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Expansion into the vacuum of stochastic gases with long-range interactions

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Pith's one-line read This paper derives exact self-similar density profiles for the expansion of long-range Riesz gases into vacuum, and finds that Coulomb gases expand as a uniformly dense ball.

desk verdict Clean exact self-similar solutions for 1D Riesz and Coulomb expansion, but the diffusion caveat for 0<s<1 should move from a section into the abstract. read the letter →

arxiv 2412.14875 v2 pith:GA6FUJNY submitted 2024-12-19 cond-mat.stat-mech math-phmath.MP

P. L. Krapivsky , Kirone Mallick This is my paper · ORCID
classification cond-mat.stat-mechmath-phmath.MP
keywords Rieszgaslong-rangeinteractionsself-similarexpansionCoulombDean-KawasakiequationDysonWignersemicirclenonlocalporousmedium
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

Krapivsky and Mallick study $N$ overdamped Brownian particles in $d$ dimensions that repel pairwise through the Riesz potential $V(r)=g/(s r^s)$ and start concentrated at a single point. Their central claim is that the subsequent expansion is self-similar: in one dimension, for every interaction exponent $s\in(-2,1)$, the density is $\rho(x,t)=\frac{B_s}{g t}(L^2-x^2)^{(s+1)/2}$ with $L(t)\sim (N g t)^{1/(s+2)}$, where $B_s$ is a known constant; for Coulomb interactions in any dimension, the density is uniform inside an expanding ball and decays as $t^{-1}$. These formulas are exact solutions of the nonlocal hydrodynamic equation obtained when Brownian diffusion is dropped from the Dean-Kawasaki current. If correct, they give a complete solvable picture of how a long-range repulsive gas fills vacuum, with the Dyson gas ($s=0$) reducing to an expanding Wigner semicircle and the infinite-particle limit existing only for Coulomb gases.

What carries the argument

Two devices carry the argument. In one dimension it is the singular integral equation obtained from the no-diffusion continuity equation; this equation belongs to the class solved by compactly supported power-law profiles, and it produces the family $F=B_s(R^2-X^2)^{(s+1)/2}$ for every $s\in(-2,1)$. In higher dimensions it is the Newton-Gauss theorem, which converts the Coulomb force at radius $r$ into a function of only the total charge inside the sphere of radius $r$, making the radial current local and the density uniform.

What would settle it

Run the particle dynamics (4) for a finite but large number of particles starting near a point, with no external potential, and measure the central density $\rho(0,t)$ as a function of time. The paper predicts $\rho(0,t)\propto t^{-1/(s+2)}$ for one-dimensional Riesz gases and a flat interior density $\rho=(\Omega_d g t)^{-1}$ for Coulomb gases; any measured exponent or radial profile that disagrees with these forms would falsify the self-similar solution.

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Extended reading notes

Core claim

The paper solves the initial-value problem $\rho(x,0)=N\delta(x)$ for the continuity equation with current $J=-D\partial_x\rho+g\rho\!\int dy\, \frac{x-y}{|x-y|^{s+2}}\rho(y,t)$. After setting $D=0$ and inserting the scaling ansatz $\rho(x,t)=\frac{N}{(N g t)^{1/(s+2)}}F(X)$, $X=x/(N g t)^{1/(s+2)}$, the scaled density obeys the singular integral equation $\frac{X}{s+2}=-\int_{-R}^{R}dY\, \frac{X-Y}{|X-Y|^{s+2}}F(Y)$ inside the support. Its compactly supported solution is $F(X)=B_s(R^2-X^2)^{(s+1)/2}$, and normalization fixes $R$; in original variables this gives the density profile of Eqs. (6)-(7). For Coulomb gases in $d$ dimensions, the nonlocal force is reduced by the Newton-Gauss theorem to a local expression, and the same scaling argument gives a flat density $\rho=(\Omega_d g t)^{-1}$ inside a ball of radius $(d N g t)^{1/d}$, with the $d=2$ case being the Ginibre gas and $d=3$ the classical Coulomb gas.

Load-bearing premise

The whole calculation treats the cloud as a continuous density evolving by the deterministic nonlocal equation (33), i.e. it assumes the random Brownian jiggling of the particles is negligible compared with their repulsion; for $0<s<1$ this is true only before the crossover time $t_*$, and after that the displayed profiles stop being the physical ones.

