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Non-equilibrium thermodynamics of gravitational objective-collapse models

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Frictionless Diosi-Penrose collapse is thermodynamically consistent only at infinite temperature; adding linear friction restores thermalization.

desk verdict A useful and honest application of the entropy-production formalism to DP collapse models — the small-β proof is clean, but the central claim is proven for the truncated equation, not the full dynamics. read the letter →

arxiv 2502.03173 v1 pith:CKYR5HMQ submitted 2025-02-05 quant-ph

Simone Artini , Gabriele Lo Monaco , Sandro Donadi , Mauro Paternostro This is my paper · ORCID
classification quant-ph
keywords Diosi-PenrosemodelobjectivecollapseentropyproductionrateFokker-PlanckequationdissipativethermalizationSecondLawofthermodynamicsWignerfunction
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

This paper asks whether gravitational objective-collapse models, which make macroscopic superpositions collapse spontaneously, can be squared with the Second Law of thermodynamics. Working with a single harmonic oscillator, the authors find that the original Diosi-Penrose model diffuses momentum without any restoring force, so its energy grows without bound; the entropy production rate turns negative whenever a finite-temperature equilibrium state is used as the reference, so the noise bath must be at infinite temperature. By contrast, the linear-friction dissipative extension reaches a genuine thermal equilibrium in the low-dissipation regime, with an entropy production rate that is non-negative at all times and vanishes only at equilibrium. The paper also derives an upper bound on the friction parameter from the uncertainty principle and examines the full non-Gaussian dynamics at short times, where apparent Second-Law violations for strong dissipation rest on an unreliable linearization. The result gives a concrete thermodynamic criterion: a collapse model is physically acceptable only if its noise can be assigned a finite temperature and the system thermalizes.

What carries the argument

The load-bearing object is the entropy production rate built from the Wigner function, the phase-space quasiprobability distribution, $\Pi(t) = -\partial_t \mathrm{KL}(W(t)\|W_{\mathrm{eq}})$, where $W_{\mathrm{eq}}$ is the asymptotic Wigner function of the dynamics. In the small-$\beta$ dissipative model this is computed from a Klein-Kramers Fokker-Planck equation $\partial_t W = \{W_H, W\}_\star + \nabla_p \cdot (f p W) + D \Delta_p W$, and the key identity is Eq. (32), which writes the entropy production rate as the integral of a squared probability current divided by the diffusion coefficient, $J^2/(D W)$. Because the integrand is manifestly non-negative, the Second Law is satisfied for every initial state with a positive Wigner function. In the frictionless model the drift term is absent, no stationary $W_{\mathrm{eq}}$ exists, and this identity has no finite-temperature counterpart, forcing the infinite-temperature interpretation.

What would settle it

Solve the exact Diosi-Penrose Lindblad equation, without the second-order momentum truncation, for a single harmonic oscillator with $m=2$, $R_0=3$ starting from the ground state, and compute the entropy production rate against a finite-temperature target state. If the exact $\Pi(t)$ never turns negative, the paper's infinite-temperature reading of the frictionless model is wrong; if it does, the claim survives. For the dissipative model, solving the full fourth-order Fokker-Planck equation at $\beta=100$ would distinguish a real early-time Second-Law violation from the artifact created by the short-time linearization.

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Extended reading notes

Core claim

The central discovery is a thermodynamic asymmetry inside the Diosi-Penrose family. In the frictionless version, the collapse Lindbladian translates into a purely diffusive Fokker-Planck equation for the Wigner function; because there is no drift toward a finite-temperature state, the position and momentum variances grow linearly in time and no asymptotic Gibbs state exists, so any finite-temperature comparison makes the entropy production rate negative. The dissipative extension changes this: for small friction strength $\beta$, the dynamics reduces to a Klein-Kramers equation with momentum drift and diffusion, whose stationary solution is an isotropic thermal Wigner function. For this equation the entropy production rate can be rewritten as $\Pi = \int dq\,dp\, J^2/(D W) \geq 0$, so it is non-negative along the whole evolution and vanishes only at equilibrium. The authors further tie $\beta$ to the equilibrium temperature, find the uncertainty bound $\beta \le \beta_c$, and show that when the full non-Gaussian dynamics is probed through a short-time linearization, negative entropy production appears at early times for large $\beta$; however, they demonstrate in an appendix that this linearization is not reliable, so the strong-dissipation regime remains unsettled.

