Pith. sign in

REVIEW 3 major objections 4 minor 51 references

Living on the edge of instability

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For the same unstable potential, 'never diverging' trajectories are exponential-tailed while 'alive at a fixed time' trajectories are power-law-tailed.

desk verdict Generalizes the authors' earlier Q-process results for the cubic potential to all monomial unstable potentials; the light/heavy-tailed dichotomy is the key new result, but the spectral-gap proof for the full class is sketched rather than rigorous and the similarity transformation has a sign typo. read the letter →

arxiv 1908.04062 v1 pith:R3RAAU33 submitted 2019-08-12 cond-mat.stat-mech math.PR

Artem Ryabov , Viktor Holubec , Ekaterina Berestneva This is my paper · ORCID
classification cond-mat.stat-mechmath.PR MSC 60J6060J7082C31 PACS 05.40.-a02.50.Ey
keywords unstablepotentialsQ-processquasi-stationarydistributionBrowniandynamicsh-transformheavytailsFokker-Planckspectrumeffective
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 studies overdamped Brownian motion in potentials that fall toward minus infinity so steeply that typical trajectories escape to infinity in finite time. It asks what happens if one keeps only the non-escaping trajectories, and it claims that the answer depends sharply on how the conditioning is done. Conditioning on divergence only in the infinitely distant future (the Q-process) yields a localized, light-tailed position law, while conditioning on mere survival up to a fixed late time (the quasi-stationary distribution) yields a heavy-tailed law whose power-law exponent is set by the leading unstable term $|x|^n$, $n>2$. Both distributions reduce to the ordinary thermal equilibrium distribution for stable potentials, so the paper reads them as two rival generalizations of equilibrium statistics to unstable dynamics, with opposite tail behavior.

What carries the argument

The argument is carried by transforming the non-Hermitian Fokker-Planck operator into a Hermitian Hamiltonian $\hat H = \mathrm{e}^{\beta V/2}\hat L^\dagger \mathrm{e}^{-\beta V/2} = -D\partial_x^2 + U(x)$ with $U(x)=[V'(x)/\gamma]^2/(4D)-V''(x)/(2\gamma)$. Since $U$ is confining whenever $V$ is highly unstable, the spectrum is discrete, and the ground state wave function $\Psi_0$ encodes the two relevant eigenfunctions through $p_0=Z^{-1/2}\mathrm{e}^{-\beta V/2}\Psi_0$ and $s_0=\sqrt{Z}\,\mathrm{e}^{\beta V/2}\Psi_0$. Substituting these relations into the two normalization conditions yields the identity that turns normalization into tail information, and differentiating the left-eigenfunction equation converts the same identity into the asymptotic formulas for both tails. The h-transform then generates an effective statistical force $F_s=2k_BT\,s_0'/s_0$ that confines non-diverging trajectories; its mean-square value is $4\gamma k_BT\lambda_0$.

What would settle it

Compute the two lowest eigenvalues of the transformed Hamiltonian for a family of monomial potentials $V(x)=-\mu|x|^n/n$ with $n$ decreasing toward $2$. If for any $n>2$ with finite mean first-passage time the gap $\lambda_1-\lambda_0$ vanishes, or the spectrum becomes continuous, then the claimed exponential tail for $\pi_{\mathrm{st}}$ and power-law tail for $Q_{\mathrm{st}}$ cannot hold in that regime.