Editorial extensions

If this is right

  • For $s\in(-2,1)$ in one dimension, the cloud always has compact support, and the radius grows as $(N g t)^{1/(s+2)}$: super-diffusively for $s<0$, diffusively for the Dyson gas at $s=0$, and sub-diffusively (until the crossover time $t_*$) for $0<s<1$.
  • The $s=0$ Dyson gas expands as an evolving Wigner semicircle, so the same profile that describes eigenvalues of Gaussian random matrices appears in an out-of-equilibrium spreading problem.
  • For Coulomb gases in any dimension, the density inside the expanding ball is exactly flat and decays as $t^{-1}$ with a dimension-dependent amplitude, and the profile remains well defined when $N=\infty$.
  • Non-Coulomb Riesz gases do not have a finite infinite-particle limit: the density at a fixed rescaled position diverges for $s>-1$ and vanishes for $s<-1$ when $N\to\infty$.
  • With an added harmonic trap, the same family of profiles describes the expansion, with the radius saturating exponentially to a stationary value; for Coulomb gases the density stays uniform and converges to a constant.

Reading between the lines

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

  • A direct finite-$N$ simulation of the particle equations (4) with weak noise could test the prefactor $B_s$ and the exponent $1/(s+2)$, especially near $s=-2$ and $s=1$ where the integral-equation solution has singular behavior; the paper does not report such numerics.
  • The $N=\infty$ Coulomb result suggests an experimental realization: a two-dimensional system of repulsively interacting colloids released from a tight spot should show an expanding disk with a flat interior and a boundary sharp on the scale of the mean interparticle spacing, measurable with video microscopy.
  • Because the no-diffusion equation is the overdamped counterpart of Coulomb explosion, the same uniform-density solutions may describe the late-time envelope of a laser-ionized cluster when inertia is negligible, a connection the paper only hints at.
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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

2 major / 6 minor

Summary. The manuscript studies the expansion into vacuum of N overdamped Brownian particles in R^d interacting through a repulsive Riesz potential. Working at the continuum level with the Dean–Kawasaki current (13) and then dropping the diffusive and noise terms, the authors derive self-similar solutions of the nonlocal continuity equation. For d=1 and -2<s<1 the density is given by Eq. (6) with support radius Eq. (7). For Coulomb gases in d dimensions, s=d-2, they obtain a uniform density inside an expanding ball, Eqs. (8)-(9) and (64), using the Gauss theorem. The paper also discusses the Dyson limit, the Ginibre gas, the N=∞ Coulomb limit, a harmonic-trap extension, and a conjectured d>1 non-Coulombic profile.

Significance. The 1D result is an elegant, parameter-free exact solution of the no-diffusion hydrodynamic equation: the support radius is fixed by normalization, and the profile solves the stated singular integral equation. The Coulomb solution is attractive and physically transparent, and the universal, N-independent t^{-1} density in arbitrary dimension is a crisp falsifiable prediction. The paper is also honest about the crossover time t* in Eq. (14), which is a strength; however, the abstract and introduction overstate the absence of diffusion effects.

major comments (2)
  1. [Abstract and Section I, Eq. (14)] The headline claim that Eqs. (6)-(7) describe the expansion for all s in (-2,1) and that the influence of noise is 'limited or negligible throughout the evolution' is too strong. For 0<s<1 the body text itself states at Eq. (14) that the diffusion length catches the deterministic span at t* ~ D^{-1}(N g/D)^{2/s}; after this time the diffusive term in Eq. (13) is no longer a small correction and the self-similar profile is not the physical solution of the full stochastic model. Please restrict the advertised domain to the no-diffusion regime t << t* (or to the N to infinity limit at fixed t) and carry this caveat into the abstract.
  2. [Section IV, Eq. (70)] The Introduction labels the d>1 non-Coulombic density profile as the functional form of the exact solution, but Section IV only states that it is 'expected' and defers the derivation to Refs. [42,43] without showing how those results apply to the present Riesz-gas initial-value problem. Either provide a derivation or state explicitly that Eq. (70) is a conjecture/open problem; as written, the section wavers between exact claim and speculation.
minor comments (6)
  1. [Section II, Eq. (6)] The prefactor B_s is of the form 0/0 at s=-1 because both cos(pi s/2) and (s+1)(s+2) vanish; since the text and Fig. 1 use the s=-1 flat profile, please state explicitly that this case is obtained by taking the limit s -> -1.
  2. [Eqs. (7) and (39)] The Gamma-function expression in the displayed formula for L(t) is typeset ambiguously; please check that the denominator is the intended Gamma function and that the factor (s+1) is correctly represented, since the normalization uses the integral of (1-u^2)^{(s+1)/2}.
  3. [Section II A] There is a typo: 'Sustituting' should be 'Substituting'.
  4. [Section IV] There is a typo: 'ampltude' should be 'amplitude'.
  5. [Section III B, N=infinity discussion] For the N=infinity Coulomb limit, the initial distribution N delta(x) has infinite total mass; please state explicitly that the result is the pointwise N to infinity limit of the finite-N density rather than a solution with a literal delta initial condition.
  6. [Section II A, Eq. (31)] The notation in Eq. (31) should be written as x_+(t) = L(t)(1 + N^{-2/3} xi_+) rather than 'x_+(t) L(t) = 1 + ...' so that the ratio is not confused with the position itself.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the self-similar profiles are derived from the no-diffusion continuity equation via standard integral-equation methods and Gauss's law, not fitted or assumed; self-citations are not load-bearing.

