[Paper Review] Chaos : Butterflies also generate phase transitions and parallel universes

Renaud Leplaideur|arXiv (Cornell University)|Jan 23, 2013

Nonlinear Dynamics and Pattern Formation13 references3 citations

TL;DR

This paper demonstrates that in mixing subshifts of finite type with specific potentials, phase transitions can occur even when the pressure function remains strictly convex and analytic—challenging the conventional link between non-analyticity and phase transitions. Surprisingly, multiple equilibrium states coexist despite analytic pressure, revealing complex dynamical behavior akin to parallel universes in statistical mechanics.

ABSTRACT

We exhibit examples of mixing subshifts of finite type and potentials such that there are phase transitions but the pressure is always strictly convex. More surprisingly, we show that the pressure can be analytic on some interval although there exist several equilibrium states.

Motivation & Objective

• To challenge the conventional belief that phase transitions in dynamical systems require non-analytic pressure functions.
• To investigate whether multiple equilibrium states can coexist when the pressure function remains analytic.
• To explore the role of local equilibrium states and their global realizability in subshifts of finite type.
• To clarify the relationship between pressure regularity and the existence of multiple equilibrium states in chaotic systems.
• To model phase transitions that are not freezing-type, where pressure remains non-affine after transition.

Proposed method

• Constructing a mixing subshift of finite type with a butterfly-shaped transition graph, using a finite alphabet including symbols {1,2,3,4,1₁,…,1ₗ} or an extended version with 3′,4′.
• Defining a potential φ(x) that depends on the first non-2 symbol in the sequence, with values based on logarithmic terms and parameters α, γ, δ, ε.
• Using thermodynamic formalism to define the pressure function P(β) = max_μ {h_μ + β∫φ dμ}, where μ ranges over σ-invariant measures.
• Applying spectral theory to the transfer operator L_Z to analyze local equilibrium states μ_{Z,[i]} on cylinders [i], using eigenfunctions and eigenmeasures.
• Establishing conditions under which local equilibrium states can be extended to global equilibrium states via finiteness of return time expectations.
• Analyzing the spectral radius λ_{Z,[i]} of the restricted transfer operator and its dependence on Z, particularly at critical values Z_c.

Experimental results

Research questions

• RQ1 Can phase transitions occur in mixing subshifts of finite type when the pressure function remains analytic?
• RQ2 Is it possible to have multiple equilibrium states while the pressure is strictly convex and analytic on an interval?
• RQ3 What conditions allow a local equilibrium state on a cylinder to extend to a global equilibrium state?
• RQ4 How does the regularity of the pressure function relate to the multiplicity of equilibrium states in chaotic systems?
• RQ5 Can non-freezing phase transitions—where pressure remains non-affine after transition—be constructed in dynamical systems?

Key findings

• The paper constructs a mixing subshift of finite type with a potential such that the pressure function is analytic on an interval despite the existence of multiple equilibrium states.
• It demonstrates that phase transitions can occur without loss of analyticity, contradicting the common assumption that non-analyticity of pressure implies phase transition.
• Multiple equilibrium states coexist even when the pressure is strictly convex, showing that the coexistence of equilibria does not imply non-analyticity.
• A critical condition for extending a local equilibrium state on a cylinder [i] to a global one is the finiteness of the expectation of the return time to [i], which depends on the derivative of the transfer operator.
• When λ_{Z_c,[i]} = 1 and the return time expectation is finite, the local equilibrium state extends to a global one, and the global pressure P(φ) equals Z_c.
• The construction reveals a dynamical mechanism where 'butterfly' structures in the transition graph generate multiple stable states, analogous to parallel universes in statistical mechanics.