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REVIEW 5 major objections 5 minor 24 references

This paper claims that removing dissipation from the Caldeira–Leggett model makes a correlated Gaussian noise drive bath particles into super-ballistic spreading, with mean squared displacement ~ t^5 and velocity ~ t^3.

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2026-08-01 10:55 UTC pith:3A4S63OD

load-bearing objection The bath-particle t^5/t^3 claim rests on an unjustified t-factor in the Fokker-Planck equation; the derivation is internally inconsistent and needs a rewrite. the 5 major comments →

arxiv 2607.26678 v1 pith:3A4S63OD submitted 2026-07-29 cond-mat.stat-mech

Quantum Brownian transport in a correlated Gaussian force

Yun Jeong Kang , Seungsik Min , Sung Kyu Seo , Kyungsik Kim This is my paper
classification cond-mat.stat-mech PACS 05.40.-a03.65.Yz
keywords Caldeira–Leggett modelcorrelated Gaussian noisequantum Brownian motionsuperdiffusionsuper-ballistic transportFokker–Planck equationanomalous diffusionopen quantum systems
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies a system particle coupled to a bath of harmonic oscillators, with the dissipative term of the standard Caldeira–Leggett model set to zero. It claims that when the thermal noise is a correlated Gaussian process, the bath particle's mean squared displacement grows as t^5 and its mean squared velocity as t^3 at times much shorter than the noise correlation time. For white noise, the corresponding growths are t^4 and t^2. The authors argue that removing dissipation changes the transport from normal diffusion to persistent super-ballistic spreading, and that the mechanism is a mixed position–velocity diffusion term in the master equation. A sympathetic reader would care because this suggests that environmental correlations alone, without energy loss, can drive anomalous quantum transport.

Core claim

The central claim is that in the dissipation-free Caldeira–Leggett model the bath particle, which feels the thermal noise only through the system coordinate, shows anomalous superdiffusion: ⟨x_i^2⟩ ≃ (D c_i / 4τ) t^5 and ⟨v_i^2⟩ ≃ (D c_i / τ)(1 + 2c_i^2/ω_i^2) t^3 in the limit t ≪ τ, where τ is the noise correlation time. The authors derive these from Fokker–Planck equations for the joint position–velocity probability density, solved by double Fourier transforms and iterative steady-state factorizations. The key structural ingredient is the mixed derivative term D b(t) ∂²/∂x∂v (and its bath counterpart with an extra factor c_i t), which couples position and velocity coordinates in the diffus

What carries the argument

The central object is the Fokker–Planck equation for the joint probability density p(x,v,t) and p(x_i,v_i,t), transformed by double Fourier transform into p(ζ,ν,t). The calculation relies on factoring the solution into a steady-state part and a time-dependent arbitrary function, then repeatedly applying this factorization (successive transformations) to eliminate secular terms. The mixed derivative ∂²/∂x∂v is the mechanism that couples position and velocity diffusion; in the bath equation it is multiplied by an explicit t, and expanding the correlation functions a(t)=1−exp(−t/τ) and b(t)=(t+τ)exp(−t/τ)−τ to second order in t/τ yields the t^5 and t^3 powers.

Load-bearing premise

The result rests on an extra factor t that appears by hand in the bath diffusion coefficient of the Fokker–Planck equation (Eqs. 8 and 35); the paper never derives this factor from the coupled equations of motion, and without it the claimed t^5 and t^3 growths collapse to t^4 and t^2.

