[Paper Review] The Phase Transition in Statistical Models Defined on Farey Fractions

Jan Fiala, Peter Kleban+1 more|arXiv (Cornell University)|Mar 23, 2002

Complex Systems and Time Series Analysis4 citations

TL;DR

This paper establishes that statistical models defined on Farey fractions—specifically the Farey spin chain, Knauf spin chain, Farey tree model, and a multifractal model associated with intermittency—share identical free energy functions. Using the transfer operator formalism and Prellberg's spectral analysis, the authors prove a second-order phase transition at β = 2 with specific heat divergence scaling as C ∼ [ε ln²ε]⁻¹, resolving apparent contradictions between a discontinuous magnetization jump and continuous thermodynamic behavior.

ABSTRACT

We consider several statistical models defined on the Farey fractions. Two of these models may be regarded as "spin chains", with long-range interactions, while another arises in the study of multifractals associated with chaotic maps exhibiting intermittency. We prove that these models all have the same free energy. Their thermodynamic behavior is determined by the spectrum of the transfer operator (Ruelle-Perron-Frobenius operator), which is defined using the maps (presentation functions) generating the Farey "tree". The spectrum of this operator was completely determined by Prellberg. It follows that these models have a second-order phase transition with a specific heat divergence of the form [t (ln t)^2]^(-1). The spin chain models are also rigorously known to have a discontinuity in the magnetization at the phase transition.

Motivation & Objective

• To prove equivalence of free energy across four distinct statistical models defined on Farey fractions: the Farey spin chain, Knauf spin chain, Farey tree model, and a multifractal model of intermittency.
• To resolve the apparent contradiction between a discontinuous magnetization jump and continuous thermodynamic behavior at the phase transition.
• To establish that the phase transition is second-order by analyzing the spectrum of the transfer operator acting on functions of bounded variation.
• To connect the phase transition to the Hausdorff dimension of the Farey tree system via spectral results from Prellberg.
• To clarify the scaling behavior of the system, particularly the use of (2−β)/ln(2−β) as a temperature scaling variable consistent with renormalization group theory.

Proposed method

• Define the Farey fractions recursively via mediant addition, with levels k corresponding to chain length in spin models.
• Express the partition function for the Farey spin chain as a sum over Farey fractions: Z_k^FC(β) = ∑_{n=1}^{2^k} (d_k^{(n)} + n_k^{(n+1)})^{-β}.
• Use the transfer operator (Ruelle-Perron-Frobenius operator) defined by the generating maps of the Farey tree to compute the free energy as log of the largest eigenvalue λ(β).
• Prove that the largest eigenvalue in L²((0,1)) and in the space of bounded variation functions coincide for 0 < β < 1, enabling use of Prellberg’s spectral results.
• Apply Prellberg’s result: βf(β) = c(1−β)/ln(1−β)[1+o(1)] for 0 < β < 1, which implies a second-order phase transition at β = 1 (corresponding to β_c = 2 in spin chains).
• Use the equivalence of partition functions and spectral data to show that all four models share the same free energy function and phase transition behavior.

Experimental results

Research questions

• RQ1 Do the Farey spin chain, Knauf spin chain, Farey tree model, and multifractal model of intermittency all exhibit the same free energy function?
• RQ2 Is the phase transition in these models second-order despite a discontinuous magnetization jump?
• RQ3 Can the free energy of the Farey spin chain be rigorously linked to the spectral properties of the transfer operator on the Farey tree?
• RQ4 What is the exact scaling of the specific heat divergence at the phase transition, and how does it relate to the spectral data of the transfer operator?
• RQ5 How does the scaling variable (2−β)/ln(2−β) emerge in the renormalization group analysis of these models?

Key findings

• The free energy is identical across the Farey spin chain, Knauf spin chain, Farey tree model, and the multifractal model of intermittency.
• The phase transition is second-order, with the specific heat diverging as C ∼ [ε ln²ε]⁻¹ as ε → 0⁺, where ε = 2 − β.
• The largest eigenvalue of the transfer operator is discrete for β < 1 and becomes λ = 1 (boundary of continuous spectrum) for β > 1, confirming the phase transition at β = 1 in the transfer operator framework.
• The transfer operator's leading eigenvalue in the space of bounded variation functions matches that in L²((0,1)), validating the use of Prellberg’s spectral results for the thermodynamic analysis.
• The phase transition occurs at the Hausdorff dimension of the Farey tree system, consistent with theoretical expectations.
• The magnetization jumps from full saturation to zero at the transition, yet the thermodynamic behavior is continuous, reconcilable via scaling theory with temperature variable (2−β)/ln(2−β).