Due to the large amount of microscopic constituents, sensible information that can be gathered about many-body systems concerns usually the behaviour of collective observables; among them, surely average observables, like the mean magnetization in quantum spin chains, but also fluctuations around mean-values. Average operators over all particles are defined with a scaling proportional to the inverse number N of considered particles; in the large N limit, the emergent collective operators form a classical algebra, with no footprints of the microscopic quantum structure they result from. On the contrary, another class of collective observables, the so-called fluctuation operators defined with a scaling proportional to square root of N , has been proved, by means of quantum central limit theorems, to retain quantum properties, giving rise to a Gaussian Bosonic system. These collective observables may thus be interpreted as witnesses of a mesoscopic behaviour positioned at the interface between macroscopic, classical behaviours and microscopic quantum ones, providing a suitable framework where to look for collective quantum phenomena in many-body systems.In this thesis we studied the dynamical behaviour of these fluctuation operators, when the many-body mesoscopic system is considered not to be isolated, but in a weak interaction with a larger environment; this is the most common situation encountered in actual experiments, where these systems can never be thought of as completely isolated from their thermal surroundings. Under some conditions on the dynamical generator, we showed that such dissipative evolution of fluctuations exists and is such that it preserves their Gaussian character. By means of a particular example, we also demonstrated that two non-interacting many-body systems can become entangled, at the level of their fluctuation operators, through the presence of a common environment usually responsible for decoherence and emergence of classical behaviours. Furthermore, the behaviour of such correlations has a neat dependence on the temperature of the heat bath, displaying a sort of phase transition, witnessed by the existence of a finite critical temperature above which entanglement is not possible.
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