full rationale

The central derivation is self-contained rather than circular. The paper starts from the microscopic Langevin dynamics (4) and obtains the Dean-Kawasaki hydrodynamic equation (13) via standard external references [24,25]. The no-diffusion reduction is an explicit modelling approximation, and the paper itself quantifies its validity domain through the crossover time t* in Eq. (14), noting that for 0<s<1 diffusion dominates at t >> t*. The self-similar ansatz (35) is fixed by the scaling invariance (34) of the governing equation together with the conservation law (11), not by the target profile. The singular integral equation (37) is solved by the textbook profile (38) with constants fixed by the integral equation and normalization (22),(39); no parameter is fitted to the answer. For Coulomb gases, the uniform-density result follows from the Newton-Gauss theorem in Eqs. (41),(55), with the radius fixed by mass conservation. Nothing is 'predicted' from a quantity that was itself fitted to the same output. The self-citations [27], [29], and [30] provide background on fluctuations, the short-range criterion, and finite-time blowup, but the load-bearing mathematical steps are either derived in-text or cited to independent external sources (e.g., [31,32] for singular integral equations, [42,43] for nonlocal porous medium equations). The abstract's statement that noise influence is 'limited or negligible throughout the evolution' is broader than the t << t* analysis supports, and Eq. (14) itself shows diffusion dominates for 0<s<1 at late times; however, that is an overstatement of the derived result's domain of applicability, not a circular reduction. Therefore no circularity is found; the score of 1 merely acknowledges the presence of minor, non-load-bearing self-citations.

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

The central derivation rests on the standard hydrodynamic limit of the Langevin system (Dean-Kawasaki), neglect of diffusion in the regime of interest, existence and uniqueness of the self-similar compactly supported solution, the Newton-Gauss theorem for Coulomb forces, and textbook inversion of singular integral equations. The d>1 non-Coulombic profile (70) is an additional conjecture not derived here. No parameters are fitted to data.

assumptions (6)
  • domain assumption The N-particle Langevin dynamics (4) is equivalent in the continuum limit to the Dean-Kawasaki equation (13) with current (12).
    Invoked in Sec. II to write the nonlocal current; derivation is formal and standard but not rigorous for singular Riesz potentials.
  • domain assumption Diffusive and noise contributions to the current can be neglected in the regime of interest.
    Used to obtain Eq. (33) and all subsequent profiles; for 0<s<1 this is valid only for t << t* given by Eq. (14), which limits the scope of the stated results.
  • domain assumption The self-similar solution of the nonlocal equation is the relevant solution selected by the delta initial condition, and the density has compact support.
    Self-similar ansatz (35) and compact support are used to derive integral equation (37) and to fix R by normalization; uniqueness is not proven.
  • standard math For Coulomb potentials the force on a particle at radius r is the same as if all mass inside radius r were concentrated at the center (Newton-Gauss theorem).
    Used in Sec. III to write the currents (41), (55), and (64).
  • standard math Solutions of the linear singular integral equation (37) are those given in textbooks [31-33].
    Eq. (38) follows from known inversion formulas; accepted as standard.
  • ad hoc to paper The proposed solution form (70) for non-Coulombic Riesz gases in d>1 is valid.
    Conjectured in Sec. IV with derivation deferred to [42,43,5,58]; used in the N=infinity classification for general d.