Load-bearing premise

The whole calculation assumes the Wigner function is smooth on the tiny momentum scale $\hbar/R_0$, so truncating the expansion of $W(q, p - 2\hbar k)$ at second order in $k$ is accurate; the paper does not quantify when that truncation fails for the parameters it uses.

Editorial extensions

If this is right

  • The frictionless Diosi-Penrose model predicts unbounded, linear-in-time heating for any isolated massive object; an experiment that observes a steady-state temperature under such dynamics would contradict this thermodynamic picture.
  • In the linear-friction extension, the equilibrium temperature is set by the dissipation parameter $\beta$, and the uncertainty-principle bound $\beta \le \beta_c$ means the coldest allowed steady state is the oscillator ground state.
  • For small $\beta$, the dissipative model is provably Second-Law compliant: the entropy production rate is non-negative for all times and all initial states with positive Wigner function, vanishing exactly at equilibrium.
  • The early-time negative entropy production seen at large $\beta$ in the full non-Gaussian treatment is not an established violation, because the paper shows the short-time linearization can fail precisely there.
  • For single particles, the DP diffusion constant and the CSL diffusion constant map onto each other through the ratio of heating powers, so thermodynamic consistency tests of one model can be translated to the other.

Reading between the lines

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

  • The infinite-temperature interpretation suggests a sharp experimental signature: in the frictionless model the relative entropy to any finite-temperature state eventually decreases, so a trapped oscillator showing no such turnover would indicate the model's noise is effectively finite-temperature and the model needs revision.
  • The non-negativity proof rests only on drift-plus-diffusion structure with a Gaussian stationary state, so it likely generalizes to any collapse model of that form; models with fourth-order derivatives or non-Gaussian equilibria, like the full dissipative DP equation, will need a separate thermodynamic test.
  • The relation $\beta_{\mathrm{eq}} \simeq 2\beta/(8-\beta\hbar\omega)$ could be used as a calibration tool: measuring the equilibrium temperature of a trapped oscillator would directly bound the dissipation parameter and, through the diffusion coefficient, the collapse strength.
  • Appendix B's demonstration that a short-time linearization can be wrong even at very small times is a caution for the wider practice of inferring thermodynamic consistency from short-time expansions; the same reliability check could be applied to other reported Second-Law violations in collapse models.
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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 / 5 minor

Summary. The paper considers a single harmonic oscillator subject to the Diósi-Penrose (DP) collapse model and to a dissipative linear-friction extension. After translating the Lindblad dynamics into an approximate Fokker-Planck equation for the Wigner function, it computes the Rényi-2 relative-entropy production rate. For the frictionless DP model it finds unbounded diffusive heating and argues that the Second Law is respected only if the target state is an infinite-temperature Gaussian. For the dissipative model in the small-dissipation regime it derives an explicit Gaussian equilibrium, proves non-negativity of the entropy production rate via a squared-current identity, and demonstrates thermalization numerically. It then studies the full (non-Gaussian) dissipative dynamics in the Q representation by a short-time linearization, reporting possible negative entropy production at early times for large dissipation, while cautioning that the linearization is uncontrolled.

