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REVIEW 4 major objections 4 minor 33 references

A Loewner-Theoretic Approach to the Nonlinear Generalized Langevin Equation: The Role of Entropy in Colored Noise Environment

T0 review · 4 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper proposes that replacing the Mori-Zwanzig projection operator with a Loewner transfer operator yields a nonlinear generalized Langevin equation whose two fluctuation-dissipation relations both dissipate energy as R ~ t^-1.

desk verdict A novel but mathematically shaky attempt to recast generalized Langevin dynamics in Loewner theory; the central projection-operator step doesn't hold, and the reported t^{-1} scaling is essentially built in. read the letter →

arxiv 2607.13384 v1 pith:BWBZNWBS submitted 2026-07-15 cond-mat.stat-mech math-phmath.MP

Yusuke K. Shibasaki This is my paper · ORCID
classification cond-mat.stat-mechmath-phmath.MP
keywords GeneralizedLangevinequationLoewnerMori-Zwanzigprojectionoperatorfluctuation-dissipationrelationcolorednoiseconformalmapcellmigrationentropy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to show that the machinery of conformal maps, specifically the chordal Loewner equation and its transfer operator, can stand in for the projection operator in the Mori-Zwanzig derivation of a generalized Langevin equation. If this works, the memory kernel and noise in a one-dimensional colored-noise system are no longer abstract terms but are computed from the geometry of a growing curve in the upper half-plane. The author derives two forms of the fluctuation-dissipation relation, one of Kubo type and one obtained directly from the Loewner time change, and claims both give energy dissipation scaling as R ~ t^-1. A numerical simulation with the cell-migration kernel K(t) = b + exp(-t/c) reproduces that scaling for both forms. A sympathetic reader would care because this suggests a new route from conformal geometry to nonequilibrium statistical mechanics, with an explicit entropy-like quantity, Loewner entropy, controlling the ensemble.

What carries the argument

The chordal Loewner equation ∂_s g_s(z) = 2/(g_s(z) - U_s) describes growth of a curve in the upper half-plane, and its discrete version gives the slit maps h_n. The transfer operator P_n f(z) = Σ_{z ∈ h^{-1}(w)} |h'_n(χ_σ(w))|^{-1} f(χ_σ(w)) is the central object: the paper substitutes P_n for the Mori-Zwanzig projection operator and defines Q_n = 1 - P_n, then applies Dyson's identity to get Eq. (11). The same operator family supplies the Loewner driving force η_s^M and the Loewner entropy S_Loew = -ln p(η_s), which together turn the memory/noise decomposition and the response formulas into conformal-geometric statements.

What would settle it

Directly compute P_n^2 f for a simple choice, such as the vertical slit map with constant ΔU_n and Δs_n, on the space of bounded functions on H equipped with a chosen inner product. If P_n^2 ≠ P_n on any non-trivial function, Eqs. (12)-(17) do not follow and the derivation collapses. Alternatively, acquire high-resolution cell-migration trajectories and fit the log-log slope of the response function; a slope that deviates from -1 over two decades would falsify the claimed universal energy-dissipation scaling.

Watch

Extended reading notes

Core claim

The central claim is that Eq. (16), ∂_t v_n(t) = h_n(x) + ∫_0^t η_s^M(t') v_n(t) dt' + F_n(t,x), is a nonlinear generalized Langevin equation obtained by replacing the Mori-Zwanzig projection operator with the Loewner transfer operator P_n built from preimages of the slit map h_n(z) = sqrt((z - ΔU_n)^2 + 4Δs_n). The paper asserts the fluctuation-dissipation relation K_n = η_s^M = ⟨F_n F_n⟩ holds, and that both the Kubo-type response function and the direct Loewner response function decay as R ~ t^-1. It further claims the Loewner entropy S_Loew = -ln p(η_s) lets the colored-noise dynamics be recast as a microcanonical ensemble, which is what makes the direct response formula possible. Numeri

Load-bearing premise

The load-bearing premise is that the Loewner transfer operator P_n of Eq. (4) is a genuine projection operator, meaning P_n^2 = P_n on a specified space of dynamical variables, so that Q_n = 1 - P_n and Dyson's identity apply to x_n and v_n; the paper does not prove this idempotence or specify the inner product and subspace.