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Cite this review

Pith. "Pith review of Expansion into the vacuum of stochastic gases with long-range interactions." pith.science (2026). https://pith.science/paper/GA6FUJNY

@misc{pith2026241214875,
  author       = {Pith},
  title        = {Pith review of: Expansion into the vacuum of stochastic gases with long-range interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GA6FUJNY}},
  note         = {Machine review of arXiv:2412.14875}
}
abstract

We study the evolution of a system of many point particles initially concentrated in a small region in $d$ dimensions. Particles undergo overdamped motion caused by pairwise interactions through the long-ranged repulsive $r^{-s}$ potential; each particle is also subject to Brownian noise. When $s<d$, the expansion is governed by non-local hydrodynamic equations. In the one-dimensional case, we deduce self-similar solutions for all $s\in (-2,1)$. The expansion of Coulomb gases remains well-defined in the infinite-particle limit: The density is spatially uniform and inversely proportional to time independent of the spatial dimension.

Figures

Figures reproduced from arXiv: 2412.14875 by the authors.

Figure 1
Figure 1. FIG. 1. The spatial dependence of the density re-scaled to be [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

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Reference graph

Works this paper leans on

72 extracted references · 68 canonical work pages · cited by 1 Pith paper

  1. [1]

    Campa, T

    A. Campa, T. Dauxois, and S. Ruffo, Statistical mechan- ics and dynamics of solvable models with long-range in- teractions, Physics Reports 480, 57 (2009)

    work page 2009

  2. [2]

    The diffusive growth of the ball indicates that the scaling ansatz (46) remains applicable even when we take diffusion into account

    Thus, at any time the density is uniform inside the growing disk ρ(r, t) = ( (2πgt)−1 r ≤ √2N gt 0 r >√2N gt (52) The radius of the disk grows diffusively as for the one- dimensional Dyson gas; the extra N factor ensures again that diffusion plays a minor role. The diffusive growth of the ball indicates that the scaling ansatz (46) remains applicable even...

    work page

  3. [3]

    Thus at any time the density is uniform inside the growing disk ρ(r, t) = ( (4πgt)−1 r ≤ 3√3N gt 0 r > 3√3N gt (63) The radius of the ball grows sub-diffusively, so the above results are valid up to a crossover time t∗ ∼ (N g)2/D3. The above calculations bear some resemblance to the well-studied problem of Coulomb explosion, an impor- tant phenomenon in l...

    work page

  4. [4]

    Thess, D

    A. Thess, D. Spirn, and B. J¨ uttner, Viscous flow at infinite Marangoni number, Phys. Rev. Lett. 75, 4614 (1995)

    work page 1995

  5. [5]

    Campa, T

    A. Campa, T. Dauxois, D. Fanelli, and S. Ruffo, Physics of Long-Range Interacting Systems (Oxford University Press, Oxford, UK, 2014)

    work page 2014

  6. [6]

    Dynamical Spreading and Memory Retention Under Power Law Potential

    I. Fanto, Y. Rosenblum, O. Harel, and N. Oppenheimer, Dynamical spreading and memory retention under power law potential, arXiv:2502.06256 (2025)

    work page Pith review arXiv 2025

  7. [7]

    Diamant and E

    H. Diamant and E. C. Oˇ guz, Fluids with power-law repul- sion: Hyperuniformity and energy fluctuations, J. Phys: Cond. Mat. 37, 065101 (2025)

    work page 2025

  8. [8]

    Bickel and F

    T. Bickel and F. Detcheverry, Exact solutions for viscous Marangoni spreading, Phys. Rev. E 106, 045107 (2022)

    work page 2022

Show all 72 references
  1. [9]

    Temprano-Coleto and H

    F. Temprano-Coleto and H. A. Stone, On the self- similarity of unbounded viscous Marangoni flows, J. Fluid Mech. 997, A95 (2024)

    2024

  2. [10]

    in the context of random matrices [10, 11] where eigenvalues played the role of particles. In two dimen- sions, a gas of Brownian particles interacting via loga- rithmic potential was introduced by Ginibre [12] also in the context of random matrices, with complex eigenval- ues...