Significance. The question whether objective-collapse models are thermodynamically consistent is timely, and the paper provides an explicit analytic non-negativity proof for the truncated small-β Klein-Kramers equation, expressed as an integral of a squared current. It also honestly documents in Appendix B that its short-time linearization can fail and can even produce spurious negative entropy production; this is a useful methodological warning. However, the analytic proof is for the approximate Fokker-Planck generator rather than for the original dissipative DP master equation, and the paper's equilibrium-state formulas contain factor-of-two and parameter-dependence inconsistencies. If these are fixed, the small-β thermalization result will be a solid contribution; the frictionless and large-β conclusions are more interpretive and presently not fully supported.

major comments (4)
  1. [Sec. IV B, Eqs. (25)–(29), (32)] The stationary state used in the proof is not a stationary state of the approximate dynamics. For the Klein-Kramers equation (25) with an isotropic Gaussian target, the stationarity condition fixes the covariance to D/f (with D and f as in Eq. (26)), whereas Eq. (27) states σ²_eq = 2D/(mωf), a formula that depends on ω through mω and coincides with the stationarity value only when mω=2. In addition, Eq. (28) writes Weq = (πσ²_eq)^{-1} exp[-(q²+p²)/σ²_eq], which has variance σ²_eq/2, not σ²_eq; the correctly normalized stationary Wigner function with covariance σ²_eq is (2πσ²_eq)^{-1} exp[-(q²+p²)/(2σ²_eq)]. Consequently the current J(Weq) in Eq. (32) does not vanish for the stated Weq, and the identity Π = ∫ J²/(DW) is not established as written. The same inconsistency propagates to the critical value in Eq. (29), which should follow from det(V_eq)≥ℏ² applied to the corrected stationary solution and, on the basis of Eq. (25), should be independent of the oscillator frequency. I recommend re-deriving Eqs. (27)–(29) and the normalization of Eq. (28) carefully.
  2. [Secs. III A, IV A, and IV B] The central thermalization claim is proven only for the truncated Fokker-Planck generator. The passage from the Lindblad equation to Eqs. (10) and (19) truncates the Taylor expansion of W(q,p-2ℏk) at second order in k, and the reduction to Eq. (25) drops the O(β²) terms D2, D3, the position diffusion, and the fourth-order term R_{ijkl} of Eq. (19). No quantitative validity condition is supplied for these truncations. The numerical demonstration in Sec. IV B uses β=3, which is close to the stated critical value β_c≈3.8, and the k-expansion parameter ℏ/(R0 σ_p) is of order 1/3 for the plotted widths; neither parameter is small in this run. Because Appendix B shows that a similar perturbative reduction can generate spurious negative entropy production, the paper should either provide an a posteriori bound on the neglected terms for the parameters used or solve the untruncated dynamics (e.g., the Q-function evolution or the full Eq. (19)) to validate the thermalization claim.
  3. [Sec. V and Appendix B] The claimed early-time negative entropy production for large β is not supported. The calculation is based on Eq. (45), Φ_t ≈ e^{L1 t}+L2 t, and Appendix B explicitly shows that this linearization can fail at arbitrarily short times and can produce negative entropy production for exact dynamics that has non-negative production (Fig. 6, D=1, f=4/3). Moreover, Sec. V studies the dissipative CSL model in one dimension, not the dissipative DP model, so the conclusions of Fig. 5 do not directly apply to the model discussed in the abstract. The Conclusion's statement that retaining all β terms 'leads to negative entropy production' and 'supports the validity of the high temperature limit' therefore overstates what the evidence permits. I recommend deleting or substantially qualifying these claims, or replacing the uncontrolled linearization with a method whose error is controlled.
  4. [Sec. III B and Fig. 2] The frictionless conclusion is conditioned on the choice of Gaussian target states. Since Eq. (10) has no normalizable stationary state, the quantity (2) computed with an arbitrary finite-temperature thermal target is not the standard entropy production of a relaxation process; the sign of this auxiliary functional is not by itself a test of the Second Law. The statement that the DP model is consistent only with an infinite-temperature noise field should therefore be presented as an interpretation of the diffusive heating and of the absence of finite-temperature stationary states, or be backed by an independent argument (for instance, a direct proof that no finite-temperature stationary solution exists and that the heating rate is strictly positive for all finite-energy states).
minor comments (5)
  1. [Figs. 3 and 4 captions] The captions state β=0.3 while the main text states β=3; the plotted equilibrium variance 1.5 corresponds to β=3 in Eq. (27), so the captions appear to be a typo, but the inconsistency should be corrected.
  2. [Sec. V, near Eq. (45)] The text 'ℏneq = [0.1, 0.15, 0.2, 0.25]' should presumably read 'neq' or 'n_eq'; please clarify the notation for the test target occupations.
  3. [Sec. V, paragraph after Eq. (45)] The sentence 'of the which cannot be assessed the regime of validity' is ungrammatical and should be rephrased.
  4. [Appendix A, around Eq. (A1)] The momentum eigenfunction is written with an exponent involving the coordinate y; the dummy variable should be the position argument, and the normalization should be checked against the convention [q,p]=2iℏ used in the main text.
  5. [General notation] The symbol σ²_eq is used sometimes as the covariance and sometimes as the width parameter of the Gaussian in Eq. (28); please define it once and use it consistently throughout.