Watch

Extended reading notes

Core claim

For potentials $V(x)\sim -|x|^n$ with $n>2$, the paper establishes a dichotomy between the two natural conditional ensembles. The limit distribution of the Q-process is $\pi_{\mathrm{st}}(x)=s_0(x)p_0(x)=\Psi_0^2(x)$, where $p_0$ and $s_0$ are the right and left eigenfunctions of the slowest Fokker-Planck mode and $\Psi_0$ is the ground state of the transformed Hamiltonian; on the unstable side it decays as $\pi_{\mathrm{st}}(x)\sim Z[\lambda_0\gamma/V'(x)]^2 e^{\beta V(x)}$, an exponential tail. The quasi-stationary distribution is $Q_{\mathrm{st}}(x)=p_0(x)$ and decays on the same side as $Q_{\mathrm{st}}(x)\sim \lambda_0\gamma/V'(x)$, which for monomial potentials means the power law $|x|^{-(n-1)}$. The two formulas are forced by a single identity, $\int e^{-\beta V(x)}s_0(x)\,dx = \int e^{-\beta V(x)}s_0^2(x)\,dx$, that follows from the simultaneous normalization of the two eigenfunctions, so the exponential-versus-power-law split is not an accident of a particular potential.

Load-bearing premise

Everything rests on the assumption that, in the long-time limit, one slowest decay mode is cleanly separated from all faster modes; the paper's Section 3 argues for such a spectral gap via the transformed Hamiltonian, but a complete proof for every potential satisfying its defining finite-divergence condition is not given.

Editorial extensions

If this is right

  • For monomial unstable potentials, the quasi-stationary left tail is $|x|^{-(n-1)}$, so for a cubic trap no integer moments exist and moment-based averages of the surviving ensemble are undefined.
  • The Q-process distribution is localized with an exponential left tail and a logarithmic effective barrier that is impenetrable for $n>2$, so never-diverging trajectories effectively avoid the unstable region.
  • Both stationary laws and the generalized partition function depend on the friction constant $\gamma$ through $s_0$, meaning they are kinetic quantities, not purely thermodynamic equilibrium distributions.
  • The mean-square effective force obeys $\langle F_s^2\rangle=4\gamma k_BT\lambda_0$, which for monomial potentials scales as $\mu^{1/n}(k_BT)^{(n-1)/n}$; stronger instability and higher temperature both strengthen the confining force.
  • On the stable right side, both distributions approach the ordinary tail $\mathrm{e}^{-\beta V(x)}/Z$, so the two conditionings differ visibly only on the unstable side.

Reading between the lines

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

  • A direct experimental test would compare two conditioning protocols on colloid trajectories in a cubic optical trap—conditioning on survival to a fixed observation time versus conditioning on eventual divergence—and check that the log-log slope of the left tail is $-(n-1)$ only in the first protocol.
  • The paper's dependence on the order of limits suggests that the crossover from the localized to the heavy-tailed law is observable: an ensemble conditioned at a finite final time should pass from $\pi_{\mathrm{st}}$ to $Q_{\mathrm{st}}$ on a time scale set by $1/(\lambda_1-\lambda_0)$.
  • The equal-area identity $\int e^{-\beta V}s_0=\int e^{-\beta V}s_0^2$ may be a useful design principle in higher-dimensional unstable systems: any variational approximation to $s_0$ would immediately yield both tails without a full spectral solution.
  • As $n\to 2^+$, the effective force weakens, and the paper itself raises the question of when the Q-process ceases to be stationary; one could test this at potentials just above $n=2$ and look for the disappearance of a normalizable $\pi_{\mathrm{st}}$.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies one-dimensional overdamped Brownian motion in potentials that decrease to -∞ so rapidly that the particle reaches infinity in finite time ('highly unstable' potentials). Using the spectral decomposition of the Fokker-Planck operator, it analyzes two conditional ensembles: the Q-process (trajectories conditioned to never diverge) and the quasi-stationary distribution (trajectories conditioned to survive up to the observation time). It derives explicit tail asymptotics: the Q-process limit distribution is light-tailed with exponential decay e^{βV} on the unstable side, while the quasi-stationary distribution is heavy-tailed with a power law ~ |x|^{-(n-1)} for monomial potentials V ~ -|x|^n with n > 2. The paper also derives an exact identity for the generalized partition function, an exact formula for the mean squared effective force, and a scaling law for that force in terms of μ, kBT, and n. The central claim is that the two natural conditional ensembles have fundamentally different statistical properties for all highly unstable systems.

