  We study the emergent dynamics resulting from the infinite volume limit of
the mean-field dissipative dynamics of quantum spin chains with clustering, but
not time-invariant states. We focus upon three algebras of spin operators: the
commutative algebra of mean-field operators, the quasi-local algebra of
microscopic, local operators and the collective algebra of fluctuation
operators. In the infinite volume limit, mean-field operators behave as
time-dependent, commuting scalar macroscopic averages while quasi-local
operators, despite the dissipative underlying dynamics, evolve unitarily in a
typical non-Markovian fashion. Instead, the algebra of collective fluctuations,
which is of bosonic type with time-dependent canonical commutation relations,
undergoes a time-evolution that retains the dissipative character of the
underlying microscopic dynamics and exhibits non-linear features. These latter
disappear by extending the time-evolution to a larger algebra where it is
represented by a continuous one-parameter semigroup of completely positive
maps. The corresponding generator is not of Lindblad form and displays mixed
quantum-classical features, thus indicating that peculiar hybrid systems may
naturally emerge at the level of quantum fluctuations in many-body quantum
systems endowed with non time-invariant states.
