper. 1 For self-gravitating systems a proper thermal equilibrium state in the usual sense does not exist, at least in three dimensions, because gravity is non-confining so that a Maxwellian velocity distribution would lead to the evaporation of the system, unless the latter has infinite mass. The relaxation time here is the time scale over which binary encounters induce a loss of memory of the initial conditions and the Boltzmann entropy grows. Gravity in spaces with dimension less than three (where a thermal equilibrium state is well defined) also has relaxation times diverging with N . arXiv:1902.02436v2 [cond-mat.stat-mech]

We study the statistics of energy fluctuations in a three-level quantum system subject to a sequence of projective quantum measurements. We check that, as expected, the quantum Jarzynski equality holds provided that the initial state is thermal. The latter condition is trivially satisfied for twolevel systems, while this is generally no longer true for N -level systems, with N > 2. Focusing on three-level systems, we discuss the occurrence of a unique energy scale factor β eff that formally plays the role of an effective inverse temperature in the Jarzynski equality. To this aim, we introduce a suitable parametrization of the initial state in terms of a thermal and a non-thermal component. We determine the value of β eff for a large number of measurements and study its dependence on the initial state. Our predictions could be checked experimentally in quantum optics.

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We derive the self-consistent harmonic approximation for the 2D XY model with non-local interactions. The resulting equation for the variational couplings holds for any form of the spin-spin coupling as well as for any dimension. Our analysis is then specialized to power-law couplings decaying with the distance r as in order to investigate the robustness, at finite σ, of the Berezinskii-Kosterlitz-Thouless (BKT) transition, which occurs in the short-range limit . We propose an ansatz for the functional form of the variational couplings and show that for any the BKT mechanism occurs. The present investigation provides an upper bound for the critical threshold above which the traditional BKT transition persists in spite of the non-local nature of the couplings.

The prominent collective character of long-range interacting quantum systems makes them promising candidates for quantum technological applications. Yet, lack of additivity overthrows the traditional picture for entanglement scaling and transport, due to the breakdown of the common mechanism based on excitations propagation and confinement. Here, we describe the dynamics of the entanglement entropy in many-body quantum systems with diverging long-range two body potentials. While in the strict thermodynamic limit entanglement dynamics is shown to be suppressed, a rich mosaic of novel scaling regimes is observed at intermediate system sizes, due to the possibility to trigger multiple resonant modes in the global dynamics. Quantitative predictions on the shape and timescales of entanglement propagation are made, paving the way to the observation of these phases in current quantum simulators. This picture is connected and contrasted with the case of local many body systems subject to Floquet driving.