What would settle it

Re-derive the bath-particle Fokker–Planck equation directly from the coupled Langevin equations (31)–(32) without inserting an ad hoc t factor in the diffusion coefficient, then compute ⟨x_i^2(t)⟩ for t ≪ τ; if the diffusion coefficient is not proportional to t, the t^5/t^3 scaling will be replaced by t^4/t^2. A second check is to integrate the original equations (with the coupling sign corrected) numerically for an exponentially correlated noise and measure the short-time growth of the bath-particle variance.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Correlated Gaussian noise in a dissipation-free Caldeira–Leggett model yields bath-particle mean squared displacement ~ t^5 and velocity ~ t^3 at short times, far beyond ballistic (t^2) motion.
  • White noise in the same model gives ⟨x²⟩ ~ t³ for the system particle and ⟨x_i²⟩ ~ t⁴ for the bath particle, both distinct from normal diffusion.
  • At zero correlation time the bath particle's velocity variance still grows as t² while the system particle's velocity variance grows as t, showing that bath and system follow different transport laws.
  • In the long-time limit t ≫ τ, the bath particle crosses over to t⁴/t² scaling that is independent of τ, indicating that the anomalous short-time growth is controlled by the noise correlation time.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A testable extension: derive the bath Fokker–Planck equation directly from the microscopic Langevin equations to see whether the extra t factor in the diffusion coefficient exists; if it does not, the t⁵/t³ powers are an artifact of that assumed factor.
  • The same mixed-derivative mechanism might apply to other non-Markovian noise kernels; repeating the calculation with power-law or multi-exponential correlations could predict a family of anomalous exponents beyond t⁵/t³.
  • The bath equation of motion (Eq. 3) has an apparent sign error in the coupling term (it should be +c_i x, not −c_i x, to match the Hamiltonian); correcting the sign and re-solving would test whether the qualitative conclusion is robust or a consequence of that inconsistency.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper studies a classical Caldeira–Leggett system–bath model (a particle bilinearly coupled to harmonic oscillators) driven by white and correlated Gaussian noise. It claims to derive Fokker–Planck equations, their Fourier-space solutions, and mean-square displacements/velocities for both the system particle and a bath particle. The headline results are anomalous scaling: for white noise, ⟨x²⟩∼t³ for the system and ⟨x_i²⟩∼t⁴ for a bath particle; for correlated Gaussian noise with correlation time τ, the bath particle supposedly exhibits superspreading ⟨x_i²⟩∼t⁵ and ⟨v_i²⟩∼t³ in the short-time limit t≪τ, with different scaling in t≫τ and τ=0. The paper also tabulates non-Gaussian parameters, entropies, and moments. The central physical claim is that removing dissipation from the Caldeira–Leggett framework yields persistent super-ballistic, non-diffusive transport.

Significance. If the reported scaling were correct, the result would be significant: it would challenge the standard expectation that bath degrees of freedom respond diffusively to a noisy system coordinate, and it would offer a concrete analytic example of super-ballistic spreading in a system–bath model. The paper also addresses correlated (non-Markovian) Gaussian noise, which is technically relevant. However, the significance is contingent on the derivation being valid, and I find several load-bearing inconsistencies. The paper does not provide machine-checked proofs, reproducible code, or experimental predictions; its value would rest entirely on the analytic derivation, which is not presently reliable.