Editorial extensions

If this is right

  • The memory kernel and colored noise in a one-dimensional GLE can be computed explicitly from conformal map data, such as the driving function and slit maps, rather than treated as purely phenomenological inputs.
  • Decomposing the dynamics via P_n yields the fluctuation-dissipation relation K_n = η_s^M = ⟨F_n F_n⟩, tying the Loewner driving force directly to the noise autocorrelation.
  • Both the Kubo-type and the direct Loewner response functions satisfy R ~ t^-1 energy dissipation, a scaling the numerical cell-migration simulation reproduces.
  • Loewner entropy provides a microcanonical-type ensemble description of the colored-noise process, offering a statistical interpretation of nonlinear response.
  • The approach gives a practical numerical route, using zipper-type slit-map algorithms, to compute nonlinear response functions for experimentally measured memory kernels.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One testable extension the author leaves implicit is applying the same decomposition to memory kernels beyond b + exp(-t/c), such as power-law or oscillatory kernels, to see whether the t^-1 scaling is universal or specific to the exponential-kernel family.
  • If the projection issue is repaired, the correspondence between Loewner driving functions and noise suggests a data-inversion scheme: estimate U_s from experimental trajectories and read off the memory kernel from the driving function's statistics.
  • The direct Loewner response formula may generalize to multi-slit or multiple-curve Loewner evolutions, which would extend the method to higher-dimensional or multi-particle systems; the paper does not develop this direction.
  • The microcanonical interpretation via S_Loew might connect to information-geometric notions of entropy production in nonequilibrium systems, but that connection remains speculative.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes a modified Mori-Zwanzig derivation of a nonlinear generalized Langevin equation in which the conventional projection operator is replaced by a transfer operator P_n constructed from the chordal Loewner equation. The author claims that this Loewner-theoretic replacement yields a decomposition into deterministic forcing, memory, and colored-noise terms, and that two different response functions (a Kubo-type and a direct Loewner-type) obey a common t^{-1} energy-dissipation scaling, confirmed by numerical simulation with a cell-migration memory kernel. The paper also introduces a 'Loewner entropy' S_Loew and claims it allows a canonical-ensemble description of the colored-noise environment.

Significance. If the central derivation were valid, the paper would offer a genuinely new conformal-geometric route to memory effects in one-dimensional stochastic systems, and the explicit comparison between two response-function formalisms would be of interest to the statistical-mechanics and biophysics communities. The numerical setup, including the use of the zipper algorithm and a concrete cell-migration memory kernel, is a positive feature. However, the paper's main theoretical claims are not supported: the operator P_n is never shown to be a projection, the Dyson identity is derived with a sign error, and the claimed t^{-1} scaling is a direct consequence of the response-function definitions rather than a dynamical law. The strengths are therefore confined to the exploratory numerical example; the formal framework that would give that example meaning is not established.