    2025 arXiv

  3. [11]

    G. R. Baker, X. Li, and A. C. Morlet, Analytic struc- ture of two 1D-transport equations with nonlocal fluxes, Physica D: Nonlinear Phenomena 91, 349 (1996)

    1996

  4. [12]

    Eggers and M

    J. Eggers and M. A. Fontelos, Selection of singular solu- tions in non-local transport equations, Nonlinearity 33, 325 (2020)

    2020

  5. [13]

    F. J. Dyson, A Brownian motion model for the eigenval- ues of a random matrix, J. Math. Phys. 3, 1191 (1962)

    1962

  6. [14]

    P. J. Forrester, Log-gases and random matrices (Prince- ton University Press, Princeton, NJ, 2010)

    2010

  7. [15]

    Ginibre, Statistical ensembles of complex, quaternion, and real matrices, J

    J. Ginibre, Statistical ensembles of complex, quaternion, and real matrices, J. Math. Phys. 6, 440 (1965)

    1965

  8. [16]

    Byun and P

    S.-S. Byun and P. J. Forrester, Progress on the study of the Ginibre ensembles (Springer, Singapore, 2024)

    2024

  9. [17]

    Lewin, Coulomb and Riesz gases: The known and the unknown, J

    M. Lewin, Coulomb and Riesz gases: The known and the unknown, J. Math. Phys. 63, 061101 (2022)

    2022

  10. [18]

    Serfaty, Lectures on Coulomb and Riesz gases, arXiv:2407.21194 (2024)

    S. Serfaty, Lectures on Coulomb and Riesz gases, arXiv:2407.21194 (2024)

    2024 arXiv

  11. [19]

    Calogero, Ground state of the one-dimensional N-body system, J

    F. Calogero, Ground state of the one-dimensional N-body system, J. Math. Phys. 10, 2197 (1969)

    1969

  12. [20]

    A. P. Polychronakos, The physics and mathematics of Calogero particles, J. Phys. A 39, 12793 (2006)

    2006

  13. [21]

    Chandrasekhar, Principles of Stellar Dynamics (Uni- versity of Chicago, Chicago, 1962)

    S. Chandrasekhar, Principles of Stellar Dynamics (Uni- versity of Chicago, Chicago, 1962)

    1962

  14. [22]

    Habib, H

    S. Habib, H. E. Kandrup, and M. E. Mahon, Chaos and noise in a truncated Toda potential, Phys. Rev. E 53, 5473 (1996)

    1996

  15. [23]

    Habib, H

    S. Habib, H. E. Kandrup, and M. E. Mahon, Chaos and noise in galactic potentials, Astrophys. J. 480, 155 (1997)

    1997

  16. [24]

    H. E. Kandrup and I. V. Sideris, Chaos and the con- tinuum limit in the gravitational N-body problem: Inte- grable potentials, Phys. Rev. E 64, 056209 (2001)

    2001

  17. [25]

    Livan, M

    G. Livan, M. Novaes, and P. Vivo, Introduction to Ran- dom Matrices (Springer, Cham, Switzerland, 2018)

    2018

  18. [26]

    Spohn, Large Scale Dynamics of Interacting Particles (Springer, Berlin, 1991)

    H. Spohn, Large Scale Dynamics of Interacting Particles (Springer, Berlin, 1991)

    1991

  19. [27]

    Kawasaki, Stochastic model of slow dynamics in su- percooled liquids and dense colloidal suspensions, Phys- ica A 208, 35 (1994)

    K. Kawasaki, Stochastic model of slow dynamics in su- percooled liquids and dense colloidal suspensions, Phys- ica A 208, 35 (1994)

    1994

  20. [28]

    D. S. Dean, Langevin equation for the density of a system of interacting Langevin processes, J. Phys. A 29, L613 (1996)

    1996

  21. [29]

    Touzo, P

    L. Touzo, P. Le Doussal, and G. Schehr, Interacting, run- ning and tumbling: The active Dyson Brownian motion, EPL 142, 61004 (2023)

    2023

  22. [30]

    Dandekar, P

    R. Dandekar, P. L. Krapivsky, and K. Mallick, Current fluctuations in the Dyson gas, Phys. Rev. E 110, 064153 (2024)

    2024

  23. [31]

    Agarwal, A

    S. Agarwal, A. Dhar, M. Kulkarni, A. Kundu, S. N. Majumdar, D. Mukamel, and G. Schehr, Harmonically confined particles with long-range repulsive interactions, Phys. Rev. Lett. 123, 100603 (2019)

    2019

  24. [32]

    Dandekar, P

    R. Dandekar, P. L. Krapivsky, and K. Mallick, Dynamical fluctuations in the Riesz gas, Phys. Rev. E 107, 044129 (2023)

    2023

  25. [33]