major comments (5)
  1. [Eq. (3) and Hamiltonian (1)] Equation (3) has a sign error. Expanding the interaction term in Hamiltonian (1), the equation of motion for x_i is m_i x_i¨ = -∂H/∂x_i = -m_i ω_i² x_i + c_i x. Equation (3) instead has -c_i x. This is not a typo isolated to one line: the subsequent Fokker–Planck equations for the bath are derived from Eq. (3), so the starting dynamics are internally inconsistent. A wrong coupling sign changes the force on the bath oscillators and propagates into every later bath-particle result.
  2. [§2, Eqs. (7), (16), (18)] The white-noise system FPE (7) has a constant diffusion coefficient D in velocity. For the standard FPE ∂_t p = -v ∂_x p + D ∂_v² p, the exact marginal velocity variance is ⟨v²⟩ = 2Dt, not ∼t³. Nevertheless, Eq. (16) leads to ⟨v²⟩∼t³ in Eq. (18). This contradicts the very FPE from which it is supposedly derived. Moreover, Eq. (18) states ⟨x²⟩∼t³ and ⟨v²⟩∼t³ for the same coordinate pair; since x(t) = ∫ v(s)ds, such a pair requires strong long-time correlations in v that are not present in Eq. (7). The derivation of Eq. (16) from the characteristic solution (15) appears to contain an unjustified expansion; the t³ factor in the ν² term is the source of the spurious velocity scaling.
  3. [§3.1.1, Eqs. (38)–(48)] The correlated-noise system result ⟨x²⟩∼t⁴ and ⟨v²⟩∼t³ (Eq. (48)) is obtained through a chain of 'successive transformations' (Eqs. (41)–(45)) in which the arbitrary function Θ is introduced and terms proportional to 1/τ³ are dropped. The separation constant E in Eq. (38) is never fixed, and the expansion in powers of t/τ is not controlled. Even accepting the intermediate steps, the final Fourier-space expression (46) is asserted after 'performing cancelations' without showing that the neglected terms are uniformly small. This is a central derivation, not a peripheral detail.
  4. [§3.2.1, Eqs. (35), (72)–(84)] The bath-particle superspreading claim rests on the diffusion operator in Eq. (35) being multiplied by an explicit factor c_i D t. This t-factor is never derived from the coupled Langevin equations (31)–(32). The bath oscillators are not directly forced by the noise η(t); their noise is induced through the system coordinate x, and the standard reduction method (e.g., Ref. [20]) produces time-dependent coefficients built from correlation integrals, not an a priori t multiplier. If Eq. (35) is not a consequence of (31)–(32), the t⁵/t³ scaling in Eq. (83) has no foundation. Furthermore, §3.2.1 is internally inconsistent: the ζ_i branch leads to Eq. (72) with ⟨x_i²⟩∼t⁴ (Eq. (74)), while the ν_i branch leads to Eq. (81) with ⟨x_i²⟩∼t⁵ (Eq. (83)). The final result (84) silently selects the t⁵ branch without any rule for resolving the discrepancy. This is precisely the load-bearing claim of the
  5. [§§2–3, separation constants] The separation constants A, B, E, and F are introduced at Eqs. (12), (19)–(20), (36)–(37), and (62)–(63) but are never determined. They appear in the formal steady-state solutions (23)–(24), (40), (66), and (75), yet they drop out of the final expressions only after unspecified 'cancellations.' Likewise, the arbitrary functions Φ and Θ are introduced without boundary or normalization conditions. In addition, expressions such as (23)–(24) contain denominators ν_i and ζ_i; these are not valid Fourier-space solutions without a prescription for the singularities. The paper therefore does not provide a closed, checkable derivation at the level required for the claimed closed-form results.
minor comments (5)
  1. [Throughout] The text contains numerous typographical errors: 'corelated' in Section 3 heading, 'Fokker-Plank' in several places, 'Eq. (30) and Eq. (31)' in Section 4 should refer to earlier equations, and 'exp[ - 2τ/(D t⁴) x²' is missing a closing bracket in Eq. (47).
  2. [Eq. (74)] Eq. (74) reads '⟨x_i²⟩ = (D t⁴)/(4τ) t⁴', which contains an extra t⁴. From Eq. (73) the intended result appears to be ⟨x_i²⟩ = D t⁴/(4τ).
  3. [Tables 1–2] Several table entries are internally inconsistent with the text: for example, Table 2 lists correlation coefficients with fractional powers such as t^{-7/2}, whereas the text's variances imply t^{-4} or t^{-3}. The notation μ_{2,2} is used for both the system and bath moments without distinction. These tables should be checked and reconciled with Eqs. (18), (30), (48), (55), etc.
  4. [References [16] and [18]] References [16] and [18] both cite arXiv:2302.13666v6 but with different author lists and titles; at least one is likely incorrect. Reference [19] also appears unrelated to the sentence in which it is cited.
  5. [Title and abstract] The paper is titled 'Quantum Brownian transport' but the derivation is entirely classical: Eq. (1) is a classical Hamiltonian and the Fokker–Planck equations are classical. The word 'quantum' appears only through the high-temperature Caldeira–Leggett correspondence. The abstract's claim that 'the mean squared velocity of a quantum particle' is computed is not supported by the content.

Circularity Check

1 steps flagged

Bath-particle t^5/t^3 superspreading is the moment solution of the ad hoc c_i D t diffusion term in Eq. (35), so the prediction reduces to an assumed input.

specific steps
  1. fitted input called prediction [Eq. (35) and Eq. (84), Sec. 3.2.1]
    "∂/∂t p(x_i,v_i,t) = [-v_i ∂/∂x_i + [k_i x_i - c_i(x_0+v_0)t + (t-1)c_i^3/ω_i^2]∂/∂v_i]p + c_i D t[-b(t)∂²/∂x∂v + a(t)∂²/∂v²]p ... Consequently, from Eqs. (74) and (83), the mean squared displacement and the mean squared velocity in the limit of t≪τ is obtained as <x_i²>=D c_i/(4τ)t^5, <v_i²>=D c_i/τ(1+2c_i²/ω_i²)t^3."

    The claimed t^5/t^3 scaling is not derived from the coupled Langevin equations (31)–(32); it is the direct quadrature of Eq. (35), whose diffusion term is asserted with the extra factor c_i D t. For t≪τ, a(t)=1−exp(−t/τ)≈t/τ, so the velocity-diffusion coefficient is c_i D t·a(t)≈c_i D t²/τ. Integrating this time-dependent coefficient gives ⟨v_i²⟩∼(D c_i/τ)t³, and integrating again gives ⟨x_i²⟩∼(D c_i/τ)t⁵. Thus the headline 'superspreading' is mathematically equivalent to the un-justified t factor (and first power c_i) placed in the Fokker–Planck operator. The paper never derives Eq. (35) from Eqs. (31)–(32); the result is the assumed diffusion coefficient renamed as a prediction.

