Microstates : Do the outliers worth

Olivier Sire

REVIEW 4 major objections 8 minor 10 references

Microstates : Do the outliers worth

T0 review · 4 major / 8 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that rare, Pareto-distributed fluctuations in a cooling gas can create microstates with locally decreasing entropy, and that adding gravity-like attraction makes these low-entropy configurations persist.

desk verdict The claimed association between Pareto outliers and low-entropy microstates is contradicted by the paper's own observables; the entropy decrease is built into the model and the two simulations are never connected. read the letter →

arxiv 2506.16080 v1 pith:PMOWSKEB submitted 2025-06-19 cond-mat.stat-mech physics.hist-ph

Olivier Sire This is my paper · ORCID

classification cond-mat.stat-mechphysics.hist-ph

keywords entropyrareeventsParetodistributionmicrostatessecondlawofthermodynamicsclusteringnearest-neighbordistancesnumericalsimulation

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 note argues that the outliers in a gas-like system are not negligible noise: in simulations of non-interacting particles, the nearest-neighbor distances develop a Pareto-distributed heavy tail, so isolated particles occur with unusually high frequency at low density. The paper then claims that such rare configurations are microstates that locally decrease entropy, even though the system as a whole continues to gain entropy in line with the second law. Introducing a gravity-like attraction, in which cells holding more particles preferentially attract more, stabilizes these low-entropy clusters and lets them persist over time. If this picture holds, rare fluctuations are the seeds of localized order, and the global rise in entropy does not forbid small regions from becoming more ordered.

What carries the argument

The argument is carried by two simulation devices plus an entropy formula. First, the nearest-neighbor distance distribution of randomly placed particles is analyzed with a Pareto-tail test to show that the tail beyond a threshold obeys a power law, so 'outsiders' are real. Second, a $32\times 32$ grid hosts $N$ particles whose migration probability into a cell is weighted by the square of that cell's current occupancy, a gravity-like preferential-attraction rule. Entropy is computed from the Boltzmann formula $S = N\ln N - \sum_i n_i\ln n_i$ via Stirling's approximation, so clustering into few cells lowers $S$. The Pareto tail supplies the rare fluctuations; the grid model supplies the dynamics that lets them persist.

What would settle it

Track the positions of the 1% most-isolated particles in the non-interacting simulation and compare them with the cells where clustering first begins under the gravity rule; if cluster seeds show no spatial correlation with Pareto-tail outliers, the claimed connection between rare events and local entropy decrease is unsupported.

Watch

Extended reading notes

Core claim

The central claim is that rare events, identified with the Pareto tail of nearest-neighbor distances in a random gas, correspond to microstates whose local entropy decreases while the bulk macrostate's entropy increases. The paper's simulations show that as particle density falls, the skewness of the distance distribution grows and the tail becomes heavy-tailed (Pareto), meaning isolated 'outsider' particles are statistically significant. In the clustering model, a simple gravity-like rule makes particles aggregate: in a rarefied system they rapidly collapse toward a single cell, driving local entropy to zero, whereas in a crowded system clustering is slower and the entropy drop is weaker. The introduction of the attraction is what stabilizes the low-entropy configurations, allowing them to persist beyond transient fluctuations.

Load-bearing premise

The load-bearing premise is that the rare particles identified by the Pareto tail of nearest-neighbor distances are the same entities as the low-entropy clusters produced by the gravity-like model, a mapping the paper never quantifies.

Editorial extensions

If this is right

If the claim is right, the second law's global increase in entropy does not prevent small, rare regions from spontaneously becoming more ordered.
The Pareto tail in nearest-neighbor distances means isolated particles are more common than Gaussian intuition suggests, so outliers should be counted as part of the system's typical behavior at low density.
Under gravity-like attraction, low-entropy clusters are stabilized and persist, so locally ordered structures may be long-lived rather than transient.
The crowding ratio $N/n$ governs how fast order emerges: rarefied regions collapse quickly to a single low-entropy cell, while crowded regions cluster more slowly and less extremely.

