{"id":"03e8f4be-795a-4397-bda6-0f6fd5ac55a4","arxiv_id":"2607.26678","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"A no-dissipation Caldeira-Leggett model is claimed to give bath-particle mean-square displacement ~t^5 and velocity ~t^3, but the derivation is internally inconsistent.","lead":"This paper claims that when a particle coupled to a bath of oscillators feels white or time-correlated random forces, the bath particles spread extremely fast—displacement growing as the fifth power of time. The derivation, however, contains internal contradictions that make the central result unreliable.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Bath-particle Fokker–Planck equation (35) carries an unexplained c_i D t factor in the diffusion term; this factor is the direct source of the t^5/t^3 superspreading and is not derived from the coupled Langevin equations.","rationale":"The paper's primary novel claim is the t^5/t^3 superspreading of bath particles under correlated Gaussian noise. For this to be true, the bath Fokker–Planck equation (35) must follow from the coupled Langevin equations (31)–(32). The system-particle equation (34) has the structure expected for an OU-driven acceleration process, but Eq. (35) introduces an extra c_i D t multiplying the diffusion operator. The paper states only 'manipulating the method of Ref. [20]' and gives no derivation. The internal inconsistency—two branches yielding t^4/t^3 and t^5/t^3, then silently selecting t^5—strengthens the concern that the t factor is an ad hoc insertion rather than a derived coefficient. A direct exact-moment computation from the linear Langevin equations would settle the matter; it requires no closure approximation and tests the actual equations of motion. This is the most load-bearing concern because it directly undermines the central claim. I find no need to adjust the reader's REJECT verdict; the concern confirms it.","tokens_in":19869,"tokens_out":8956,"duration_ms":96614,"concrete_test":"Derive the exact closed second-moment equations from the Langevin equations (31)–(32) without substituting Eq. (35). Use the physically correct bath coupling m_i x_i_ddot = -m_i ω_i² x_i + c_i x (not the sign in Eq. (3)), write x_i(t) and v_i(t) as linear functionals of η(s), insert the OU correlation (33), and evaluate the leading powers of ⟨x_i²(t)⟩ and ⟨v_i²(t)⟩ for t≪τ. If those powers are not t^5 and t^3, the unexplained c_i D t factor in Eq. (35) is the source of the claimed superspreading and the central claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—bath ⟨x_i²⟩~t^5 and ⟨v_i²⟩~t^3 for t≪τ—is obtained from Eq. (35), whose diffusion operator is multiplied by c_i D t. This t factor is never derived from the coupled Langevin equations (31)–(32). The bath feels no direct noise; diffusion must be induced by the system coordinate, and the standard Gaussian-noise reduction (e.g., the Heinrichs method cited as Ref. [20]) yields time-dependent coefficients built from correlation integrals, not an a priori t multiplier. Moreover, the calculation is internally inconsistent: in §3.2.1, the ζ_i branch gives Eq. (72) with ⟨x_i²⟩~t^4 and ⟨v_i²⟩~t^3, while the ν_i branch gives Eq. (81) with ⟨x_i²⟩~t^5 and ⟨v_i²⟩~t^3; the final result (83) silently selects t^5 for displacement. The c_i D t factor is exactly what converts the lower-power branch into t^5. If Eq. (35) is not a consequence of (31)–(32), the superspreading claim has no foundation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20177,"tokens_out":5444,"duration_ms":48132,"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":[{"comment":"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.","section":"Eq. (3) and Hamiltonian (1)"},{"comment":"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.","section":"§2, Eqs. (7), (16), (18)"},{"comment":"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.","section":"§3.1.1, Eqs. (38)–(48)"},{"comment":"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","section":"§3.2.1, Eqs. (35), (72)–(84)"},{"comment":"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.","section":"§§2–3, separation constants"}],"minor_comments":[{"comment":"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).","section":"Throughout"},{"comment":"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).","section":"Eq. (74)"},{"comment":"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.","section":"Tables 1–2"},{"comment":"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.","section":"References [16] and [18]"},{"comment":"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.","section":"Title and abstract"}],"recommendation":"reject","confidential_remarks":"The central claims of the paper—especially the t⁵/t³ superspreading for the bath particle—are not supported by a sound derivation. The explicit t multiplier in Eq. (35) appears to be introduced ad hoc, and the two branches in §3.2.1 give conflicting scalings. I see no straightforward fix within the manuscript's current scope, because the issue is at the level of the basic Fokker–Planck equations, not a presentation problem. I also note that the method is taken from Ref. [20] and several of the system-particle results overlap with the authors' own Refs. [23]–[25]; the novelty relative to those prior works is not clearly delineated. I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing to know: the headline bath-particle scaling (⟨x_i²⟩ ~ t⁵ for correlated noise) is produced by a term in the Fokker-Planck equation that the paper asserts but never derives. Until that term is shown to follow from the coupled equations of motion, the central result has no foundation.