Significance. If correct, the dichotomy established here is conceptually important and practically useful: it distinguishes two natural conditional ensembles in unstable stochastic dynamics and gives explicit, parameter-free asymptotic predictions. The derivations are elegant and largely self-contained: the identity (24) follows from eigenfunction normalization, the tail asymptotics follow from the adjoint eigenvalue equation, and the mean-squared-force result (50) is an exact integration-by-parts identity. The paper provides falsifiable predictions, such as the power-law exponent n-1 and the scaling (54), which are checkable in the colloidal experiments mentioned in the introduction. The connection to Q-process theory and Doob's h-transform is well explained. The monomial results are clean and convincing; the main weakness is the generality of the spectral-gap claim.

major comments (3)
  1. [Sec. 3, Eqs. (18a)-(18c)] The similarity transformation is mis-signed. Acting on a smooth test function yields e^{βV/2} L† e^{-βV/2} = -(-D∂² + U), not (-D∂² + U) as written; the same sign error is present in Eq. (18b). The correct statement is H = -e^{βV/2} L† e^{-βV/2} = -e^{-βV/2} L e^{βV/2} = -D∂² + U. As written, Eq. (18a) combined with Eq. (20) (H Ψ_n = λ_n Ψ_n) would imply Fokker-Planck eigenvalues +λ_n, contradicting Eq. (6). This is a local sign error, but it appears in the proof of the discrete spectrum and should be corrected.
  2. [Sec. 3, Eq. (19) and the sentence 'U(x) is always confining for all highly unstable potentials'] The assertion that U(x) is confining for all highly unstable potentials is not proved from the defining finite-mean-first-passage-time condition in Sec. 2. Finite MFPT does not by itself rule out oscillatory potentials for which U(x) has infinitely many deep negative wells and the associated Schrödinger operator has a non-discrete spectrum. Consequently, the discrete-spectrum claim (5), which is load-bearing for the propagator asymptotics (16)-(17), the Q-process propagator (31), and the tail results (48) and (62), is not established for the full class as defined. The monomial potentials (for which U ~ |x|^{2n-2}) are fine, but the 'for all' claims in the abstract and Sec. 7 need either a rigorous proof of discrete spectrum under the stated definition or a more precisely restricted class of potentials.
  3. [Sec. 6, Eq. (62)] The tail expression Q_st(x) ~ λ0 γ / V'(x) is written without an absolute value. For a potential that decreases to -∞ as x → -∞, V'(x) is negative on that side, which would make the right-hand side negative. The subsequent monomial formula in Eq. (63) correctly uses |x|^{n-1}. Please replace V'(x) by |V'(x)| (or explicitly state that V'(x) is taken in the direction where it is positive).
minor comments (4)
  1. [Sec. 3, Eq. (18b)] This equation has the same sign issue as Eq. (18a); both should be corrected as described in the major comment.
  2. [Sec. 6, after Eq. (59)] The cross-reference to '(eq:Qsts0)' is a broken LaTeX label; it should refer to Eq. (59).
  3. [Sec. 4 and Sec. 6] 'Chapmann-Kolmogorov' should be 'Chapman-Kolmogorov', and 'absorbed Marov chains' should be 'absorbed Markov chains'.
  4. [Figure 1 caption] The caption labels both the left and right axes of panel (d) with '(d)'; it should read '(d) Left axis: ...' and '(d) Right axis: ...'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; central tail predictions follow from spectral decomposition with no fitted constants, though the Sec. 3 spectral-gap argument is incomplete (a correctness issue, not circularity).