    P. L. Krapivsky and B. Meerson, Finite-time blowup of a Brownian particle in a repulsive potential, arXiv:2502.11796 (2025)

    2025 arXiv

  26. [34]

    Estrada and R

    R. Estrada and R. P. Kanwal,Singular Integral Equations (Birkh¨ auser, Boston, MA, 2000)

    2000

  27. [35]

    A. D. Polyanin and A. V. Manzhirov, Handbook of Inte- gral Equations (Chapman and Hall/CRC, Boca Raton, FL, 2008)

    2008

  28. [36]

    J. A. Carrillo and Y. Huang, Explicit equilibrium solu- tions for the aggregation equation with power-law poten- tials, Kinet. Relat. Models 10, 171 (2017)

    2017

  29. [37]

    C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978)

    1978

  30. [38]

    S. N. Majumdar and G. Schehr, Top eigenvalue of a ran- dom matrix: large deviations and third order phase tran- sition, J. Stat. Mech. 2014, P01012 (2014)

    2014

  31. [39]

    A. E. Kaplan, B. Y. Dubetsky, and P. L. Shkolnikov, Shock shells in coulomb explosions of nanoclusters, Phys. Rev. Lett. 91, 143401 (2003)

    2003

  32. [40]

    V. F. Kovalev and V. Y. Bychenkov, Kinetic description of the Coulomb explosion of a spherically symmetric clus- ter, JETP 101, 212 (2005)

    2005

  33. [41]

    Grech, R

    M. Grech, R. Nuter, A. Mikaberidze, P. Di Cintio, L. Gremillet, E. Lefebvre, U. Saalmann, J. M. Rost, and S. Skupin, Coulomb explosion of uniformly charged spheroids, Phys. Rev. E 84, 056404 (2011)

    2011

  34. [42]

    Zandi and R

    O. Zandi and R. M. van der Veen, General formulation of coulomb explosion dynamics of highly symmetric charge distributions, J. Phys. Commun. 6, 085003 (2022)

    2022

  35. [43]

    Liddle, An Introduction to Modern Cosmology (Wiley, Chichester, UK, 2015)

    A. Liddle, An Introduction to Modern Cosmology (Wiley, Chichester, UK, 2015)

    2015

  36. [44]

    Mukhanov, Physical Foundations of Cosmology (Cam- bridge University Press, Cambridge, UK, 2005)

    V. Mukhanov, Physical Foundations of Cosmology (Cam- bridge University Press, Cambridge, UK, 2005)

    2005

  37. [45]

    Biler, C

    P. Biler, C. Imbert, and G. Karch, Barenblatt profiles for a nonlocal porous medium equation, C. R. Math. Acad. 10 Sci. Paris 349, 641 (2011)

    2011

  38. [46]

    Biler, C

    P. Biler, C. Imbert, and G. Karch, The nonlocal porous medium equation: Barenblatt profiles and other weak solutions, Arch. Rational Mech. Anal. 215, 497 (2015)

    2015

  39. [47]

    Ispolatov and E

    I. Ispolatov and E. G. D. Cohen, Phase transitions in systems with 1 /rα attractive interactions, Phys. Rev. E 64, 056103 (2001)

    2001

  40. [48]

    Di Cintio, L

    P. Di Cintio, L. Ciotti, and C. Nipoti, Relaxation of n- body systems with additive r−α interparticle forces, Mon. Not. R. Astron. Soc. 431, 3177 (2013)

    2013

  41. [49]

    P. F. Di Cintio, L. Ciotti, and C. Nipoti, Radially anisotropic systems with r−α forces: equilibrium states, J. Plasma Phys. 81, 495810504 (2015)

    2015

  42. [50]

    Sire and P.-H

    C. Sire and P.-H. Chavanis, Thermodynamics and col- lapse of self-gravitating Brownian particles in D dimen- sions, Phys. Rev. E 66, 046133 (2002)

    2002

  43. [51]

    K. R. Yawn and B. N. Miller, Ergodic properties and equilibrium of one-dimensional self-gravitating systems, Phys. Rev. E 56, 2429 (1997)

    1997

  44. [52]

    A. E. Schultz, W. Dehnen, G. Jungman, and S. Tremaine, Gravitational collapse in one dimension, Mon. Not. R. Astron. Soc. 431, 49 (2013)

    2013

  45. [53]

    R. B. Mann, One-dimensional relativistic self-gravitating systems, Entropy 26, 612 (2024)

    2024

  46. [54]