major comments (4)
  1. [§2, Eq. (4) and Eqs. (10)–(14)] The entire Mori-Zwanzig structure depends on P_n being a projection operator, i.e., P_n^2 = P_n on a specified subspace of dynamical variables. Eq. (4) defines a Loewner transfer operator acting on functions of a complex variable by summing over preimages, but the paper never proves idempotency, never defines the inner product or relevant subspace, and does not connect P_n to observables x_n and v_n. Consequently Q_n = 1 − P_n in Eq. (10) is not shown to be the complementary projection required for the Dyson identity, and the decomposition in Eqs. (12)–(14) — and hence the nonlinear GLE in Eqs. (15)–(16) and the FDR in Eq. (17) — does not follow.
  2. [Appendix A, Eq. (A.4)] The Dyson identity is derived with the wrong sign. Starting from Eq. (A.1)–(A.2), integrating gives A(t) = 1 − ∫_0^t exp(−t' L) P_n L exp(t' Q_n L) dt'. Multiplying by exp(t L) yields exp(t Q_n L) = exp(t L) − ∫_0^t exp[(t−t')L] P_n L exp(t' Q_n L) dt'. Eq. (A.4) has a plus sign before the integral. This sign error propagates into the main decomposition Eq. (11) and into the memory term of the GLE. Even if P_n were a valid projection, the Dyson identity as written is incorrect.
  3. [§3, Eqs. (22), (23), (26), and (31)] The claimed universal t^{-1} scaling is not a prediction of the dynamics; it is built into the response-function definitions. Eq. (22) defines d(x,t) = 2t'/(v_n(t')^2 + t'^2), which is asymptotically 2/t for large t. Both response functions R(t,t+h) in Eq. (21) and R(s,s+h) in Eq. (29) are multiplied by this same factor d(x,t), so Eqs. (23) and (31) follow from the prefactor alone, independent of the memory kernel or the GLE. The numerical confirmation in Figs. 1–2 therefore verifies the defining prefactor, not a dynamical dissipation law.
  4. [§2, Eq. (17) and §3, Eq. (21)] The fluctuation-dissipation relation Eq. (17), K_n(t',x) = η_s^M(t') = ⟨F_n(t,x)F_n(t',x)⟩, is asserted rather than derived. The paper does not construct an ensemble over the Loewner driving force, and the Loewner entropy in Eq. (24) is introduced only later. Moreover, in Eq. (21) the manipulation from the correlation term ⟨F_n(t)F_n(t+h)⟩ to ⟨η_s^M(t+h)⟩ is not justified; the last term in the final expression contains a first-order expectation ⟨η_s^M(t+h)⟩, which is not equal to the second-order correlation function under any ensemble specified in the paper.
minor comments (4)
  1. [§2, Eqs. (15)–(16)] The two GLE forms are inconsistent: Eq. (15) has the memory integral ∫_0^t η_s^M(t') dt' without a factor v_n, while Eq. (16) has ∫_0^t η_s^M(t') v_n(t) dt'. The discretization in Eq. (34) uses v_n(j−1), whereas Eq. (16) uses v_n(t). The notation should be harmonized and the time argument in v_n inside the memory term clarified.
  2. [§4, Eq. (32)] The memory kernel K_n(t',x) = b + exp(−t/c) is dimensionally inconsistent unless b carries the same units as exp(−t/c), which is dimensionless. In Eq. (16) K multiplies v_n and is integrated over t', so b should have units of inverse time. Please specify the units and ensure dimensional consistency in Eq. (34).
  3. [Throughout] There are numerous typos and unclear expressions: 'demotes a suitable constant' should be 'denotes'; Eq. (22) uses t' without clear definition; the phrase 'In Sec. 4 and Sec. 5' in the introduction mismatches the actual section numbering; reference 8 has a malformed volume/pagination, and reference 18 has an invalid DOI. These presentation issues are secondary to the technical concerns above.
  4. [§4, paragraph after Eq. (36)] The numerical section states that 100 realizations were used but reports no error bars, convergence checks, or sensitivity analysis for b, c, k. Given that the main numerical claim is the t^{-1} scaling, the absence of statistical uncertainty makes it difficult to assess the strength of the agreement.

Circularity Check

3 steps flagged · score 8.0 of 10

The claimed t^{-1} dissipation law is built into the definition of the response functions via the d(x,t) prefactor; the Loewner-entropy ensemble is a relabeling of p, and the 'direct' FDR is imported from the author's own prior papers.

  1. self definitional [Sec. 3, Eqs. (21)-(23) and (29)-(31)]
    "d(x,t) = 2t'/(v_n(t')^2 + t'^2). ... R(t,t+h) ~ C(h) t^{-1} ... R(s,s+h)=α'd(x,t)<v_n(s) ∂/∂s v_n(s+h)> ... R(t,t+h) ~ C'(h) t^{-1}."

    Both response functions are defined with the explicit factor d(x,t)=2t/(v_n^2+t^2), which is asymptotically 2/t. Therefore the 'scaling law' R~t^{-1} follows from the definition of R before any dynamics of the GLE is used. The numerical test in Sec. 4 evaluates these same formulas, so Figs. 1-2 confirm the inserted prefactor rather than an independent physical prediction.

  2. renaming known result [Sec. 3, Eqs. (24)-(25); Sec. 5 discussion]
    "S_Loew = -ln p(η_s(t)). ... R(t,t+h)=αd(x,t) Σ [h_n h_n + h_n η_s^M + h_n η_s^M + F_n F_n] exp(-S_Loew). ... By defining the Loewner entropy, the diffusion process with memory effects are deduced to that having the microcanonical ensemble."

    Since S_Loew is defined as -ln p, the factor exp(-S_Loew) is exactly the original probability p. Replacing p by exp(-S_Loew) is a change of notation, not a derivation. The claimed deduction of a microcanonical/canonical ensemble from Loewner entropy is therefore a definitional restatement of the input probability distribution.