Reading between the lines

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

The paper leaves the link between the two simulations implicit; a direct tracking of the Pareto-tail particles into the clustering model would test whether the same entities that are statistically rare are the ones that nucleate low-entropy clusters.
The clustering rule is a preferential-attachment mechanism, so the cluster-size distribution at late times should follow a power law; measuring it across densities would turn the qualitative 'four worlds' observation into a quantitative prediction.
The zero-entropy collapse in the rarefied case (all particles in one cell) is likely an artifact of the deterministic, noiseless rule; adding thermal fluctuations would clarify whether low-entropy states are stabilized or merely metastable.
The analogy to conformational drift suggests a testable distinction: isolated microstates should evolve deterministically on internal forces, while crowded ones are buffeted by neighbors; comparing trajectory divergences from identical initial densities would operationalize this.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this note doesn't hold together. The abstract says rare events—the Pareto tail of large nearest-neighbor distances—are associated with microstates that locally decrease entropy. But the paper's own low-entropy microstates are dense clusters in the second simulation, the opposite of isolated outsiders. The association is not unsupported; it's contradicted by the observables the paper defines.

What's actually there? The first simulation does a clean enough job of showing that nearest-neighbor distances in random point patterns have a right tail that fits a Pareto law above a threshold. That's a standard result, but it's correctly demonstrated with the extremefit package, and the skewness curve as density decreases is a reasonable illustration. The second simulation is a simple preferential-attachment model (cell recruiting probability ∝ n_i²) and the entropy formula S = N ln N − Σ n_i ln n_i is correct for that model. Given that rule, entropy decreases and the system clumps; that's a mathematical consequence, not a discovery.

The soft spots are large. The clustering rule builds in the entropy decrease: there's no other outcome. The paper never measures any local entropy; it computes only a single global S for the whole grid. The two simulations are completely disconnected numerically—there's no attempt to track the particles in the first simulation into the second, or to show that the rare outliers are the ones that form clumps. The Pareto analysis is based on a single run with no error bars. The model has several free parameters and no sensitivity analysis.

The most generous reading is that this is a philosophical essay with toy numerics. The references to Jill North and the Janus model are fine as motivation, but the technical content doesn't support the conclusions. The paper even acknowledges its own qualitative language. I would desk-reject this without sending it to a referee. It might be acceptable as a blog post, but it doesn't meet the bar for a statistical-mechanics archive.

Recommendation: reject. Not worth a referee's time.

Referee Report

4 major / 8 minor

Summary. The paper presents two loosely connected numerical experiments. In the first, random non-interacting particle configurations on a 2D surface are analyzed through nearest-neighbor distances; the author reports positive skewness that increases at low density and, for one case (N=4096, n=8192), identifies a Pareto-distributed right tail of large nearest-neighbor distances, interpreting the corresponding isolated particles as rare "outsiders" or "microstates." In the second, a 32x32 grid hosts N particles whose occupancy evolves by a rule in which the probability that a cell gains particles is proportional to the square of its current occupancy. The paper computes a global entropy S = N ln N - sum_i n_i ln n_i over 8 time steps, showing that entropy decreases as particles cluster, and claims that rare events are associated with local entropy decreases that coexist with global entropy increases, with gravity-like attraction stabilizing low-entropy configurations. The manuscript concludes with speculative analogies to gravitational collapse, protein conformational drift, and the emergence of complex structures.

Significance. If the central claim were established, the paper would provide a toy demonstration that rare statistical fluctuations can produce local entropy reductions without violating the second law, and that an attractive interaction can stabilize such reductions. However, the manuscript does not establish that claim: the connection between the Pareto tail of nearest-neighbor distances and the low-entropy clustered states is not demonstrated and is, in fact, contradicted by the paper's own observables; the entropy model computes only global entropy, not local entropy; and the "gravity-like attraction" is a rule that forces clustering by construction. The topic is of potential interest, and the qualitative questions about rare events and local entropy fluctuations are legitimate, but the present treatment is too underdeveloped and internally inconsistent to support the abstract's conclusions. The paper does not ship code, data, or rigorous statistical validation, so the numerical results are not verifiable from the manuscript alone.

major comments (4)

