Gibbs-non-Gibbs dynamical transitions for mean-field interacting Brownian motions

Abstract

We consider a system of real-valued spins interacting with each other through a mean-field Hamiltonian that depends on the empirical magnetization of the spins via a general potential. The system is subjected to a stochastic dynamics where the spins perform independent Brownian motions. As in \cite{FedHoMa13}, which considers the Curie-Weiss model with Ising spins interacting via a quadratic potential and subjected to independent spins flips, we follow the program outlined in \cite{vEFedHoRe10}. We show that in the thermodynamic limit the system is non-Gibbs at time $t\in (0,\infty )$ if and only if there exists an $\alpha \in \mathbb{R}$ such that the large deviation rate function for the trajectory of the magnetization conditional on hitting the value $\alpha$ at time $t$ has multiple global minimizers. We further show that different minimizing trajectories are different at time $t=0$. We give conditions on the potential under which the system is Gibbs at time $t=0$, classify the possible scenarios of being Gibbs at time $t\in (0,\infty )$ in terms of the second difference quotient of the potential, and show that the system cannot become Gibbs once it has become non-Gibbs, i.e., there is a unique and 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 of Gibbsianness, 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 cross-over time time $t=t_c$.

References in corpus (2)

• Gibbs-non-Gibbs transitions via large deviations: computable examples
• On the prevalence of non-Gibbsian states in mathematical physics

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• Dynamical Gibbs-non-Gibbs transitions in lattice Widom-Rowlinson models with hard-core and soft-core interactions
• Dynamical Gibbs-non-Gibbs transitions in the Curie-Weiss Potts model in the regime beta<3
• Attractor properties of non-reversible dynamics w.r.t invariant Gibbs measures on the lattice
• Large deviations of continuous regular conditional probabilities
• Gibbs Properties of the Bernoulli field on inhomogeneous trees under the removal of isolated sites