1 more flagged steps
  1. self citation load bearing [Sec. 3, Eqs. (26)-(29)]
    "Contrary to the above approach, the previous studies 30,31 have suggested a method of deriving the FDR directly from the one-dimensional dynamics using the conformal time transformation. According to this method ... R(s,s+h)=α'd(x,t)<v_n(s) ∂/∂s v_n(s+h)>_eq."

    The 'direct Loewner-theoretic' response function, which is one of the two FDR types the paper compares, is not derived in this manuscript; it is imported from Refs. 30 and 31, both authored by Shibasaki. This self-citation is load-bearing because the paper's central comparison and the shared t^{-1} scaling rest on that prior result, with no independent derivation shown here.

full rationale

The central Mori-Zwanzig decomposition (Eqs. 10-17) is not itself circular: if P_n were a genuine projection, Dyson's identity would yield the GLE structure. The serious problem there is a correctness gap (P_n^2=P_n and the relevant inner product are never established; Appendix A also contains a sign error in Eq. A.4), not a definitional reduction. The actual circularity is in the paper's two main 'predictions'. First, the t^{-1} energy-dissipation law is forced by the prefactor d(x,t)=2t/(v_n^2+t^2) that appears in both response functions by definition, so the numerical confirmation verifies the formula. Second, the Loewner entropy S_Loew=-ln p makes the canonical-ensemble weighting a relabeling of the probability p. Third, the direct FDR on which the comparison depends is taken from the author's own prior works (Refs. 30,31). These are definitional/self-citational reductions of the central claimed scaling and ensemble results, giving partial but substantial circularity. The score is 8 rather than 10 because the skeleton of the GLE derivation and the Kubo-type FDR formula retain independent formal content once the projection assumption is granted.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The paper's own accounting is dominated by assumptions. P_n is assumed to be a projection; X(s)=v_n is assumed; an equilibrium measure for eta_s is assumed; and the vanishing-drift limit is asserted to produce a Boltzmann weight. The simulation uses hand-picked constants b,c,k, and the claimed universal scaling is encoded in the chosen d(x,t). The only genuinely invented object, Loewner entropy, carries no independent evidence and reduces to the probability p.

free parameters (4)
  • alpha, alpha' = unspecified; 'suitable constant'
    Proportionality constants in the response functions (Eqs 21 and 29) set the magnitude of R but are never fixed or derived.
  • d(x,t) conformal-time factor = 2t/(v_n^2+t^2)
    Introduced via the Loewner time change (Appendix B, Eq 22); because d -> 2/t, the t^-1 scaling in Eqs 23 and 31 is forced by this choice, not by the dynamics.
  • b, c, k = b=0.5, c=2.0, k=1.0
    Simulation constants for the cell-migration kernel and potential (Sec 4, Eq 36); chosen by hand, with no fitting or sensitivity analysis.
  • C(h), C'(h) = unspecified functions of h
    Scaling prefactors in Eqs 23 and 31 are left undetermined.
assumptions (5)
  • ad hoc to paper P_n (Eq 4) is a projection operator
    Used to define Q_n=1-P_n and to apply the Dyson/Mori-Zwanzig identity in Eqs 10-17; idempotence and the relevant subspace are never established.
  • domain assumption X(s)=v_n and Y(s)=t(s)
    Appendix B identifies the Loewner curve coordinate X(s) with particle velocity and Y(s) with time; dimensional and mechanical justification is absent, yet this substitution underlies Eqs 22 and 26.
  • ad hoc to paper An equilibrium ensemble of eta_s exists with weight exp(-S_Loew)
    S_Loew=-ln p, so exp(-S_Loew)=p; using it as a canonical/microcanonical weight assumes the very distribution the paper claims to derive (Eqs 24-28).
  • ad hoc to paper Vanishing drift as s -> infinity yields peq=(1/Z)exp(-ln S)
    In Eq 26 the RHS tends to zero as s -> infinity, which implies a frozen variable, not a Boltzmann distribution; no derivation of Eq 27 is supplied.
  • domain assumption Kernel b+exp(-t/c) and noise exp(-t/2c) satisfy the FDR
    Taken from cell-migration experiments (Refs 16,18) and imposed in Eq 17; treated as input, not derived.
invented entities (1)
  • Loewner entropy S_Loew = -ln p(eta_s)
    purpose: Reformulate the response functions as sums over exp(-S_Loew) and estimate canonical/microcanonical ensembles in colored noise (Eqs 24-30).
    It is a relabeling of the probability density of the Loewner driving force; no falsifiable prediction is attached to calling -ln p an entropy.