    Gabrielli, M

    A. Gabrielli, M. Joyce, and B. Marcos, Quasistationary states and the range of pair interactions, Phys. Rev. Lett. 105, 210602 (2010)

    2010

  47. [55]

    Gabrielli, M

    A. Gabrielli, M. Joyce, and F. Sicard, One-dimensional gravity in infinite point distributions, Phys. Rev. E 80, 041108 (2009)

    2009

  48. [56]

    Gabrielli, M

    A. Gabrielli, M. Joyce, B. Marcos, and F. Sicard, A dy- namical classification of the range of pair interactions, J. Stat. Phys. 141, 970 (2010)

    2010

  49. [57]

    Chavanis, Hamiltonian and Brownian systems with long-range interactions: II

    P.-H. Chavanis, Hamiltonian and Brownian systems with long-range interactions: II. kinetic equations and stabil- ity analysis, Physica A 361, 81 (2006)

    2006

  50. [58]

    Chavanis, Kinetic equations for systems with long- range interactions: a unified description, J

    P.-H. Chavanis, Kinetic equations for systems with long- range interactions: a unified description, J. Stat. Mech.: Theor. and Exp. 2010, P05019 (2010)

    2010

  51. [59]

    Chavanis, Kinetic theory of long-range interacting systems with angle-action variables and collective effects, Physica A 391, 3680 (2012)

    P.-H. Chavanis, Kinetic theory of long-range interacting systems with angle-action variables and collective effects, Physica A 391, 3680 (2012)

    2012

  52. [60]

    Fouvry, P.-H

    J.-B. Fouvry, P.-H. Chavanis, and C. Pichon, Kinetic the- ory of one-dimensional homogeneous long-range interact- ing systems with an arbitrary potential of interaction, Phys. Rev. E 102, 052110 (2020)

    2020

  53. [61]

    J. L. V´ azquez, The mathematical theories of diffusion: Nonlinear and fractional diffusion, in Nonlocal and Non- linear Diffusions and Interactions: New Methods and Di- rections: Cetraro, Italy 2016 , edited by M. Bonforte and G. Grillo (Springer, Cham, 2017) pp. 205–278

    2016

  54. [62]

    D. S. Dean and S. N. Majumdar, Large deviations of extreme eigenvalues of random matrices, Phys. Rev. Lett. 97, 160201 (2006)

    2006

  55. [63]

    D. S. Dean and S. N. Majumdar, Extreme value statistics of eigenvalues of Gaussian random matrices, Phys. Rev. E 77, 041108 (2008)

    2008

  56. [64]

    Sandier and S

    E. Sandier and S. Serfaty, 2D Coulomb gases and the renormalized energy, Ann. Probab. 43, 2026 (2015)

    2015

  57. [65]

    Petrache and S

    M. Petrache and S. Serfaty, Crystallization for Coulomb and Riesz interactions as a consequence of the Cohn- Kumar conjecture, Proc. Amer. Math. Soc. 148, 3047 (2020)

    2020

  58. [66]

    E. H. Lieb and H. Narnhofer, The thermodynamic limit for jellium, J. l’Ecole Polytechnique 12, 291 (1975)

    1975

  59. [67]

    Lewin, E

    M. Lewin, E. H. Lieb, and R. Seiringer, Statistical me- chanics of the uniform electron gas, J. l’Ecole Polytech- nique 4, 79 (2018)

    2018

  60. [68]

    Lewin, E

    M. Lewin, E. H. Lieb, and R. Seiringer, Floating Wigner crystal with no boundary charge fluctuations, Phys. Rev. B 100, 035127 (2019)

    2019

  61. [69]

    E. B. Kolomeisky, Natural analog to cosmology in basic condensed matter physics, Phys. Rev. B 100, 140301(R) (2019)

    2019

  62. [70]

    E. B. Kolomeisky, Normal modes of vibrations around Hubble flow in jellium, Phys. Rev. B 101, 174304 (2020)

    2020

  63. [71]

    E. B. Kolomeisky, Analog de Sitter space in a controlled Coulomb explosion, Phys. Rev. A 103, 043101 (2021)

    2021

  64. [72]

    Faraoni, Cosmic Analogies: How Natural Systems Emulate the Universe (World Scientific, London, 2022)

    V. Faraoni, Cosmic Analogies: How Natural Systems Emulate the Universe (World Scientific, London, 2022)

    2022

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