full rationale

The paper's derivation chain starts from the Fokker-Planck spectral problem: propagator expansion (9), long-time asymptotics (16)-(17), Q-process propagator (31), and definitions of pi_st and Q_st. The two limit distributions are computed directly from the leading eigenfunction pair (p0, s0) and are not assumed as inputs. The tail asymptotics (48) and (62)-(63) are obtained by integrating the eigenvalue equation for s0 and using the normalization identity (24); the power-law exponent n-1 follows from the assumed monomial form V(x) = mu x^n/n without any fitted parameters. Self-citations ([2,5,24,46]) supply context, earlier cubic-potential results, and references, but do not carry the general derivation; the load-bearing spectral and normalization arguments are carried out in the text. The reviewer-flagged sign inconsistency around Eqs. (18a)-(18b) and the unproved assertion that U(x) is confining for all finite-MFPT potentials are mathematical-rigor and correctness concerns, not circularity: even if the proof of the spectral gap is incomplete, the target results are not being assumed as premises. No equation in the paper reduces to its own input by construction, and no fitted parameter is renamed as a prediction. Hence a low score is appropriate.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters: the generalized partition function Z and the decay rate λ0 are derived quantities from the spectral problem, not fitted. The effective potential and force are derived from the eigenfunction s0. The central results rest on standard spectral theory of Fokker-Planck operators plus the domain assumption defining highly unstable potentials. No new particles, forces, or conserved quantities are postulated; the effective force Fs is a derived statistical quantity.

assumptions (4)
  • domain assumption The Fokker-Planck operator for a highly unstable potential has a discrete, well-separated spectrum with positive gap.
    Invoked in Sec. 2-3 and used for the long-time asymptotics (16)-(17) and for the existence of the Q-process and QSD limits. Argued via the similarity transform to a Schrödinger operator with a confining potential U(x), but the argument is a sketch rather than a rigorous proof for the entire class.
  • domain assumption The potential V(x) decreases to -∞ at least as -|x|^n with n>2, so that the particle reaches infinity in finite time.
    This defines 'highly unstable' and is used in the deterministic blow-up solution (3) and in the asymptotic tails.
  • standard math The eigenfunction expansion (9) converges and the boundary terms in integrations by parts vanish.
    Used in deriving the identity (24), the tail asymptotics (46)-(48), and the mean-squared force (50)-(51).
  • standard math The ground-state wave function Ψ0 is positive and belongs to L^2, so that s0 and p0 are well-defined via (21)-(22).
    Needed for the probabilistic interpretation of the Q-process and for the identity π_st = Ψ0^2.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Living on the edge of instability." pith.science (2026). https://pith.science/paper/R3RAAU33

@misc{pith2026190804062,
  author       = {Pith},
  title        = {Pith review of: Living on the edge of instability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R3RAAU33}},
  note         = {Machine review of arXiv:1908.04062}
}
read the original abstract

Statistical description of stochastic dynamics in highly unstable potentials is strongly affected by properties of divergent trajectories, that quickly leave meta-stable regions of the potential landscape and never return. Using ideas from theory of Q-processes and quasi-stationary distributions, we analyze position statistics of non-diverging trajectories. We discuss two limit distributions which can be considered as (formal) generalizations of the Gibbs canonical distribution to highly unstable systems. Even though the associated effective potentials differ only slightly, properties of the two distributions are fundamentally different for all highly unstable system. The distribution for trajectories conditioned to diverge in an infinitely distant future is localized and light-tailed. The other distribution, describing trajectories surviving in the meta-stable region at the instant of conditioning, is heavy-tailed. The exponent of the corresponding power-law tail is determined by the leading divergent term of the unstable potential. We discuss different equivalent forms of the two distributions and derive properties of the effective statistical force arising in the ensemble of non-diverging trajectories after the Doob h-transform. The obtained explicit results generically apply to non-linear dynamical models with meta-stable states and fast kinetic transitions.

Figures

Figures reproduced from arXiv: 1908.04062 by the authors.