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Cite this review

Pith. "Pith review of A Loewner-Theoretic Approach to the Nonlinear Generalized Langevin Equation: The Role of Entropy in Colored Noise Environment." pith.science (2026). https://pith.science/paper/BWBZNWBS

@misc{pith2026260713384,
  author       = {Pith},
  title        = {Pith review of: A Loewner-Theoretic Approach to the Nonlinear Generalized Langevin Equation: The Role of Entropy in Colored Noise Environment},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BWBZNWBS}},
  note         = {Machine review of arXiv:2607.13384}
}
read the original abstract

In this study, the formal derivation of a one-dimensional nonlinear generalized Langevin equation is demonstrated using a decomposition method based on the conformal transformation governed by the chordal Loewner equation. Here, we used a modified Mori-Zwanzig method whose operator is substituted by that is derived from the discrete Loewner evolution. By this approach, the different types of fluctuation-dissipation relation (FDR) were reformulated using mathematical terms affected by the conformal maps. Dealing with a memory kernel that models the cell migration experiment, the numerical simulation was performed to obtain the specific scaling law of energy dissipation that is common among the two obtained types of FDRs. In addition, the concept of Loewner entropy is used for the estimation of the canonical ensemble in the colored noise environment throughout the theoretical analyses.

Figures

Figures reproduced from arXiv: 2607.13384 by the authors.

Figure 1
Figure 1. Plots of nonlinear response functions of two different types of FDRs. (a) Time-dependent [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Exponential decay of nonlinear response functions. (a) Log-log plot of the time-dependent [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

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Reference graph

Works this paper leans on

33 extracted references · 8 canonical work pages

  1. [1]

    D. T. Limmer,Statistical Mechanics and Stochastic Thermodynamics, (Oxford Univer- sity Press, 2024)

    2024

  2. [2]

    M´ ezard and A

    M. M´ ezard and A. Montanari,Information, Physics, and Computation, (Oxford Uni- versity Press, 2009)

    2009

  3. [3]

    Boffetta and A

    G. Boffetta and A. Vulpiani,Probability in Physics, (Springer, 2026)

    2026

  4. [4]

    Risken,The Fokker-Planck equation: methods of solution and applications, (Berlin, Heidelberg: Springer Berlin Heidelberg, 1989)

    H. Risken,The Fokker-Planck equation: methods of solution and applications, (Berlin, Heidelberg: Springer Berlin Heidelberg, 1989)

    1989

  5. [5]

    Nelson,Dynamical Theories of Brownian Motion, (Princeton University Press, 1967)

    E. Nelson,Dynamical Theories of Brownian Motion, (Princeton University Press, 1967)

    1967

  6. [6]

    R. Kubo, J. Phys. Soc. Jpn.12(6), 570 (1957),https://doi.org/10.1143/JPSJ.12. 570

    work page doi:10.1143/jpsj.12 1957

  7. [7]

    U. M. B. Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani,Phys. Rep.461, 111 (2008). https://doi.org/10.1016/j.physrep.2008.02.002

    doi:10.1016/j.physrep.2008.02.002 2008

  8. [8]

    Sarracino and A

    A. Sarracino and A. Vulpiani,Chaos,29083132 (2019).https://doi.org/10.1063/ 1.5110262. July 16, 2026 0:29 BioPhys˙L 12Yusuke K. Shibasaki

    2019

Show all 33 references
  1. [9]

    Mori,Prog

    H. Mori,Prog. Theor. Phys.33423 (1965),https://doi.org/10.1143/PTP.33.423

    1965 doi

  2. [10]

    Zwanzig,J

    R. Zwanzig,J. Stat. Phys.9215 (1973),https://doi.org/10.1007/BF01008729

    1973 doi

  3. [11]

    M. T. Vrugt and R. Wittkowski,Euro. J. Phys41, 045101 (2020).https://doi.org/ 10.1088/1361-6404/ab8e28

    2020 doi

  4. [12]