Circularity Check

1 steps flagged · score 3.0 of 10

Dissipative DP thermalization proof is self-contained; the frictionless 'infinite-temperature bath' conclusion is a restatement of the diffusion-only input.

  1. renaming known result [Section III B, text following Eq. (10) and Fig. 2]
    "Using the entropy production rate as the figure of merit computed as defined in eq. (2), it is possible to show that this dynamic must be the result of the system being in contact with an infinite temperature bath, never reaching equilibrium."

    For the frictionless DP model the input phase-space generator is pure momentum diffusion: Eq. (10) has a momentum Laplacian but no friction term, so no finite-temperature stationary state exists. The entropy production rate, Eq. (2), is defined relative to a Gaussian thermal target Weq chosen by the authors; scanning target temperatures, they find that any finite target eventually gives negative Π and only an infinite-temperature target avoids this. That outcome is forced by the diffusion-only structure, because no finite-variance Gaussian can be a stationary state of an unboundedly heating process.

full rationale

Aside from the frictionless 'infinite-temperature' interpretation, the derivation chain is self-contained. The dissipative small-β result starts from the linear-friction Lindblad operators (12), derives the Klein-Kramers equation (25) from the phase-space expansion, finds the stationary thermal state (27)-(28), and proves non-negativity of the entropy production rate via the explicit identity (32), which is re-derived in the text by integration by parts. No fitted parameter is renamed as a prediction, and no conclusion is imported solely from a self-citation: Refs. [23], [25], and [29] are confirmatory or methodological, and Eq. (32) is proved inline. The frictionless DP unbounded-heating statement also follows directly from the pure diffusion in Eq. (10) and does not depend on the cited Ref. [23]. The main caveats are approximation-validity issues: the small-k truncation leading to Eq. (10), the small-β truncation leading to Eq. (25), and the early-time linearization in Sec. V whose unreliability is demonstrated in Appendix B. These concern whether the results apply to the exact Lindblad dynamics, not whether the results are circular. The central thermalization claim for the dissipative model retains independent, parameter-free content, so the circularity score is low.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper uses no fitted parameters; all numerical values are chosen for readability. The main extra assumptions are the truncation of the Wigner expansion at second order in k, the choice of Gaussian thermal targets in the frictionless case, and the assumption that the full dissipative dynamics thermalizes. No new entities are introduced.

