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Gibbs-non-Gibbs dynamical transitions for mean-field interacting Brownian motions. (English) Zbl 1320.60160

Summary: We consider a system of real-valued spins interacting with each other through a mean-field Hamiltonian that depends on the empirical magnetisation of the spins. The system is subject to stochastic dynamics where the spins perform independent Brownian motions. Using large deviation theory, we show that there exists an explicitly computable crossover time \(t_c \in [0, \infty]\) from Gibbs to non-Gibbs. We give examples of immediate loss of Gibbsianness (\(t_c = 0\)), short-time conservation and large-time loss of Gibbsianness (\(t_c \in(0, \infty)\)), and preservation of Gibbsianness (\(t_c = \infty\)). Depending on the potential, the system can be Gibbs or non-Gibbs at the crossover time \(t = t_c\).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J65 Brownian motion
60F10 Large deviations
82C22 Interacting particle systems in time-dependent statistical mechanics
82C27 Dynamic critical phenomena in statistical mechanics
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