Figure 1
Figure 1. (a) Sketch of the Brownian particle in the highly unstable cubic optical potential discussed in Refs. [1,2]. (b) Characteristic feature of highly unstable dynamics are rapidly diverging trajectories that never return to finite x. (c) Ensemble of trajectories that do not diverge at least up to time t = 15. Their dynamics exhibits two transient (I and III) and a stationary regime (II). (d) Right axis: The cubic potent… view at source ↗
Figure 2
Figure 2. Left: The ground state wave function (dashed line) in the potential U(x), compared with the limit distribution of the Q-process (38) (dot-dashed line) and the quasi-stationary distribution (59) (solid line). The function are related through Eqs. (21) and (22). Right: the two functions occurring under integrals in the identity (24) enclose the same areas. Up to the normalization constant Z, the heavy￾tailed function … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

51 extracted references · 41 canonical work pages

  1. [1]

    ˇSiler M, J´ akl P, Brzobohat O, Ryabov A, Filip R and Zem´ anek P 2017 Sci. Rep. 7 1697 URL https://doi.org/10.1038/s41598-017-01848-4

    work page doi:10.1038/s41598-017-01848-4 2017

  2. [2]

    ˇSiler M, Ornigotti L, Brzobohat´ y O, J´ akl P, Ryabov A, Holubec V, Zem´ anek P and Filip R 2018 Phys. Rev. Lett. 121 230601 URL https://doi.org/10.1103/PhysRevLett.121.230601

    work page doi:10.1103/physrevlett.121.230601 2018

  3. [3]

    Filip R and Zem´ anek P 2016J. Opt. 18 065401 URL https://doi.org/10.1088/2040-8978/18/ 6/065401

    work page doi:10.1088/2040-8978/18/

  4. [4]

    Zem´ anek P,ˇSiler M, Brzobohat´ y O, J´ akl P and Filip R 2016 J. Opt. 18 065402 URL https: //doi.org/10.1088/2040-8978/18/6/065402

    work page doi:10.1088/2040-8978/18/6/065402 2016

  5. [5]

    Ornigotti L, Ryabov A, Holubec V and Filip R 2018 Phys. Rev. E 97 032127 URL http: //doi.org/10.1103/PhysRevE.97.032127

    work page doi:10.1103/physreve.97.032127 2018

  6. [6]

    Henri Poincar´ e16 2005 URL https://doi.org/10.1007/ s00023-014-0375-8

    Chetrite R and Touchette H 2015 Ann. Henri Poincar´ e16 2005 URL https://doi.org/10.1007/ s00023-014-0375-8

    work page 2015

  7. [7]

    Collet P, Mart´ ınez S and San Mart´ ın J 2013 Quasi-Stationary Distributions: Markov Chains, Diffusions and Dynamical Systems (Springer-Verlag Berlin Heidelberg) ISBN 978-3-642-33130-5 URL https://doi.org/10.1007/978-3-642-33131-2

    doi:10.1007/978-3-642-33131-2 2013

  8. [8]

    Doob J L 2001 Classical Potential Theory and Its Probabilistic Counterpart (Springer-Verlag Berlin Heidelberg) ISBN 978-3-540-41206-9 URL https://doi.org/10.1007/978-3-642-56573-1

    doi:10.1007/978-3-642-56573-1 2001

Show all 51 references
  1. [9]

    Redner S 2001 A Guide to First-Passage Processes (Cambridge University Press) ISBN 9780511606014 URL https://doi.org/10.1017/CBO9780511606014

    2001 doi

  2. [10]

    H¨ anggi P, Talkner P and Borkovec M 1990Rev. Mod. Phys. 62 251–341 URL https://doi.org/ 10.1103/RevModPhys.62.251

    doi

  3. [11]

    Arecchi F T, Politi A and Ulivi L 1982 Il Nuovo Cimento B (1971-1996) 71 119–154 URL http://dx.doi.org/10.1007/BF02721698

    1982 doi

  4. [12]

    Young M R and Singh S 1985 Phys. Rev. A 31(2) 888–891 URL https://doi.org/10.1103/ PhysRevA.31.888

    1985

  5. [13]