[Abstract; §3, Figs. 3–5] The central association between Pareto-distributed rare events and entropy-decreasing microstates is contradicted by the paper's own definitions. In §3, the Pareto tail identified for N=4096 (threshold τ=0.0139) consists of large nearest-neighbor distances, and the text describes these as 'isolated' particles or 'lone tiny worlds.' In contrast, the entropy decrease in Figs. 4–5 arises precisely because particles cluster into a few cells, which corresponds to short nearest-neighbor distances, i.e. the left tail of the distance distribution, not the right tail. The manuscript never shows that the particles in the Pareto tail are the same entities that undergo local entropy decrease; the two categories point in opposite directions. This is a load-bearing gap in the abstract's claim.
[§3, entropy equations and Figs. 4–5] The paper claims that rare events are associated with 'microstates that locally decrease entropy' while 'global entropy may continue to increase,' but no local entropy is measured anywhere. The entropy formula S = N ln N - sum_i n_i ln n_i is applied to the whole 32x32 grid, and Figs. 4b and 5b show the total entropy of the entire system decreasing monotonically. There is no bulk region whose entropy increases, no subdivision into local and non-local parts, and no reservoir or environment whose entropy change could offset the decrease. Thus the assertion of coexistence of local decrease and global increase is not simulated; the model as described actually shows a global entropy decrease, which would require an external compensating entropy increase elsewhere to be consistent with the second law.
[§3, clustering rule and Fig. 4–5] The claim that 'the introduction of gravity-like attraction stabilizes these low-entropy configurations' is a restatement of the update rule rather than an independent finding. The transition probability is chosen to be proportional to the square of the occupancy of the destination cell, so populated cells preferentially attract more particles by construction; any reasonable entropy functional that rewards concentration will then decrease. The first simulation is non-interacting and the second simulation has no non-attractive control with the same entropy measure, so the effect of 'gravity' is not isolated. The statement that attraction stabilizes low-entropy states is therefore built into the model and does not provide independent evidence for the paper's thesis.
[§3, Pareto analysis] The Pareto-tail identification rests on a single simulation run for N=4096 and n=8192, with no error bars, no variation of the random seed, and no report of the number of independent repetitions. The adaptive threshold τ=0.0139 is quoted as a point value without uncertainty, and the log-log plot in Fig. 3b is presented without a formal goodness-of-fit comparison against exponential or other thin-tailed distributions. The claim that the tail 'is better fitted by a power law than by an exponential distribution' is therefore not supported by the statistics shown. Multiple runs with confidence intervals and a quantitative model comparison would be needed for the Pareto claim to bear the weight placed on it.

minor comments (8)

[Title; throughout] The title 'Do the outsiders worth ?' is ungrammatical; it should read 'Do the outsiders matter?' or 'Do the outsiders matter for entropy?'.
[§2 and §3] The symbol n is used with different meanings: n=8192 iterations in the first simulation, n=1024 cells in the second, and also as the generic particle count in the entropy derivation. Distinct symbols would remove ambiguity.
[§2 and §3] The text says '33x32 matrix' near the definition of the grid but otherwise refers to a 32x32 grid with 1024 cells; the dimensions should be made consistent.
[§2, first simulation] The phrase 'ranges logarithmically from 30 to104' is unclear; it should be written as 10^4 or 10,000, and the same notation should be used consistently throughout the density discussion.
[Figure 4 caption] The caption contains a typo: 'at t7 ant t8' should read 'at t7 and t8.'
[References] Reference [3] (Albert, Time and Chance) is never cited in the body of the text; please cite it where it is relevant or remove it from the reference list.
[Methods, entropy equations] The entropy expression omits the explicit Boltzmann constant k_B; although it is common to set k_B=1, this should be stated for clarity.
[General] No code or data availability statement is provided. Given the numerical nature of the paper, making the R scripts and seed values available would substantially aid reproducibility.

Circularity Check

1 steps flagged · score 6.0 of 10

Gravity-stabilization conclusion restates the model's clustering rule; the Pareto/entropy link is asserted, not derived.

self definitional [Section 3, entropy model (32x32 grid), Figs 4-5; abstract]
"the probability that a particular cell hires more particles at each step is weighted by the square of particles yet hosted at the previous one. It yields a more or less fast particle’s clustering that depends on the initial partition and on the N/n particles/cells ratio... The equations quantify how much entropy decreases as particles cluster, reflecting how order increases in the corresponding system."

The update rule is defined as rich-get-richer: cells with more particles are more likely to gain particles, i.e., clustering. The entropy formula S = N ln N - sum_i n_i ln n_i is a measure of concentration, minimized when all particles occupy one cell. Therefore the 'result' that gravity-like attraction stabilizes low-entropy configurations is not an emergent prediction; it is the update rule restated in entropy language. The paper does not simulate an alternative dynamics or measure local entropy; it runs a clustering rule and reports that the entropy diagnostic decreases. The abstract's additional claim connecting these states to the Pareto-tail 'rare events' is not derived from either simulation, and the two simulations are never quantitatively linked.

