Paper · Jun 12, 2026

Stochastic Thermodynamics on Time-Evolving Curved Spaces

Rihito Nagase, Shoki Sugimoto, Asuka Takatsu, Takahiro Sagawa

cond-mat.stat-mecharXiv:2606.14186
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As on arXiv

Abstract

We construct stochastic thermodynamics of overdamped Langevin systems on nonrelaticvistic curved spaces with time-dependent metrics. The time dependence of the metric contributes to the energy balance by performing work on the kinetic energy, which is instantaneously dissipated as heat in the overdamped regime. This contribution makes our framework thermodynamically consistent so that entropy production satisfies the second law of thermodynamics. As a special case, when the metric evolves according to backward Ricci flow, the entropy balance exhibits a structure similar to Perelman's entropy functional. Our framework provides a way to quantify thermodynamic costs in dynamics on time-evolving spaces such as diffusion on membranes.

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Submitted
Jun 12, 2026
Subjects
cond-mat.stat-mech, math-ph

Comments: (7+6) pages, 3 figures

Ricci flowPerelman entropy

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper A local Sobolev inequality on Ricci flow and its applications (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)3Chan 2021 Evolution of Lifshitz metric anisotropies in Einstein-Proca theory under the Ricci-DeTurck flow (Roberto Cartas-Fuentevilla, Manuel de la Cruz, Alfredo Herrera-Aguilar et al., 2022)8Cartas-Fuent 2022 Local smooth convergence of F-limit flows (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2023)5Chan 2023 Monotonicity of RG flow in emergent dual holography of worldsheet nonlinear σ model (Ki-Seok Kim, Arpita Mitra, Debangshu Mukherjee et al., 2024)2Kim 2024 Innovative Dynamics: Utilizing Perelman's Entropy and Ricci Flow for Settler Position Models on Manifolds (Zeraoulia Rafik and Sobhan Sobhan Allah, 2024)1Rafik 2024 Nonassicative cosmological solitonic R-flux deformations in gauge gravity and G. Perelman geometric flow thermodynamics (L. Bubuianu, J. O. Seti, S. Vacaru et al., 2024)4Bubuianu 2024 The fundamental physical importance of generic off-diagonal solutions and Grigori Perelman entropy in the Einstein gravity theory (Sergiu I. Vacaru and Elşen V. Veliev, 2025)7Vacaru 2025 On the singularity formation of gauge fields coupled to Ricci flow (Andoni Royo Abrego, 2026)6Abrego 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Innovative Dynamics: Utilizing Perelman's Entropy and Ricci Flow for Settler Position Models on Manifolds

    Zeraoulia Rafik and Sobhan Sobhan Allah

    Topics Perelman entropy · Shared terms “contributes”, “quantify”

  2. Monotonicity of RG flow in emergent dual holography of worldsheet nonlinear model

    Ki-Seok Kim, Arpita Mitra, Debangshu Mukherjee et al.

    Topics Perelman entropy · Shared terms “entropy production”, “entropy functional”

  3. A local Sobolev inequality on Ricci flow and its applications

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Perelman entropy

  4. Nonassicative cosmological solitonic R-flux deformations in gauge gravity and G. Perelman geometric flow thermodynamics

    L. Bubuianu, J. O. Seti, S. Vacaru et al.

    Topics Perelman entropy · Shared terms “thermodynamics”

  5. Local smooth convergence of -limit flows

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Perelman entropy

  6. On the singularity formation of gauge fields coupled to Ricci flow

    Andoni Royo Abrego

    Topics Perelman entropy

  7. The fundamental physical importance of generic off-diagonal solutions and Grigori Perelman entropy in the Einstein gravity theory

    Sergiu I. Vacaru and Elşen V. Veliev

    Topics Perelman entropy · Shared terms “thermodynamic”

  8. Evolution of Lifshitz metric anisotropies in Einstein-Proca theory under the Ricci-DeTurck flow

    Roberto Cartas-Fuentevilla, Manuel de la Cruz, Alfredo Herrera-Aguilar et al.

    Topics Perelman entropy · Shared terms “entropy functional”, “consistent”, “contribution”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Perelman entropy Perelman found that Ricci flow is a gradient flow. His 𝓕-functional and 𝓦-entropy increase along the flow, and their monotonicity rules out… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

  1. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.

  2. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  3. 6B.16
    Eigenvalue Problems and Perelman’s λBook 6B · Linear PDE: Characteristics, Waves and Regularity

    Rayleigh quotients and min–max, Faber–Krahn and Cheeger’s inequality, ground states of Schrödinger operators, spectral clustering, and the first eigenvalue of −4Δ + R that begins Perelman’s first preprint.