    Colet P, San Miguel M, Casademunt J and Sancho J M 1989 Phys. Rev. A 39 149–156 URL https://doi.org/10.1103/PhysRevA.39.149

    1989 doi

  6. [14]

    Hirsch J E, Huberman B A and Scalapino D J 1982 Phys. Rev. A 25 519–532 URL https: //doi.org/10.1103/PhysRevA.25.519

    1982 doi

  7. [15]

    Sigeti D and Horsthemke W 1989 J. Stat. Phys. 54 1217–1222 URL http://dx.doi.org/10. 1007/BF01044713

    1989

  8. [16]

    Ram´ ırez-Piscina L and Sancho J M 1991 Phys. Rev. A 43 663–668 URL https://doi.org/10. 1103/PhysRevA.43.663

    1991

  9. [17]

    C´ aceres M O, Budde C E and Sibona G J 1995 J. Phys. A: Math. Gen. 28 3877 URL https://doi.org/10.1088/0305-4470/28/14/009

    1995 doi

  10. [18]

    Mantegna R N and Spagnolo B 1996 Phys. Rev. Lett. 76 563–566 URL https://doi.org/10. 1103/PhysRevLett.76.563

    1996

  11. [19]

    Agudov N V 1998 Phys. Rev. E 57 2618–2625 URL https://doi.org/10.1103/PhysRevE.57. 2618 Living on the edge of instability 18

    1998 doi

  12. [20]

    15 1761–1788 URL https://doi.org/ 10.1162/08997660360675035

    Lindner B, Longtin A and Bulsara A 2003 Neural Comput. 15 1761–1788 URL https://doi.org/ 10.1162/08997660360675035

    2003 doi

  13. [21]

    15 2281–2306 URL https://doi.org/10.1162/ 089976603322362365

    Brunel N and Latham P E 2003 Neural Comput. 15 2281–2306 URL https://doi.org/10.1162/ 089976603322362365

    2003

  14. [22]

    Fiasconaro A, Spagnolo B and Boccaletti S 2005 Phys. Rev. E 72 061110 URL https://doi. org/10.1103/PhysRevE.72.061110

    2005 doi

  15. [23]

    C´ aceres M O 2008 J. Stat. Phys. 132 487–500 URL http://dx.doi.org/10.1007/ s10955-008-9554-7

    2008

  16. [24]

    Ryabov A, Zem´ anek P and Filip R 2016Phys. Rev. E 94 042108 URL http://doi.org/10.1103/ PhysRevE.94.042108

  17. [25]

    Agudov N V and Malakhov A N 1999 Phys. Rev. E 60 6333–6342 URL https://doi.org/10. 1103/PhysRevE.60.6333

    1999

  18. [26]

    Dubkov A A, Agudov N V and Spagnolo B 2004 Phys. Rev. E 69 061103

    2004

  19. [27]

    Gardiner C 2009 Stochastic Methods: A Handbook for the Natural and Social Sciences 4th ed (Springer, Berlin, Heidelberg) ISBN 978-3-540-70712-7 URL http://www.springer.com/ 978-3-540-70712-7

    2009

  20. [28]

    Haken H 2004 Synergetics: Introduction and Advanced Topics (Springer, Berlin, Heidelberg) ISBN 978-3-642-07405-9 URL https://doi.org/10.1007/978-3-662-10184-1

    2004 doi

  21. [29]

    Risken H 1996 The Fokker-Planck Equation: Methods of Solution and Applications 2nd ed (Springer, Berlin, Heidelberg) ISBN 978-3-540-61530-9 URL https://doi.org/10.1007/ 978-3-642-61544-3

    1996

  22. [30]

    Van Kampen N G 2007 Stochastic Processes in Physics and Chemistry 3rd ed (North Holland) ISBN 978-0-444-52965-7 URL https://doi.org/10.1016/B978-0-444-52965-7.X5000-4

    2007 doi

  23. [31]

