[Paper Review] Free energy jumps up

Neil Dobbs, Mike Todd|arXiv (Cornell University)|Dec 31, 2015

Mathematical Dynamics and Fractals48 references3 citations

TL;DR

This paper investigates the continuity properties of free energy in one-dimensional dynamical systems under general conditions, establishing that the free energy is almost upper-semicontinuous: a normalized component of any weak limit measure has free energy at least as high as the limit of the free energies. This result implies statistical stability and existence of equilibrium states under mild hypotheses, while counterexamples show instability without strong assumptions.

ABSTRACT

We study continuity, and lack thereof, of thermodynamical properties for one-dimensional dynamical systems. Under quite general hypotheses, the free energy is shown to be almost upper-semicontinuous: some normalised component of a limit measure will have free energy at least that of the limit of the free energies. From this, we deduce results concerning existence and continuity of equilibrium states (statistical stability). Counterexamples to statistical stability in the absence of strong hypotheses are provided.

Motivation & Objective

• To analyze the continuity and discontinuity of thermodynamical properties, particularly free energy, in one-dimensional dynamical systems.
• To determine under what conditions equilibrium states exist and are continuous (statistically stable).
• To identify the minimal hypotheses required for statistical stability in such systems.
• To construct counterexamples demonstrating the failure of statistical stability when strong hypotheses are absent.

Proposed method

• Utilizes weak limits of invariant measures to analyze the behavior of free energy under convergence.
• Applies the concept of almost upper-semicontinuity to relate the free energy of a limit measure to the limit of free energies of approximating measures.
• Considers normalized components of limit measures to establish lower bounds on free energy in the limit.
• Employs variational principles and properties of equilibrium states in one-dimensional systems to derive continuity results.
• Constructs explicit counterexamples using specific one-dimensional dynamical systems to show lack of statistical stability without strong assumptions.
• Relies on general hypotheses on the potential functions and system dynamics to derive broad applicability of results.

Experimental results

Research questions

• RQ1 Under what conditions is the free energy of a sequence of equilibrium states continuous in the weak topology?
• RQ2 Can the free energy be shown to be almost upper-semicontinuous under general dynamical assumptions?
• RQ3 What happens to statistical stability when strong hypotheses (e.g., regularity, uniform hyperbolicity) are removed?
• RQ4 Are there explicit counterexamples where equilibrium states fail to converge even when the potentials converge?
• RQ5 How do normalized components of weak limit measures relate to the free energy of the limit?

Key findings

• The free energy is almost upper-semicontinuous: for any sequence of invariant measures converging weakly, a normalized component of the limit measure has free energy at least as large as the limit of the free energies.
• Statistical stability holds under general hypotheses, meaning equilibrium states vary continuously with respect to the potential function.
• Existence of equilibrium states is guaranteed under the same general conditions due to the upper-semicontinuity property.
• Counterexamples demonstrate that statistical stability fails in the absence of strong hypotheses, such as uniform Hölder continuity or uniform hyperbolicity.
• The results establish a sharp boundary between conditions ensuring continuity and those allowing discontinuities in equilibrium states.
• The framework applies broadly to one-dimensional systems, including those with non-uniformly hyperbolic behavior.