full rationale

The paper contains no parameter fitting, no self-citation chain, and no imported uniqueness theorem; the Pareto-tail identification is a standard extreme-value fit and the entropy formula is standard. The circularity is confined to the gravity/entropy part: the 'stabilization of low-entropy configurations' is the update rule itself (cells with more particles preferentially gain particles), and the entropy measure is defined to decrease under concentration. The simulation therefore cannot provide independent evidence for that claim. Additionally, the abstract's association between Pareto-tail rare events (large nearest-neighbor distances) and low-entropy states (particles clustered in few cells) is never operationalized; if anything the paper's own definitions put the two in opposite tails. That is a coherence gap rather than a reduction, so it does not by itself raise the circularity score, but it means the central narrative is not independently supported. Overall, one central claim reduces to its input by construction, hence the score of 6.

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

The paper relies on a data-derived threshold, a hand-chosen clustering exponent, and two ad hoc assumptions linking statistical tails to entropy dynamics. There are no genuinely invented physical entities.

free parameters (4)

Pareto tail threshold tau = 0.0139
Determined by the hill.adapt function on a single run with N=4096, n=8192; used to claim a heavy-tailed distribution.
Clustering exponent = 2
The rule that a cell's chance of gaining particles is proportional to the square of its current occupancy is chosen by hand to mimic gravity; no justification is given.
Grid size = 32x32
Simulation domain chosen without sensitivity analysis.
Number of time steps = 8
Entropy evolution is shown over only 8 steps; no convergence check.

assumptions (4)

standard math The entropy of the lattice is S = N ln N - sum n_i ln n_i (Stirling approximation)
Standard Boltzmann entropy with Stirling's approximation; used in Section 3 to compute entropy evolution.
domain assumption The system is isolated with a constant number of particles N
The model keeps N constant and evolves the occupancy without exchange with an external reservoir.
ad hoc to paper The clustering rule (transition probability proportional to n_i^2) represents gravity-like attraction
This rule is introduced in Section 2 and 3 without derivation from gravitational physics; the central conclusion depends on it.
ad hoc to paper Skewness and Pareto tail of nearest-neighbor distances indicate physically significant rare microstates
The paper interprets statistical tail behavior as evidence for 'outsiders' that affect entropy dynamics, but no mechanism links them.

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Cite this review

Pith. "Pith review of Microstates : Do the outliers worth." pith.science (2026). https://pith.science/paper/PMOWSKEB

@misc{pith2026250616080,
  author       = {Pith},
  title        = {Pith review of: Microstates : Do the outliers worth},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PMOWSKEB}},
  note         = {Machine review of arXiv:2506.16080}
}

read the original abstract

This note addresses the relevance of rare events in system dynamics, inspired by Jill North reflections on the origin of the arrow of time in thermodynamics. After identifying the existence of rare events, characterized by a Pareto distribution, within a simple gas particle simulation, we investigate their impact on entropy evolution. These rare events are associated with microstates that locally decrease entropy, in contrast to the overall entropy increase observed in the bulk of the system. We present numerical simulations of gas particles, both without and with a gravity-like attractive force, to explore the fate of these rare events. Our results show that, while rare events can transiently generate local decreases in entropy, global entropy may continue to increase in accordance with the second law of thermodynamics. The introduction of gravity-like attraction stabilizes these low-entropy configurations, allowing them to persist longer. This study highlights the interplay between rare statistical fluctuations and macroscopic thermodynamic behavior, providing new insights into the emergence and stability of order in complex systems.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

Stirling’s Approximation for a large number of particlesn : ln(n!) ≈n ln(n) −n

work page
[2]

Total Entropy of the System : the entropyS is proportional to the logarithm of the number of microstates Ω : S =kB ln(Ω) where Ω = N! ∏1024 i=1 ni! with : — N = ∑ 1024 i=1 ni : total number of particles in the matrix, — ni : number of particles in celli. Using Stirling’s approximation, the entropy becomes : S =N ln(N) −N − 1024∑ i=1 [ni ln(ni) −ni] Simpli...

work page
[3]

exception

in the René Thom meaning 5 (a) Particles distribution (b) Entropy evolution over time Figure 4 – N n = 0.5 Model (a) Particles distribution (b) Entropy evolution over time Figure 5 – N n = 4 Model Figures 4a and 5a displays how many particles are hosted in each cell over time through 8 time steps. Figures 4b and 5b display the corresponding entropy evolut...

work page
[4]