\n\nWhat the paper does well: it asks a genuine question—what happens to transport in the Caldeira-Leggett model when dissipation is dropped and the noise has finite correlation time. The system-particle results (⟨x²⟩ ~ t³ for white noise, ⟨x²⟩ ~ t⁴ and ⟨v²⟩ ~ t³ for correlated noise) are consistent with a random-acceleration picture and follow the same technique as Heinrichs (Ref. [20]). The paper also computes moments, non-Gaussian parameters, and entropies, which could be useful for other studies.\n\nThe soft spots are load-bearing. First, Eq. (3) has a sign error in the bath coupling: expanding the Hamiltonian gives the force on the bath as +c_i x, not −c_i x. Second, Eq. (35) for the bath contains a c_i D t factor multiplying the diffusion operator. The bath feels no direct noise; its diffusion must be induced by the system coordinate. No derivation of that t-factor is given, and if it is removed, the t⁵ scaling disappears. Third, the short-time bath calculation has two branches: the ζ_i branch produces ⟨x_i²⟩ ~ t⁴ (Eqs. 72–74), while the ν_i branch produces ⟨x_i²⟩ ~ t⁵ (Eqs. 81–83). The paper silently selects the t⁵ result without explaining why one branch is physical. Fourth, the separation constants A, B, E, F are introduced and then vanish via unexplained cancellations.\n\nOne of the reader's specific objections is not quite right: ⟨x²⟩ ~ t⁴ and ⟨v²⟩ ~ t³ are not automatically incompatible, since a velocity process with variance growing as t³ can generate displacement variance growing as t⁴. But that is a minor point compared with the unjustified t-factor and the branch inconsistency.\n\nWho this is for: a specialist in anomalous diffusion or open quantum systems might find the system-particle results worth a look, but the bath-particle claim should not be quoted. The underlying question is interesting enough that I would not desk-reject on topic alone, but the derivation needs to be redone: fix the sign, derive (35) from (31)–(32), and resolve the two branches. My recommendation: if the editor wants to take a chance on a major revision, send it to a referee with that expectation; otherwise a desk reject with resubmission encouraged is defensible. As it stands, the central claim is unsupported.","headline":"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.","tokens_in":20706,"tokens_out":5478,"would_cite":false,"duration_ms":43052,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["05.40.-a","03.65.Yz"],"model":"deepseek-v4-flash","headline":"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.","keywords":["Caldeira–Leggett model","correlated Gaussian noise","quantum Brownian motion","superdiffusion","super-ballistic transport","Fokker–Planck equation","anomalous diffusion","open quantum systems"],"falsifier":"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.","tokens_in":19692,"feed_emoji":"📈","tokens_out":7264,"duration_ms":63399,"temperature":0.7,"pith_summary":"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.","feed_headline":"Correlated noise yields t^5 bath-particle spreading","feed_subtitle":"Without dissipation, a Caldeira–Leggett bath shows super-ballistic diffusion; short correlation times make it faster.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Bath particle superdiffuses: t^5 growth without friction","No friction? Bath still spreads as t^5 from correlated noise","Correlated noise alone gives bath t^5 ballistic spread","Dissipation-free bath shows anomalous t^5 diffusion","t^5 bath spreading: frictionless Caldeira-Leggett model"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Bath particle superdiffuses: t^5 growth without friction","No friction? Bath still spreads as t^5 from correlated noise","Correlated noise alone gives bath t^5 ballistic spread","Dissipation-free bath shows anomalous t^5 diffusion","t^5 bath spreading: frictionless Caldeira-Leggett model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000253,"raw_usage":{"total_tokens":1422,"prompt_tokens":783,"completion_tokens":639,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":561}},"tokens_in":527,"tokens_out":639,"duration_ms":7296,"temperature":1.0,"reasoning_tokens":561,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T10:55:06.900217+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}