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<p>We investigate the connection between thermodynamic phase transitions and chaotic dynamics in AdS-black hole spacetimes in <i>f</i>(<i>R</i>) gravity by analyzing the Lyapunov exponents of massless and massive particles orbiting unstable circular orbits. We find that the Lyapunov exponent exhibits multivalued behavior across a the phase transition, with each branch uniquely corresponding to a distinct black hole phase–small, intermediate, or large. Furthermore, the discontinuity in the Lyapunov exponent, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Delta \lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> </mrow> </math></EquationSource> </InlineEquation>, serves as an effective order parameter: it remains finite away from criticality and vanishes continuously at the critical point. Notably, near the critical point, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Delta \lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> </mrow> </math></EquationSource> </InlineEquation> follows a square-root scaling with reduced temperature, characterized by a critical exponent of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(1/2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, consistent with mean-field behavior. These demonstrate that the Lyapunov exponent provides a robust dynamical framework for probing and characterizing black hole phase structure in more generic framework of gravity. Finally, we explore the chaos bound and its violation during the phase transition.</p>

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Lyapunov Exponent and Phase Transitions of AdS Black Hole in f(R) Gravity

  • 期刊论文

  • 发表日期：

    2026年2月12日

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  • Long Cheng,
  • Hong-Guang Li

摘要

We investigate the connection between thermodynamic phase transitions and chaotic dynamics in AdS-black hole spacetimes in f(R) gravity by analyzing the Lyapunov exponents of massless and massive particles orbiting unstable circular orbits. We find that the Lyapunov exponent exhibits multivalued behavior across a the phase transition, with each branch uniquely corresponding to a distinct black hole phase–small, intermediate, or large. Furthermore, the discontinuity in the Lyapunov exponent, \(\Delta \lambda \) Δ λ , serves as an effective order parameter: it remains finite away from criticality and vanishes continuously at the critical point. Notably, near the critical point, \(\Delta \lambda \) Δ λ follows a square-root scaling with reduced temperature, characterized by a critical exponent of \(1/2\) 1 / 2 , consistent with mean-field behavior. These demonstrate that the Lyapunov exponent provides a robust dynamical framework for probing and characterizing black hole phase structure in more generic framework of gravity. Finally, we explore the chaos bound and its violation during the phase transition.