Jill North, ‘Time in Thermodynamics, in The Oxford Handbook of Philosophy of Time, edited by Craig Callender, Oxford University Press, 2011, pp. 312–350

work page 2011
[5]

A bimetric cosmological model based on Andrei Sakharov’s twin universe approach

Petit, J.-P.; Margnat, F.; Zejli, H. (novembre 2024). “A bimetric cosmological model based on Andrei Sakharov’s twin universe approach”,European Physical Journal C, 84 : 1226. DOI : 10.1140/epjc/s10052-024-13569-w

work page doi:10.1140/epjc/s10052-024-13569-w 2024
[6]

Time and Chance

Albert, David Z. (2000).“Time and Chance”. Cambridge, Mass. : Harvard University Press

work page 2000
[7]

(2018).extremefit : A Package for Extreme Quantiles

Durrieu, G., Grama, I., Jaunatre, K., Pham, Q.-K., Tricot, J.-M. (2018).extremefit : A Package for Extreme Quantiles. Journal of Statistical Software, Vol. 50, Issue 9. DOI : 10.32614/CRAN.package.extremefit

work page doi:10.32614/cran.package.extremefit 2018
[8]

Cancer Associated E17K Mutation Causes Rapid Confor- mational Drift in AKT1 Pleckstrin Homology (PH) Domain

Kumar, A., Purohit, R. (2013). “Cancer Associated E17K Mutation Causes Rapid Confor- mational Drift in AKT1 Pleckstrin Homology (PH) Domain”.PLoS ONE, 8(5), e64364. DOI : 10.1371/journal.pone.0064364

work page doi:10.1371/journal.pone.0064364 2013

Show all 10 references

[9]

Here we consider that a particle may feature a small world on its own, submitted to its own laws, regardless its nature
[10]

Genetically Modified Organisms 8

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Stirling’s Approximation for a large number of particlesn : ln(n!) ≈n ln(n) −n

work page

[2] [2]

Total Entropy of the System : the entropyS is proportional to the logarithm of the number of microstates Ω : S =kB ln(Ω) where Ω = N! ∏1024 i=1 ni! with : — N = ∑ 1024 i=1 ni : total number of particles in the matrix, — ni : number of particles in celli. Using Stirling’s approximation, the entropy becomes : S =N ln(N) −N − 1024∑ i=1 [ni ln(ni) −ni] Simpli...

work page

[3] [3]

exception

in the René Thom meaning 5 (a) Particles distribution (b) Entropy evolution over time Figure 4 – N n = 0.5 Model (a) Particles distribution (b) Entropy evolution over time Figure 5 – N n = 4 Model Figures 4a and 5a displays how many particles are hosted in each cell over time through 8 time steps. Figures 4b and 5b display the corresponding entropy evolut...

work page

[4] [4]

Jill North, ‘Time in Thermodynamics, in The Oxford Handbook of Philosophy of Time, edited by Craig Callender, Oxford University Press, 2011, pp. 312–350

work page 2011

[5] [5]

A bimetric cosmological model based on Andrei Sakharov’s twin universe approach

Petit, J.-P.; Margnat, F.; Zejli, H. (novembre 2024). “A bimetric cosmological model based on Andrei Sakharov’s twin universe approach”,European Physical Journal C, 84 : 1226. DOI : 10.1140/epjc/s10052-024-13569-w

work page doi:10.1140/epjc/s10052-024-13569-w 2024

[6] [6]

Time and Chance

Albert, David Z. (2000).“Time and Chance”. Cambridge, Mass. : Harvard University Press

work page 2000

[7] [7]

(2018).extremefit : A Package for Extreme Quantiles

Durrieu, G., Grama, I., Jaunatre, K., Pham, Q.-K., Tricot, J.-M. (2018).extremefit : A Package for Extreme Quantiles. Journal of Statistical Software, Vol. 50, Issue 9. DOI : 10.32614/CRAN.package.extremefit

work page doi:10.32614/cran.package.extremefit 2018

[8] [8]

Cancer Associated E17K Mutation Causes Rapid Confor- mational Drift in AKT1 Pleckstrin Homology (PH) Domain

Kumar, A., Purohit, R. (2013). “Cancer Associated E17K Mutation Causes Rapid Confor- mational Drift in AKT1 Pleckstrin Homology (PH) Domain”.PLoS ONE, 8(5), e64364. DOI : 10.1371/journal.pone.0064364

work page doi:10.1371/journal.pone.0064364 2013