free parameters (5)
  • Oscillator mass m = 2 (dimensionless, with ℏ=G=1)
    Chosen for readability of plots in Secs. III and IV; not fitted to data.
  • DP cutoff length R0 = 3 (dimensionless) in Secs. III-IV; 0.9 in Sec. V
    Model parameter; set to arbitrary values for visualization. Physical range is 4e-10 m to 1e-4 m.
  • Dissipation strength β = 0.3 in Figs. 3-4; 3 for the σ²_eq=1.5 curve; 0.01, 1, 100 in Fig. 5
    Model parameter of the linear-friction extension from Ref. [22]; chosen to probe small-beta and large-beta regimes.
  • CSL rate γ (Sec. V) = 1
    Set to 1 for readability in the Q-function analysis.
  • Target asymptotic occupations n_eq = 0.1, 0.15, 0.2, 0.25 (Fig. 5)
    Chosen as test target temperatures for the Wehrl entropy production rate; not fixed by the dynamics.
assumptions (6)
  • domain assumption The collapse dynamics is governed by a Lindblad master equation with the Diosi-Penrose dissipator given in Eq. (4).
    The DP model is taken as the starting point; the paper does not derive it from first principles.
  • ad hoc to paper The Wigner function expansion in powers of k is truncated at second order (Sec. III A, Eq. (9); Sec. IV A, Eq. (19)).
    Valid only for states whose momentum distribution is broad compared to ℏ/R0; the paper assumes 'well-localized states' without a quantitative validity bound.
  • ad hoc to paper In the frictionless DP case, the relative-entropy target W_eq is chosen as a Gaussian thermal state at increasing temperature (Sec. III B, Fig. 2).
    No stationary state exists for the frictionless dynamics, so the choice of target is not dictated by the model; the infinite-temperature conclusion depends on this choice.
  • domain assumption The full dissipative DP dynamics is assumed to thermalize for long times, following Ref. [22] (Sec. V, paragraph after Eq. (45)).
    Used to choose target Wehrl-entropy states; the paper does not prove thermalization for the full non-Gaussian dynamics.
  • standard math Standard phase-space correspondences and the Wigner-Weyl transform are used without re-derivation (Secs. II, III A, Appendix A).
    Background mathematical toolkit for converting master equations to Fokker-Planck equations.
  • domain assumption The Rényi-2/Wigner entropy and Wehrl entropy are taken as the entropy measure; the entropy production rate in Eq. (2) is assumed to reflect the Second Law.
    The choice of entropy measure affects quantitative results; the paper uses it for all conclusions.

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Cite this review

Pith. "Pith review of Non-equilibrium thermodynamics of gravitational objective-collapse models." pith.science (2026). https://pith.science/paper/CKYR5HMQ

@misc{pith2026250203173,
  author       = {Pith},
  title        = {Pith review of: Non-equilibrium thermodynamics of gravitational objective-collapse models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CKYR5HMQ}},
  note         = {Machine review of arXiv:2502.03173}
}
read the original abstract

We investigate the entropy production in the Di\'osi-Penrose (DP) model, one of the most extensively studied gravity-related collapse mechanisms, and one of its dissipative extensions. To this end, we analyze the behavior of a single harmonic oscillator, subjected to such collapse mechanisms, focusing on its phase-space dynamics and the time evolution of the entropy production rate, a central quantity in non-equilibrium thermodynamics. Our findings reveal that the original DP model induces unbounded heating, producing dynamics consistent with the Second Law of thermodynamics only under the assumption of an infinite-temperature noise field. In contrast, its dissipative extension achieves physically consistent thermalization in the regime of low dissipation strength. We further our study to address the complete dynamics of the dissipative extension, thus including explicitly non-Gaussian features in the state of the system that lack from the low-dissipation regime, using a short-time approach.

Figures

Figures reproduced from arXiv: 2502.03173 by the authors.

Figure 1
Figure 1. Evolution of the covariance matrix entries over time. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. (a): Relative entropy between the state of the system at time t and target states of increasing temperatures. (b): Corresponding entropy production rate. The relative entropy becomes null at finite time, becoming then negative. This can be avoided only by assuming an infinite temperature target state. All parameters as in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Evolution of the covariance matrix entries over time and their asymptotic equilibrium values. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Relative entropy (a) and entropy production rate (b) varying the temperature of the initial data. In the linear fric￾tion model the system is driven towards thermal equilibrium as witnessed by the zero entropy production rate, which is otherwise always non-negative. Al…
Figure 5
Figure 5. Figure 5: Wehrl entropy production rate over time for the linearized dynamics at very early times keeping all the terms in [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Comparison between the entropy production rate computed with the exact solution (solid lines) and the one computed [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: L2-norm distance between the exact and the linearized state, computed as DL2 = qR dqdp(Wlin(q, p) − Wex(q, p))2 For D = 1, f = 4/3 the approximate Wigner diverges from the exact one much faster than for D = 3/4, f = 1. In particular, the stationary state is not station…

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Reviewed August 9, 2026 · model on record in the stance chip above.