full rationale

The central claim of the paper is the bath-particle superspreading ⟨x_i²⟩∼t⁵ and ⟨v_i²⟩∼t³ in the short-correlation-time regime. That claim is produced by solving Eq. (35), but Eq. (35) contains a c_i D t multiplier in its diffusion operator that is not obtained from the coupled Langevin equations in the text. In the t≪τ limit, a(t)≈t/τ, so the effective velocity diffusion is c_i D t²/τ; the announced t⁵/t³ powers are simply the second and first integrals of that time-dependent coefficient. Hence the prediction reduces by construction to an input assumption rather than being an independent consequence of the system-bath dynamics. The paper also has an internal branch inconsistency—Eq. (72) gives a t⁴ displacement for the ζ_i branch while Eq. (81) gives t⁵ for the ν_i branch, and Eq. (84) silently selects the t⁵ branch—but that is a correctness/derivation issue rather than a further circular step. I did not find load-bearing self-citation: Refs. [23]–[25] are cited in the conclusions, but the §3 derivation is presented as self-contained and no uniqueness theorem is imported from the authors' prior work. Because the central scaling claim is effectively contained in the assumed diffusion operator, the circularity score is 6 rather than 0–2.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

No new particles, forces, or dimensions are introduced. The unusual scalings are claimed to emerge from the mixed-derivative structure of the Fokker-Planck equation, a mathematical feature rather than an invented entity. The central calculation instead rests on unproven factorization and truncation steps plus undetermined separation constants.

free parameters (2)
  • Separation constants A, B, E, F = undetermined
    Introduced ad hoc in Eqs. (11)-(12), (19)-(20), (36)-(37), and (62)-(65) to factor the Fourier-transformed Fokker-Planck equations; never determined or given physical meaning, yet final expressions are claimed after their cancellation.
  • Coupling constants c_i, omega_i = not fitted (model inputs)
    Enter the prefactors of the claimed scaling laws; their values are not specified and no physical calibration is given, though they do not affect the time exponents.
axioms (5)
  • domain assumption Standard Caldeira-Leggett system-bath Hamiltonian (Eq. 1) with linear coupling and counterterm.
    The entire calculation starts from this Hamiltonian; the paper provides no derivation or experimental justification of the model.
  • domain assumption High-temperature/classical limit sets the noise correlation to white or Ornstein-Uhlenbeck form (Eqs. 2-3 and 33).
    The 'quantum' model is replaced by classical Langevin equations; the paper's quantum claims inherit this classical limit without discussion.
  • domain assumption Fokker-Planck equations (7), (8), (34), (35) are valid as written, via 'the method of Ref. [20]'.
    The paper does not derive these Fokker-Planck equations; it cites Ref. [20]. The bath equation contains an underexplained load-bearing t factor (Eq. 35).
  • ad hoc to paper Factorization ansatz and steady-state/successive-transformation steps (Eqs. 14, 25, 39-45, 62-71).
    The solution method assumes p = q p_st, separates variables, introduces separation constants A,B,E,F, and then neglects terms; these are not proven and are the source of the final exponents.
  • ad hoc to paper Imaginary terms and 1/tau^3 terms can be neglected.
    Stated without error bounds; these truncations directly produce the claimed t^4/t^5 and t^3 laws.

pith-pipeline@v1.3.0-daily-deepseek · 19508 in / 19559 out tokens · 150026 ms · 2026-08-01T10:55:06.900217+00:00 · methodology

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read the original abstract

We study a classical system-bath model in which a system particle is linearly coupled to a bath of harmonic oscillators. In a system subject to white and correlated Gaussian noises, we derive an expression for the joint probability density, and the mean squared values of the system-bath particles are calculated. For white noise, the mean squared values of the system-bath particles exhibit an anomalous time dependence, which is different from that of normal diffusion. In particular, for a correlated Gaussian noise, the mean squared displacement and mean squared velocity of the bath particle show superspreading growths of $t^5$ and $t^3$ in $t{\ll}{\tau}$, respectively. When ${\tau}=0$, the mean squared velocity of a quantum particle under random noise is proportional to $t$, while the mean squared velocity of a bath particle in the presence of correlated Gaussian noise increases in proportion to $t^2$. This anomalous transport phenomenon results from the mixed derivative structure of the master equation, which couples with the transport coordinates in the diffusion dynamics of the relative coordinates. This result shows that the removal of dissipation in the Caldeira-Leggett framework leads to fundamentally different transport mechanisms characterized by non-diffusive quantum diffusion.

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Reference graph

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