    Berezin F A and Shubin M 1991 The Schr¨ odinger Equation(Springer Netherlands) ISBN 978-0- 7923-1218-5 URL https://doi.org/10.1007/978-94-011-3154-4

    1991 doi

  24. [32]

    Lambert A 2007 Electron. J. Probab. 12 420 URL https://doi.org/10.1214/EJP.v12-402

    2007 doi

  25. [33]

    Surveys 9 340 URL https://doi.org/10.1214/ 11-PS191

    M´ el´ eard S and Villemonais D 2012 Probab. Surveys 9 340 URL https://doi.org/10.1214/ 11-PS191

    2012

  26. [34]

    Jack R L and Sollich P 2010 Prog. Theor. Phys. Suppl. 184 304 URL https://doi.org/10.1143/ PTPS.184.304

    2010

  27. [35]

    Chetrite R and Touchette H 2013 Phys. Rev. Lett. 111 120601 URL https://doi.org/10.1103/ PhysRevLett.111.120601

    2013

  28. [36]

    Nyawo P T and Touchette H 2016 Phys. Rev. E 94 032101 URL https://doi.org/10.1103/ PhysRevE.94.032101

    2016

  29. [37]

    Nyawo P T and Touchette H 2018 Phys. Rev. E 98 052103 URL https://doi.org/10.1103/ PhysRevE.98.052103

    2018

  30. [38]

    Tiz´ on-Escamilla N, Lecomte V and Bertin E 2019 J. Stat. Mech. 2019 013201 URL https: //doi.org/10.1088/1742-5468/aaeda3

    2019 doi

  31. [39]

    Lazarescu A, Cossetto T, Falasco G and Esposito M 2019 arXiv e-prints arXiv:1902.08416 URL https://arxiv.org/abs/1902.08416

    2019 arXiv

  32. [40]

    Basu U, Maes C and Netoˇ cn´ y K 2015Phys. Rev. Lett. 114 250601 URL https://doi.org/10. 1103/PhysRevLett.114.250601

  33. [41]

    Basu U, Krger M, Lazarescu A and Maes C 2015 Phys. Chem. Chem. Phys. 17 6653 URL http://dx.doi.org/10.1039/C4CP04977B

    2015 doi

  34. [42]

    Maes C 2016 Math. Mech. Complex Syst. 4 275 URL http://dx.doi.org/10.2140/memocs.2016. 4.275

    2016 doi

  35. [43]

    Maes C 2018 Non-Dissipative Effects in Nonequilibrium Systems (Springer International Publishing) ISBN 978-3-319-67779-8 URL https://doi.org/10.1007/978-3-319-67780-4

    2018 doi

  36. [44]

    Rold´ an´E and Vivo P 2019 arXiv e-prints arXiv:1903.08271 URL https://arxiv.org/abs/1903. 08271

    2019 arXiv

  37. [45]

    Bray A J 2000 Phys. Rev. E 62 103 URL https://doi.org/10.1103/PhysRevE.62.103 Living on the edge of instability 19

    2000 doi

  38. [46]

    Ryabov A, Berestneva E and Holubec V 2015 J. Chem. Phys. 143 114117 URL https://doi. org/10.1063/1.4931474

    2015 doi

  39. [47]

    Yaglom A M 1947 Dokl. Acad. Nauk SSSR (in Russian) 56 795

    1947

  40. [48]

    Mandl P 1961 Czechoslovak Math. J. 11 558 URL http://eudml.org/doc/12097

    1961

  41. [49]

    Darroch J N and Seneta E 1965 J. Appl. Probab. 2 88 URL https://doi.org/10.2307/3211876

    1965 doi

  42. [50]

    Seneta E and Vere-Jones D 1966 J. Appl. Probab. 3 403 URL https://doi.org/10.2307/3212128

    1966 doi

  43. [51]

    Aghion E, Kessler D A and Barkai E 2019 Phys. Rev. Lett. 122 010601 URL https://doi.org/ 10.1103/PhysRevLett.122.010601

    2019 doi

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.