1,917 research outputs found
Spin-Flip Dynamics of the Curie-Weiss Model: Loss of Gibbsianness with Possibly Broken Symmetry
We study the conditional probabilities of the Curie-Weiss Ising model in vanishing external field under a symmetric independent stochastic spin-flip dynamics and discuss their set of points of discontinuity (bad points). We exhibit a complete analysis of the transition between Gibbsian and non-Gibbsian behavior as a function of time, extending the results for the corresponding lattice model, where only partial answers can be obtained. For initial temperature Ī²^ā1 ā„ 1, we prove that the time-evolved measure is always Gibbsian. For ā
ā¤ Ī²^ā1 < 1, the time-evolved measure loses its Gibbsian character at a sharp transition time. For Ī²^ā1 < ā
, we observe the new phenomenon of symmetry-breaking in the set of points of discontinuity: Bad points corresponding to non-zero spin-average appear at a sharp transition time and give rise to biased non-Gibbsianness of the time-evolved measure. These bad points become neutral at a later transition time, while the measure stays non-Gibbs. In our proof we give a detailed description of the phase-diagram of a Curie-Weiss random field Ising model with possibly non-symmetric random field distribution based on bifurcation analysis.
Stratified Rotating Boussinesq Equations in Geophysical Fluid Dynamics: Dynamic Bifurcation and Periodic Solutions
The main objective of this article is to study the dynamics of the stratified
rotating Boussinesq equations, which are a basic model in geophysical fluid
dynamics. First, for the case where the Prandtl number is greater than one, a
complete stability and bifurcation analysis near the first critical Rayleigh
number is carried out. Second, for the case where the Prandtl number is smaller
than one, the onset of the Hopf bifurcation near the first critical Rayleigh
number is established, leading to the existence of nontrivial periodic
solutions. The analysis is based on a newly developed bifurcation and stability
theory for nonlinear dynamical systems (both finite and infinite dimensional)
by two of the authors [16]
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