We study spontaneous scalarization of electrically charged extremal black holes in $D\geq 4$ spacetime dimensions. Such a phenomenon is caused by the symmetry breaking due to quartic interactions scalar -- Higgs potential and Stueckelberg interaction with electromagnetic gravitational fields, characterized couplings $a$ $b$, respectively. use entropy representation states vicinity horizon, appl...

We consider Ising-spin systems starting from an initial Gibbs measure ν and evolving under a spin-flip dynamics towards a reversible Gibbs measure μ 6= ν. Both ν and μ are assumed to have a finite-range interaction. We study the Gibbsian character of the measure νS(t) at time t and show the following: (1) For all ν and μ, νS(t) is Gibbs for small t. (2) If both ν and μ have a high or infinite t...