2024/12/07 by C. Evans Hedges, Hedges, C. Evans · 1 citation
Physics and Astronomy · Mathematics · Computer Science · #Advanced Thermodynamics and Statistical Mechanics #Mathematical Biology Tumor Growth #Nonlinear Dynamics and Pattern Formation
paper · pdf · doi:10.48550/arxiv.2412.05639
This paper is concerned with freezing phase transitions in general dynamical systems. A freezing phase transition is one in which, for a given potential ϕ, there exists some inverse temperature β0 > 0 such that for all α, β> β0, the collection of equilibrium states for αϕ and βϕ coincide. In this sense, below the temperature 1 / β0, the system "freezes" on a fixed collection of equilibrium states. We show that for a given invariant measure μ, it is no more restrictive that μ is the freezing state for some potential than it is for μ to be the equilibrium state for some potential. In fact, our main result applies to any collection of equilibrium states with the same entropy. In the case where the entropy map h is upper semi-continuous, we show any ergodic measure μ can be obtained as a freezing state for some potential. In this upper semi-continuous setting, we additionally show that the collection of potentials that freeze at a single state is dense in the space of all potentials. However, in the \Z action setting where the dynamical system satisfies specification, the collection of potentials that do not freeze contains a dense Gδ.