    S. H. Chung and M. Roper,Biophys. Rev. Lett.14171 (2019),https://doi.org/10. 1142/S1793048019300019

    2019

  5. [13]

    Di Cairano,J

    L. Di Cairano,J. Phys. Comm.6015002 (2022),https://doi.org/10.1088/ 2399-6528/ac438d

    2022

  6. [14]

    G. Y. Hu and R. F. O’Connell,Phys. Rev. B,41, 5586 (1990),https://doi.org/10. 1103/PhysRevB.41.5586

    1990

  7. [15]

    Kaulakys and J

    B. Kaulakys and J. Ruseckas,Phys. Rev. E,70, 020101 (2004),https://doi.org/ 10.1103/PhysRevE.70.020101

    2004 doi

  8. [16]

    B. G. Mitterwallner, C. Schreiber, J. O. Daldrop, J. O., R¨ adler, and R. R. Netz,Phys. Rev. E,101032408 (2020),https://doi.org/10.1103/PhysRevE.101.032408

    2020 doi

  9. [17]

    R. R. Netz,Phys. Rev. E,110, 014123 (2024),https://doi.org/10.1103/PhysRevE. 110.014123

    2024 doi

  10. [18]

    Klimek, J.C

    A. Klimek, J.C. Heyn, D. Mondal, S. Schwartz, J. O. R¨ adler, P. Sharma, ... and R. R. Netz,PRX Life,3, 023015, (2025),https://doi.org/10.1103/77t1-2qvv

    2025 doi

  11. [19]

    Vroylandt,Europhy

    H. Vroylandt,Europhy. Lett.,140, 62003 (2022),https://doi.org/10.1209/ 0295-5075/acab7d

    2022

  12. [20]

    K. P. Hadeler, T. Hillen, and F. Lutscher,Math.l Mod. Meth. App. Sci.,14, 156 (2004),https://doi.org/10.1142/S0218202504003726

    2004 doi

  13. [21]

    S. A. McKinley and H. D. Nguyen,SIAM J. Math. An.505119 (2018),https: //doi.org/10.1137/17M115517X

    2018 doi

  14. [22]

    Cui and A

    B. Cui and A. ZacconePhys. Rev. E,97, 060102 (2018),https://doi.org/10.1103/ PhysRevE.97.060102

    2018

  15. [23]

    Huang,Biophys

    K. Huang,Biophys. Rev. Lett.,3, 1 (2008).https://doi.org/10.1142/ S1793048008000599

    2008

  16. [24]

    Taloni and F

    A. Taloni and F. Marchesoni, (2014).Biophys. Rev. Lett.9381 (2014).https://doi. org/10.1142/S1793048014400050

    2014 doi

  17. [25]

    G. F. Lawler,Conformally invariant processes in the plane. (American Mathematical Society, 2008)

    2008

  18. [26]

    Rohde, S and O

    S. Rohde, S and O. Schramm,Ann. Math.161, 883 (2005)

    2005

  19. [27]

    Shibasaki and M

    Y. Shibasaki and M. Saito,Chaos30, 113130 (2020)https://doi.org/10.1063/5. 0023261

    2020 doi

  20. [28]

    Shibasaki,Prog

    Y. Shibasaki,Prog. Theor. Exp. Phys.2025, 023A02 (2025),https://doi.org/10. 1093/ptep/ptaf017

    2025

  21. [29]

    Kennedy,J

    T. Kennedy,J. Stat. Phys.131, 803 (2008),https://doi.org/10.1007/ s10955-008-9535-x

    2008

  22. [30]

    Shibasaki,Europhys

    Y. Shibasaki,Europhys. Lett.139, 31001 (2022),https://doi.org/10.1209/ 0295-5075/ac7b44

    2022

  23. [31]

    Shibasaki,Biophys

    Y. Shibasaki,Biophys. Rev. Lett.19, 183 (2024),https://doi.org/10.1142/ S1793048024500048

    2024

  24. [32]

    Beck and F

    C. Beck and F. Sch¨ ogl,Thermodynamics of chaotic systems, (Cambridge University Press, 1995)

    1995

  25. [33]

    Ruelle,Thermodynamic formalism: the mathematical structure of equilibrium sta- tistical mechanics, (Cambridge University Press, 2004)

    D. Ruelle,Thermodynamic formalism: the mathematical structure of equilibrium sta- tistical mechanics, (Cambridge University Press, 2004)

    2004

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